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batch nfs 2003 shamir tromer 2003 lenstra tromer shamir d

Batch NFS 2003 ShamirTromer, 2003 LenstraTromerShamir D. J. - PowerPoint PPT Presentation

Sep 07, 2022 β€’194 likes β€’1.21k views

Batch NFS 2003 ShamirTromer, 2003 LenstraTromerShamir D. J. Bernstein KortsmitDodsonHughes University of Illinois at Chicago & Leyland, 2005 Geiselmann Technische Universiteit Eindhoven

  1. Shamir–Tromer, 2003 Example: The IP address of Users seem Lenstra–Tromer–Shamir– dnssec-deployment.org 1. β€œThe rtsmit–Dodson–Hughes– is signed by an RSA-1024 key more than Leyland, 2005 Geiselmann– signed by an RSA-2048 key 2. β€œThe Shamir–Steinwandt–Tromer, 2005 signed by org ’s RSA-1024 key off-the-shelf; e–Kleinjung–Paar–Pelzl– signed by an RSA-2048 key attackers Priplata–Stahlke, etc.: RSA-1024 signed by a root RSA-1024 key 3. For signatures: reakable in a year by an attack signed by an RSA-2048 key. switch keys machine costing ❁ 10 9 dollars. Most β€œDNSSEC” signatures the attack Internet switched to follow a similar pattern. RSA-2048, and we no longer care Another example: SSL has used RSA-1024 security, right? many millions of RSA-1024 keys. rong! Imagine that an attacker has recorded tons of SSL traffic.

  2. romer, 2003 Example: The IP address of Users seem unconcerned: romer–Shamir– dnssec-deployment.org 1. β€œThe attack machine dson–Hughes– is signed by an RSA-1024 key more than this RSA Geiselmann– signed by an RSA-2048 key 2. β€œThe attack machine andt–Tromer, 2005 signed by org ’s RSA-1024 key off-the-shelf; it’s only ung–Paar–Pelzl– signed by an RSA-2048 key attackers building e, etc.: RSA-1024 signed by a root RSA-1024 key 3. For signatures: year by an attack signed by an RSA-2048 key. switch keys every month, ❁ 10 9 dollars. Most β€œDNSSEC” signatures the attack machine switched to follow a similar pattern. we no longer care Another example: SSL has used security, right? many millions of RSA-1024 keys. Imagine that an attacker has recorded tons of SSL traffic.

  3. 2003 Example: The IP address of Users seem unconcerned: dnssec-deployment.org 1. β€œThe attack machine costs dson–Hughes– is signed by an RSA-1024 key more than this RSA key is w Geiselmann– signed by an RSA-2048 key 2. β€œThe attack machine isn’t romer, 2005 signed by org ’s RSA-1024 key off-the-shelf; it’s only for elzl– signed by an RSA-2048 key attackers building ASICs.” RSA-1024 signed by a root RSA-1024 key 3. For signatures: β€œWe an attack signed by an RSA-2048 key. switch keys every month, and ❁ dollars. Most β€œDNSSEC” signatures the attack machine takes a y to follow a similar pattern. longer care Another example: SSL has used right? many millions of RSA-1024 keys. Imagine that an attacker has recorded tons of SSL traffic.

  4. Example: The IP address of Users seem unconcerned: dnssec-deployment.org 1. β€œThe attack machine costs is signed by an RSA-1024 key more than this RSA key is worth.” signed by an RSA-2048 key 2. β€œThe attack machine isn’t signed by org ’s RSA-1024 key off-the-shelf; it’s only for signed by an RSA-2048 key attackers building ASICs.” signed by a root RSA-1024 key 3. For signatures: β€œWe signed by an RSA-2048 key. switch keys every month, and Most β€œDNSSEC” signatures the attack machine takes a year.” follow a similar pattern. Another example: SSL has used many millions of RSA-1024 keys. Imagine that an attacker has recorded tons of SSL traffic.

  5. Example: The IP address of Users seem unconcerned: dnssec-deployment.org 1. β€œThe attack machine costs is signed by an RSA-1024 key more than this RSA key is worth.” signed by an RSA-2048 key 2. β€œThe attack machine isn’t signed by org ’s RSA-1024 key off-the-shelf; it’s only for signed by an RSA-2048 key attackers building ASICs.” signed by a root RSA-1024 key 3. For signatures: β€œWe signed by an RSA-2048 key. switch keys every month, and Most β€œDNSSEC” signatures the attack machine takes a year.” follow a similar pattern. Real quote: β€œDNSSEC signing Another example: SSL has used keys should be large enough to many millions of RSA-1024 keys. avoid all known cryptographic Imagine that an attacker has attacks during the effectivity recorded tons of SSL traffic. period of the key.”

  6. Example: The IP address of Users seem unconcerned: Continuation despite huge dnssec-deployment.org 1. β€œThe attack machine costs signed by an RSA-1024 key broken a more than this RSA key is worth.” by an RSA-2048 key fact, the 2. β€œThe attack machine isn’t by org ’s RSA-1024 key estimated off-the-shelf; it’s only for by an RSA-2048 key of a 700-bit attackers building ASICs.” by a root RSA-1024 key breaking 3. For signatures: β€œWe by an RSA-2048 key. would need switch keys every month, and amounts β€œDNSSEC” signatures the attack machine takes a year.” power in a similar pattern. be detected Real quote: β€œDNSSEC signing Another example: SSL has used single key keys should be large enough to millions of RSA-1024 keys. estimated avoid all known cryptographic Imagine that an attacker has safely use attacks during the effectivity rded tons of SSL traffic. least the period of the key.”

  7. address of Users seem unconcerned: Continuation of quote: despite huge efforts, dnssec-deployment.org 1. β€œThe attack machine costs RSA-1024 key broken a regular 1024-bit more than this RSA key is worth.” RSA-2048 key fact, the best completed 2. β€œThe attack machine isn’t RSA-1024 key estimated to be the off-the-shelf; it’s only for RSA-2048 key of a 700-bit key. An attackers building ASICs.” RSA-1024 key breaking a 1024-bit 3. For signatures: β€œWe RSA-2048 key. would need to exp switch keys every month, and amounts of network β€œDNSSEC” signatures the attack machine takes a year.” power in a way that pattern. be detected in order Real quote: β€œDNSSEC signing example: SSL has used single key. Because keys should be large enough to RSA-1024 keys. estimated that most avoid all known cryptographic attacker has safely use 1024-bit attacks during the effectivity SSL traffic. least the next ten period of the key.”

  8. of Users seem unconcerned: Continuation of quote: β€œTo despite huge efforts, no one 1. β€œThe attack machine costs key broken a regular 1024-bit key; more than this RSA key is worth.” ey fact, the best completed attack 2. β€œThe attack machine isn’t key estimated to be the equivalent off-the-shelf; it’s only for ey of a 700-bit key. An attacker attackers building ASICs.” RSA-1024 key breaking a 1024-bit signing k 3. For signatures: β€œWe ey. would need to expend phenomenal switch keys every month, and amounts of networked computing signatures the attack machine takes a year.” power in a way that would not be detected in order to break Real quote: β€œDNSSEC signing as used single key. Because of this, it keys should be large enough to RSA-1024 keys. estimated that most zones c avoid all known cryptographic has safely use 1024-bit keys for at attacks during the effectivity traffic. least the next ten years.” period of the key.”

  9. Users seem unconcerned: Continuation of quote: β€œTo date, despite huge efforts, no one has 1. β€œThe attack machine costs broken a regular 1024-bit key; in more than this RSA key is worth.” fact, the best completed attack is 2. β€œThe attack machine isn’t estimated to be the equivalent off-the-shelf; it’s only for of a 700-bit key. An attacker attackers building ASICs.” breaking a 1024-bit signing key 3. For signatures: β€œWe would need to expend phenomenal switch keys every month, and amounts of networked computing the attack machine takes a year.” power in a way that would not be detected in order to break a Real quote: β€œDNSSEC signing single key. Because of this, it is keys should be large enough to estimated that most zones can avoid all known cryptographic safely use 1024-bit keys for at attacks during the effectivity least the next ten years.” period of the key.”

  10. seem unconcerned: Continuation of quote: β€œTo date, Goal of ou despite huge efforts, no one has analyze the β€œThe attack machine costs broken a regular 1024-bit key; in specifically than this RSA key is worth.” fact, the best completed attack is ratio , of β€œThe attack machine isn’t estimated to be the equivalent off-the-shelf; it’s only for β€œMany”: of a 700-bit key. An attacker ers building ASICs.” β€œPrice-perfo breaking a 1024-bit signing key signatures: β€œWe area-time would need to expend phenomenal keys every month, and amounts of networked computing β€œRAM” attack machine takes a year.” power in a way that would not bit integers be detected in order to break a quote: β€œDNSSEC signing accessing single key. Because of this, it is should be large enough to realistic; β†βš estimated that most zones can all known cryptographic β€œAsymptotic”: safely use 1024-bit keys for at attacks during the effectivity suppress least the next ten years.” of the key.” speedups

  11. unconcerned: Continuation of quote: β€œTo date, Goal of our paper: despite huge efforts, no one has analyze the asymptotic machine costs broken a regular 1024-bit key; in specifically price-perfo RSA key is worth.” fact, the best completed attack is ratio , of breaking many machine isn’t estimated to be the equivalent only for β€œMany”: e.g. millions. of a 700-bit key. An attacker ing ASICs.” β€œPrice-performance breaking a 1024-bit signing key signatures: β€œWe area-time product would need to expend phenomenal every month, and amounts of networked computing β€œRAM” metric (adding machine takes a year.” power in a way that would not bit integers has same be detected in order to break a β€œDNSSEC signing accessing array of single key. Because of this, it is large enough to realistic; β€œ β†βš ” metric estimated that most zones can cryptographic β€œAsymptotic”: We safely use 1024-bit keys for at the effectivity suppress polynomial least the next ten years.” ey.” speedups are superp

  12. Continuation of quote: β€œTo date, Goal of our paper: despite huge efforts, no one has analyze the asymptotic cost, costs broken a regular 1024-bit key; in specifically price-performance worth.” fact, the best completed attack is ratio , of breaking many RSA isn’t estimated to be the equivalent β€œMany”: e.g. millions. of a 700-bit key. An attacker β€œPrice-performance ratio”: breaking a 1024-bit signing key area-time product for chips. would need to expend phenomenal and amounts of networked computing β€œRAM” metric (adding two a year.” power in a way that would not bit integers has same cost as accessing array of size 2 64 ) is be detected in order to break a signing single key. Because of this, it is ough to realistic; β€œ β†βš ” metric is realistic. estimated that most zones can cryptographic β€œAsymptotic”: We systematically safely use 1024-bit keys for at effectivity suppress polynomial factors. least the next ten years.” speedups are superpolynomial.

  13. Continuation of quote: β€œTo date, Goal of our paper: despite huge efforts, no one has analyze the asymptotic cost, broken a regular 1024-bit key; in specifically price-performance fact, the best completed attack is ratio , of breaking many RSA keys. estimated to be the equivalent β€œMany”: e.g. millions. of a 700-bit key. An attacker β€œPrice-performance ratio”: breaking a 1024-bit signing key area-time product for chips. would need to expend phenomenal amounts of networked computing β€œRAM” metric (adding two 64- power in a way that would not bit integers has same cost as accessing array of size 2 64 ) is not be detected in order to break a single key. Because of this, it is realistic; β€œ β†βš ” metric is realistic. estimated that most zones can β€œAsymptotic”: We systematically safely use 1024-bit keys for at suppress polynomial factors. Our least the next ten years.” speedups are superpolynomial.

  14. Continuation of quote: β€œTo date, Goal of our paper: Best result time β–² 1 ✿ 185632 despite huge efforts, no one has analyze the asymptotic cost, β–² ✿ a regular 1024-bit key; in specifically price-performance using chip β†βš is β–² 1 ✿ the best completed attack is ratio , of breaking many RSA keys. estimated to be the equivalent β€œMany”: e.g. millions. Our main 700-bit key. An attacker a batch of β–² ✿ β€œPrice-performance ratio”: reaking a 1024-bit signing key time β–² 1 ✿ 022400 area-time product for chips. need to expend phenomenal β–² ✿ using chip amounts of networked computing β€œRAM” metric (adding two 64- β–² ✿ β†βš per k in a way that would not bit integers has same cost as This pap accessing array of size 2 64 ) is not detected in order to break a at β–² ♦ (1) , key. Because of this, it is realistic; β€œ β†βš ” metric is realistic. speedup estimated that most zones can β€œAsymptotic”: We systematically Results a use 1024-bit keys for at suppress polynomial factors. Our guess from the next ten years.” speedups are superpolynomial.

  15. quote: β€œTo date, Goal of our paper: Best result known time β–² 1 ✿ 185632 efforts, no one has analyze the asymptotic cost, using chip area β–² 0 ✿ 1024-bit key; in specifically price-performance β†βš is β–² 1 ✿ 976052 . completed attack is ratio , of breaking many RSA keys. the equivalent β€œMany”: e.g. millions. Our main result fo An attacker a batch of β–² 0 ✿ 5 keys: β€œPrice-performance ratio”: 1024-bit signing key time β–² 1 ✿ 022400 area-time product for chips. expend phenomenal using chip area β–² 1 ✿ orked computing β€œRAM” metric (adding two 64- β†βš per key is β–² 1 ✿ 704000 that would not bit integers has same cost as This paper also looks accessing array of size 2 64 ) is not rder to break a at β–² ♦ (1) , analyzing Because of this, it is realistic; β€œ β†βš ” metric is realistic. speedup from early-ab most zones can β€œAsymptotic”: We systematically Results are not what 1024-bit keys for at suppress polynomial factors. Our guess from 1982 P ten years.” speedups are superpolynomial.

  16. o date, Goal of our paper: Best result known for one key time β–² 1 ✿ 185632 one has analyze the asymptotic cost, using chip area β–² 0 ✿ 790420 ; key; in specifically price-performance β†βš is β–² 1 ✿ 976052 . attack is ratio , of breaking many RSA keys. equivalent β€œMany”: e.g. millions. Our main result for attacker a batch of β–² 0 ✿ 5 keys: β€œPrice-performance ratio”: signing key time β–² 1 ✿ 022400 area-time product for chips. phenomenal using chip area β–² 1 ✿ 181600 ; computing β€œRAM” metric (adding two 64- β†βš per key is β–² 1 ✿ 704000 . not bit integers has same cost as This paper also looks more closely accessing array of size 2 64 ) is not reak a at β–² ♦ (1) , analyzing asymptotic this, it is realistic; β€œ β†βš ” metric is realistic. speedup from early-abort ECM. can β€œAsymptotic”: We systematically Results are not what one would r at suppress polynomial factors. Our guess from 1982 Pomerance. speedups are superpolynomial.

  17. Goal of our paper: Best result known for one key time β–² 1 ✿ 185632 analyze the asymptotic cost, using chip area β–² 0 ✿ 790420 ; specifically price-performance β†βš is β–² 1 ✿ 976052 . ratio , of breaking many RSA keys. β€œMany”: e.g. millions. Our main result for a batch of β–² 0 ✿ 5 keys: β€œPrice-performance ratio”: time β–² 1 ✿ 022400 area-time product for chips. using chip area β–² 1 ✿ 181600 ; β€œRAM” metric (adding two 64- β†βš per key is β–² 1 ✿ 704000 . bit integers has same cost as This paper also looks more closely accessing array of size 2 64 ) is not at β–² ♦ (1) , analyzing asymptotic realistic; β€œ β†βš ” metric is realistic. speedup from early-abort ECM. β€œAsymptotic”: We systematically Results are not what one would suppress polynomial factors. Our guess from 1982 Pomerance. speedups are superpolynomial.

  18. of our paper: Best result known for one key Asymptotic time β–² 1 ✿ 185632 analyze the asymptotic cost, 1. Attack using chip area β–² 0 ✿ 790420 ; ecifically price-performance is reduced, β†βš is β–² 1 ✿ 976052 . of breaking many RSA keys. can targe β€œMany”: e.g. millions. Our main result for 2. Prima a batch of β–² 0 ✿ 5 keys: β€œPrice-performance ratio”: memory time β–² 1 ✿ 022400 ime product for chips. for off-the-shelf using chip area β–² 1 ✿ 181600 ; β€œRAM” metric (adding two 64- β†βš per key is β–² 1 ✿ 704000 . 3. Attack integers has same cost as (and can This paper also looks more closely accessing array of size 2 64 ) is not breaking at β–² ♦ (1) , analyzing asymptotic realistic; β€œ β†βš ” metric is realistic. speedup from early-abort ECM. β€œAsymptotic”: We systematically Results are not what one would ress polynomial factors. Our guess from 1982 Pomerance. eedups are superpolynomial.

  19. er: Best result known for one key Asymptotic consequ time β–² 1 ✿ 185632 asymptotic cost, 1. Attack cost per using chip area β–² 0 ✿ 790420 ; rice-performance is reduced, so attack β†βš is β–² 1 ✿ 976052 . reaking many RSA keys. can target lower-value millions. Our main result for 2. Primary bottleneck a batch of β–² 0 ✿ 5 keys: rmance ratio”: memory factorizationβ€”w time β–² 1 ✿ 022400 duct for chips. for off-the-shelf graphics using chip area β–² 1 ✿ 181600 ; (adding two 64- β†βš per key is β–² 1 ✿ 704000 . 3. Attack time is reduced same cost as (and can be reduced This paper also looks more closely of size 2 64 ) is not breaking key rotation. at β–² ♦ (1) , analyzing asymptotic β†βš metric is realistic. speedup from early-abort ECM. e systematically Results are not what one would olynomial factors. Our guess from 1982 Pomerance. erpolynomial.

  20. Best result known for one key Asymptotic consequences: time β–² 1 ✿ 185632 cost, 1. Attack cost per key using chip area β–² 0 ✿ 790420 ; rmance is reduced, so attacker β†βš is β–² 1 ✿ 976052 . RSA keys. can target lower-value keys. Our main result for 2. Primary bottleneck is low- a batch of β–² 0 ✿ 5 keys: ratio”: memory factorizationβ€”well suited time β–² 1 ✿ 022400 chips. for off-the-shelf graphics cards. using chip area β–² 1 ✿ 181600 ; o 64- β†βš per key is β–² 1 ✿ 704000 . 3. Attack time is reduced as (and can be reduced more), This paper also looks more closely ) is not breaking key rotation. at β–² ♦ (1) , analyzing asymptotic β†βš realistic. speedup from early-abort ECM. systematically Results are not what one would rs. Our guess from 1982 Pomerance. olynomial.

  21. Best result known for one key Asymptotic consequences: time β–² 1 ✿ 185632 1. Attack cost per key using chip area β–² 0 ✿ 790420 ; is reduced, so attacker β†βš is β–² 1 ✿ 976052 . can target lower-value keys. Our main result for 2. Primary bottleneck is low- a batch of β–² 0 ✿ 5 keys: memory factorizationβ€”well suited time β–² 1 ✿ 022400 for off-the-shelf graphics cards. using chip area β–² 1 ✿ 181600 ; β†βš per key is β–² 1 ✿ 704000 . 3. Attack time is reduced (and can be reduced more), This paper also looks more closely breaking key rotation. at β–² ♦ (1) , analyzing asymptotic speedup from early-abort ECM. Results are not what one would guess from 1982 Pomerance.

  22. Best result known for one key Asymptotic consequences: time β–² 1 ✿ 185632 1. Attack cost per key using chip area β–² 0 ✿ 790420 ; is reduced, so attacker β†βš is β–² 1 ✿ 976052 . can target lower-value keys. Our main result for 2. Primary bottleneck is low- a batch of β–² 0 ✿ 5 keys: memory factorizationβ€”well suited time β–² 1 ✿ 022400 for off-the-shelf graphics cards. using chip area β–² 1 ✿ 181600 ; β†βš per key is β–² 1 ✿ 704000 . 3. Attack time is reduced (and can be reduced more), This paper also looks more closely breaking key rotation. at β–² ♦ (1) , analyzing asymptotic β€œDo the asymptotics really kick in speedup from early-abort ECM. before 1024 bits?” β€” Maybe not, Results are not what one would but no basis for confidence. guess from 1982 Pomerance.

  23. result known for one key Asymptotic consequences: Eratosthenes β–² 1 ✿ 185632 1. Attack cost per key Sieving small ✐ ❃ chip area β–² 0 ✿ 790420 ; is reduced, so attacker using prim ❀ ❀ ❀ β–² 1 ✿ 976052 . β†βš can target lower-value keys. 1 2 2 main result for 3 3 2. Primary bottleneck is low- batch of β–² 0 ✿ 5 keys: 4 2 2 5 memory factorizationβ€”well suited 6 2 3 β–² 1 ✿ 022400 7 for off-the-shelf graphics cards. 8 2 2 2 chip area β–² 1 ✿ 181600 ; 9 3 3 10 2 er key is β–² 1 ✿ 704000 . 3. Attack time is reduced β†βš 11 12 2 2 3 (and can be reduced more), 13 paper also looks more closely 14 2 breaking key rotation. 15 3 β–² ♦ (1) , analyzing asymptotic 16 2 2 2 2 17 β€œDo the asymptotics really kick in eedup from early-abort ECM. 18 2 3 3 19 before 1024 bits?” β€” Maybe not, Results are not what one would 20 2 2 but no basis for confidence. from 1982 Pomerance. etc.

  24. wn for one key Asymptotic consequences: Eratosthenes for smo β–² ✿ 1. Attack cost per key Sieving small integers ✐ ❃ β–² 0 ✿ 790420 ; is reduced, so attacker using primes 2 ❀ 3 ❀ 5 ❀ β–² ✿ β†βš can target lower-value keys. 1 2 2 for 3 3 2. Primary bottleneck is low- β–² ✿ keys: 4 2 2 5 5 memory factorizationβ€”well suited 6 2 3 β–² ✿ 7 7 for off-the-shelf graphics cards. 8 2 2 2 β–² 1 ✿ 181600 ; 9 3 3 10 2 5 β–² ✿ 704000 . 3. Attack time is reduced β†βš 11 12 2 2 3 (and can be reduced more), 13 looks more closely 14 2 7 breaking key rotation. 15 3 5 β–² ♦ analyzing asymptotic 16 2 2 2 2 17 β€œDo the asymptotics really kick in rly-abort ECM. 18 2 3 3 19 before 1024 bits?” β€” Maybe not, what one would 20 2 2 5 but no basis for confidence. Pomerance. etc.

  25. key Asymptotic consequences: Eratosthenes for smoothness β–² ✿ 1. Attack cost per key Sieving small integers ✐ ❃ 0 β–² ✿ is reduced, so attacker using primes 2 ❀ 3 ❀ 5 ❀ 7: β–² ✿ β†βš can target lower-value keys. 1 2 2 3 3 2. Primary bottleneck is low- 4 2 2 β–² ✿ 5 5 memory factorizationβ€”well suited 6 2 3 β–² ✿ 7 7 for off-the-shelf graphics cards. 8 2 2 2 β–² ✿ 9 3 3 10 2 5 β–² ✿ 3. Attack time is reduced β†βš 11 12 2 2 3 (and can be reduced more), 13 re closely 14 2 7 breaking key rotation. 15 3 5 β–² ♦ ptotic 16 2 2 2 2 17 β€œDo the asymptotics really kick in ECM. 18 2 3 3 19 before 1024 bits?” β€” Maybe not, would 20 2 2 5 but no basis for confidence. omerance. etc.

  26. Asymptotic consequences: Eratosthenes for smoothness 1. Attack cost per key Sieving small integers ✐ ❃ 0 is reduced, so attacker using primes 2 ❀ 3 ❀ 5 ❀ 7: can target lower-value keys. 1 2 2 3 3 2. Primary bottleneck is low- 4 2 2 5 5 memory factorizationβ€”well suited 6 2 3 7 7 for off-the-shelf graphics cards. 8 2 2 2 9 3 3 10 2 5 3. Attack time is reduced 11 12 2 2 3 (and can be reduced more), 13 14 2 7 breaking key rotation. 15 3 5 16 2 2 2 2 17 β€œDo the asymptotics really kick in 18 2 3 3 19 before 1024 bits?” β€” Maybe not, 20 2 2 5 but no basis for confidence. etc.

  27. Asymptotic consequences: Eratosthenes for smoothness The Q sieve ttack cost per key Sieving small integers ✐ ❃ 0 Sieving ✐ ✐ ✐ reduced, so attacker using primes 2 ❀ 3 ❀ 5 ❀ 7: using prime ❀ ❀ ❀ rget lower-value keys. 1 1 2 2 2 2 3 3 3 3 Primary bottleneck is low- 4 2 2 4 2 2 5 5 5 ry factorizationβ€”well suited 6 2 3 6 2 3 7 7 7 -the-shelf graphics cards. 8 2 2 2 8 2 2 2 9 3 3 9 3 3 10 2 5 10 2 ttack time is reduced 11 11 12 2 2 3 12 2 2 3 can be reduced more), 13 13 14 2 7 14 2 reaking key rotation. 15 3 5 15 3 16 2 2 2 2 16 2 2 2 2 17 17 the asymptotics really kick in 18 2 3 3 18 2 3 3 19 19 1024 bits?” β€” Maybe not, 20 2 2 5 20 2 2 basis for confidence. etc. etc.

  28. consequences: Eratosthenes for smoothness The Q sieve er key Sieving small integers ✐ ❃ 0 Sieving ✐ and 611 + ✐ ✐ attacker using primes 2 ❀ 3 ❀ 5 ❀ 7: using primes 2 ❀ 3 ❀ 5 ❀ er-value keys. 1 1 612 2 2 2 2 2 613 3 3 3 3 614 2 ottleneck is low- 4 2 2 4 2 2 615 5 5 5 5 616 2 rizationβ€”well suited 6 2 3 6 2 3 617 7 7 7 7 618 2 graphics cards. 8 2 2 2 8 2 2 2 619 9 3 3 9 3 3 620 2 10 2 5 10 2 5 621 is reduced 11 11 622 2 12 2 2 3 12 2 2 3 623 reduced more), 13 13 624 2 14 2 7 14 2 7 625 rotation. 15 3 5 15 3 5 626 2 16 2 2 2 2 16 2 2 2 2 627 17 17 628 2 asymptotics really kick in 18 2 3 3 18 2 3 3 629 19 19 630 2 bits?” β€” Maybe not, 20 2 2 5 20 2 2 5 631 confidence. etc. etc.

  29. Eratosthenes for smoothness The Q sieve Sieving small integers ✐ ❃ 0 Sieving ✐ and 611 + ✐ for sm ✐ using primes 2 ❀ 3 ❀ 5 ❀ 7: using primes 2 ❀ 3 ❀ 5 ❀ 7: eys. 1 1 612 2 2 3 3 2 2 2 2 613 3 3 3 3 614 2 low- 4 2 2 4 2 2 615 3 5 5 5 5 5 616 2 2 2 ell suited 6 2 3 6 2 3 617 7 7 7 7 618 2 3 cards. 8 2 2 2 8 2 2 2 619 9 3 3 9 3 3 620 2 2 5 10 2 5 10 2 5 621 3 3 3 11 11 622 2 12 2 2 3 12 2 2 3 623 re), 13 13 624 2 2 2 2 3 14 2 7 14 2 7 625 5 15 3 5 15 3 5 626 2 16 2 2 2 2 16 2 2 2 2 627 3 17 17 628 2 2 kick in 18 2 3 3 18 2 3 3 629 19 19 630 2 3 3 5 ybe not, 20 2 2 5 20 2 2 5 631 confidence. etc. etc.

  30. Eratosthenes for smoothness The Q sieve Sieving small integers ✐ ❃ 0 Sieving ✐ and 611 + ✐ for small ✐ using primes 2 ❀ 3 ❀ 5 ❀ 7: using primes 2 ❀ 3 ❀ 5 ❀ 7: 1 1 612 2 2 3 3 2 2 2 2 613 3 3 3 3 614 2 4 2 2 4 2 2 615 3 5 5 5 5 5 616 2 2 2 7 6 2 3 6 2 3 617 7 7 7 7 618 2 3 8 2 2 2 8 2 2 2 619 9 3 3 9 3 3 620 2 2 5 10 2 5 10 2 5 621 3 3 3 11 11 622 2 12 2 2 3 12 2 2 3 623 7 13 13 624 2 2 2 2 3 14 2 7 14 2 7 625 5 5 5 5 15 3 5 15 3 5 626 2 16 2 2 2 2 16 2 2 2 2 627 3 17 17 628 2 2 18 2 3 3 18 2 3 3 629 19 19 630 2 3 3 5 7 20 2 2 5 20 2 2 5 631 etc. etc.

  31. Eratosthenes for smoothness The Q sieve Have complet the congruences ✐ βœ‘ ✐ Sieving small integers ✐ ❃ 0 Sieving ✐ and 611 + ✐ for small ✐ for some ✐ primes 2 ❀ 3 ❀ 5 ❀ 7: using primes 2 ❀ 3 ❀ 5 ❀ 7: 14 ✁ 625 1 612 2 2 3 3 2 2 613 64 ✁ 675 3 3 3 614 2 4 2 2 615 3 5 75 ✁ 686 5 5 5 616 2 2 2 7 3 6 2 3 617 7 7 7 618 2 3 14 ✁ 64 ✁ 75 ✁ ✁ ✁ 8 2 2 2 619 3 3 9 3 3 620 2 2 5 = 2 8 3 4 5 8 5 10 2 5 621 3 3 3 11 622 2 3 12 2 2 3 623 7 ✟ ✠ gcd 611 ❀ ✁ ✁ οΏ½ 13 624 2 2 2 2 3 7 14 2 7 625 5 5 5 5 = 47. 3 5 15 3 5 626 2 16 2 2 2 2 627 3 17 628 2 2 611 = 47 ✁ 3 3 18 2 3 3 629 19 630 2 3 3 5 7 5 20 2 2 5 631 etc.

  32. smoothness The Q sieve Have complete facto the congruences ✐ βœ‘ ✐ integers ✐ ❃ 0 Sieving ✐ and 611 + ✐ for small ✐ for some ✐ ’s. ❀ ❀ 5 ❀ 7: using primes 2 ❀ 3 ❀ 5 ❀ 7: 14 ✁ 625 = 2 1 3 0 5 4 7 1 612 2 2 3 3 2 2 613 64 ✁ 675 = 2 6 3 3 5 2 7 3 3 614 2 4 2 2 615 3 5 75 ✁ 686 = 2 1 3 1 5 2 7 5 5 616 2 2 2 7 6 2 3 617 7 7 618 2 3 14 ✁ 64 ✁ 75 ✁ 625 ✁ 675 ✁ 8 2 2 2 619 = 2 8 3 4 5 8 7 4 = (2 4 3 9 3 3 620 2 2 5 10 2 5 621 3 3 3 11 622 2 12 2 2 3 623 7 ✟ ✠ gcd 611 ❀ 14 ✁ 64 ✁ 75 οΏ½ 13 624 2 2 2 2 3 14 2 7 625 5 5 5 5 = 47. 15 3 5 626 2 16 2 2 2 2 627 3 17 628 2 2 611 = 47 ✁ 13. 18 2 3 3 629 19 630 2 3 3 5 7 20 2 2 5 631 etc.

  33. othness The Q sieve Have complete factorization the congruences ✐ βœ‘ 611 + ✐ ✐ ❃ 0 Sieving ✐ and 611 + ✐ for small ✐ for some ✐ ’s. ❀ ❀ ❀ using primes 2 ❀ 3 ❀ 5 ❀ 7: 14 ✁ 625 = 2 1 3 0 5 4 7 1 . 1 612 2 2 3 3 2 2 613 64 ✁ 675 = 2 6 3 3 5 2 7 0 . 3 3 614 2 4 2 2 615 3 5 75 ✁ 686 = 2 1 3 1 5 2 7 3 . 5 5 616 2 2 2 7 6 2 3 617 7 7 618 2 3 14 ✁ 64 ✁ 75 ✁ 625 ✁ 675 ✁ 686 8 2 2 2 619 = 2 8 3 4 5 8 7 4 = (2 4 3 2 5 4 7 2 ) 2 . 9 3 3 620 2 2 5 10 2 5 621 3 3 3 11 622 2 611 ❀ 14 ✁ 64 ✁ 75 οΏ½ 2 4 3 2 5 12 2 2 3 623 7 ✟ ✠ gcd 13 624 2 2 2 2 3 14 2 7 625 5 5 5 5 = 47. 15 3 5 626 2 16 2 2 2 2 627 3 17 628 2 2 611 = 47 ✁ 13. 18 2 3 3 629 19 630 2 3 3 5 7 20 2 2 5 631 etc.

  34. The Q sieve Have complete factorization of the congruences ✐ βœ‘ 611 + ✐ Sieving ✐ and 611 + ✐ for small ✐ for some ✐ ’s. using primes 2 ❀ 3 ❀ 5 ❀ 7: 14 ✁ 625 = 2 1 3 0 5 4 7 1 . 1 612 2 2 3 3 2 2 613 64 ✁ 675 = 2 6 3 3 5 2 7 0 . 3 3 614 2 4 2 2 615 3 5 75 ✁ 686 = 2 1 3 1 5 2 7 3 . 5 5 616 2 2 2 7 6 2 3 617 7 7 618 2 3 14 ✁ 64 ✁ 75 ✁ 625 ✁ 675 ✁ 686 8 2 2 2 619 = 2 8 3 4 5 8 7 4 = (2 4 3 2 5 4 7 2 ) 2 . 9 3 3 620 2 2 5 10 2 5 621 3 3 3 11 622 2 611 ❀ 14 ✁ 64 ✁ 75 οΏ½ 2 4 3 2 5 4 7 2 ✠ 12 2 2 3 623 7 ✟ gcd 13 624 2 2 2 2 3 14 2 7 625 5 5 5 5 = 47. 15 3 5 626 2 16 2 2 2 2 627 3 17 628 2 2 611 = 47 ✁ 13. 18 2 3 3 629 19 630 2 3 3 5 7 20 2 2 5 631 etc.

  35. sieve Have complete factorization of The numb the congruences ✐ βœ‘ 611 + ✐ Sieving ✐ and 611 + ✐ for small ✐ Generalize ✐ βœ‘ ✐ β—† β—† for some ✐ ’s. primes 2 ❀ 3 ❀ 5 ❀ 7: ✦ ❛ βœ‘ ❛ βœβ—† β—† 14 ✁ 625 = 2 1 3 0 5 4 7 1 . ✦ ❛ οΏ½ βœβ™  βœ‘ ❛ οΏ½ βœβ˜› β™  οΏ½ β˜› 612 2 2 3 3 613 64 ✁ 675 = 2 6 3 3 5 2 7 0 . for root β˜› ✷ 3 614 2 615 3 5 75 ✁ 686 = 2 1 3 1 5 2 7 3 . of nonzero 5 616 2 2 2 7 3 617 7 618 2 3 14 ✁ 64 ✁ 75 ✁ 625 ✁ 675 ✁ 686 For any β™  β˜› 619 = 2 8 3 4 5 8 7 4 = (2 4 3 2 5 4 7 2 ) 2 . 3 3 620 2 2 5 so that facto β™  οΏ½ β˜› 5 621 3 3 3 622 2 produces β—† 611 ❀ 14 ✁ 64 ✁ 75 οΏ½ 2 4 3 2 5 4 7 2 ✠ 3 623 7 ✟ gcd 624 2 2 2 2 3 7 625 5 5 5 5 = 47. Optimal β™  3 5 626 2 ❂ ❂ 627 3 ( βœ– + ♦ (1))(log β—† β—† 628 2 2 611 = 47 ✁ 13. 3 3 629 630 2 3 3 5 7 5 631

  36. Have complete factorization of The number-field sieve the congruences ✐ βœ‘ 611 + ✐ ✐ 611 + ✐ for small ✐ Generalize ✐ βœ‘ ✐ + β—† β—† for some ✐ ’s. ❀ ❀ 5 ❀ 7: ✦ ❛ βœ‘ ❛ + βœβ—† (mo β—† 14 ✁ 625 = 2 1 3 0 5 4 7 1 . ✦ ❛ οΏ½ βœβ™  βœ‘ ❛ οΏ½ βœβ˜› β™  οΏ½ β˜› 2 2 3 3 64 ✁ 675 = 2 6 3 3 5 2 7 0 . for root β˜› ✷ C 2 3 5 75 ✁ 686 = 2 1 3 1 5 2 7 3 . of nonzero integer 2 2 2 7 2 3 14 ✁ 64 ✁ 75 ✁ 625 ✁ 675 ✁ 686 For any β™  can find β˜› = 2 8 3 4 5 8 7 4 = (2 4 3 2 5 4 7 2 ) 2 . 2 2 5 so that factoring β™  οΏ½ β˜› 3 3 3 2 produces factorization β—† 611 ❀ 14 ✁ 64 ✁ 75 οΏ½ 2 4 3 2 5 4 7 2 ✠ 7 ✟ gcd 2 2 2 2 3 5 5 5 5 = 47. Optimal choice of β™  2 ( βœ– + ♦ (1))(log β—† ) 2 ❂ ❂ 3 β—† 2 2 611 = 47 ✁ 13. 2 3 3 5 7

  37. Have complete factorization of The number-field sieve the congruences ✐ βœ‘ 611 + ✐ ✐ ✐ small ✐ Generalize ✐ βœ‘ ✐ + β—† (mod β—† for some ✐ ’s. ❀ ❀ ❀ ✦ ❛ βœ‘ ❛ + βœβ—† (mod β—† ) 14 ✁ 625 = 2 1 3 0 5 4 7 1 . ✦ ❛ οΏ½ βœβ™  βœ‘ ❛ οΏ½ βœβ˜› (mod β™  οΏ½ β˜› 64 ✁ 675 = 2 6 3 3 5 2 7 0 . for root β˜› ✷ C 5 75 ✁ 686 = 2 1 3 1 5 2 7 3 . of nonzero integer poly. 7 14 ✁ 64 ✁ 75 ✁ 625 ✁ 675 ✁ 686 For any β™  can find β˜› = 2 8 3 4 5 8 7 4 = (2 4 3 2 5 4 7 2 ) 2 . 5 so that factoring β™  οΏ½ β˜› produces factorization of β—† . 611 ❀ 14 ✁ 64 ✁ 75 οΏ½ 2 4 3 2 5 4 7 2 ✠ 7 ✟ gcd 5 5 5 5 = 47. Optimal choice of log β™  is ( βœ– + ♦ (1))(log β—† ) 2 ❂ 3 (log log β—† ❂ 611 = 47 ✁ 13. 5 7

  38. Have complete factorization of The number-field sieve the congruences ✐ βœ‘ 611 + ✐ Generalize ✐ βœ‘ ✐ + β—† (mod β—† ) for some ✐ ’s. ✦ ❛ βœ‘ ❛ + βœβ—† (mod β—† ) 14 ✁ 625 = 2 1 3 0 5 4 7 1 . ✦ ❛ οΏ½ βœβ™  βœ‘ ❛ οΏ½ βœβ˜› (mod β™  οΏ½ β˜› ) 64 ✁ 675 = 2 6 3 3 5 2 7 0 . for root β˜› ✷ C 75 ✁ 686 = 2 1 3 1 5 2 7 3 . of nonzero integer poly. 14 ✁ 64 ✁ 75 ✁ 625 ✁ 675 ✁ 686 For any β™  can find β˜› = 2 8 3 4 5 8 7 4 = (2 4 3 2 5 4 7 2 ) 2 . so that factoring β™  οΏ½ β˜› produces factorization of β—† . 611 ❀ 14 ✁ 64 ✁ 75 οΏ½ 2 4 3 2 5 4 7 2 ✠ ✟ gcd = 47. Optimal choice of log β™  is ( βœ– + ♦ (1))(log β—† ) 2 ❂ 3 (log log β—† ) 1 ❂ 3 . 611 = 47 ✁ 13.

  39. complete factorization of The number-field sieve RAM cost congruences ✐ βœ‘ 611 + ✐ Generalize ✐ βœ‘ ✐ + β—† (mod β—† ) 1993 Buhler–Lenstra–P ome ✐ ’s. β–² ✿ ✦ ❛ βœ‘ ❛ + βœβ—† (mod β—† ) Smoothness Sieve β–² 1 ✿ ✁ 625 = 2 1 3 0 5 4 7 1 . ✦ ❛ οΏ½ βœβ™  βœ‘ ❛ οΏ½ βœβ˜› (mod β™  οΏ½ β˜› ) ❛❀ ❜ ✁ 675 = 2 6 3 3 5 2 7 0 . Find β–² 0 ✿ 961500 for root β˜› ✷ C ✁ 686 = 2 1 3 1 5 2 7 3 . of nonzero integer poly. with ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ˜› β–² ✿ Total RAM ✁ 75 ✁ 625 ✁ 675 ✁ 686 For any β™  can find β˜› ✁ 5 8 7 4 = (2 4 3 2 5 4 7 2 ) 2 . so that factoring β™  οΏ½ β˜› 1993 Copp β–² ✿ produces factorization of β—† . Total RAM 611 ❀ 14 ✁ 64 ✁ 75 οΏ½ 2 4 3 2 5 4 7 2 ✠ ✟ using multiple Optimal choice of log β™  is ( βœ– + ♦ (1))(log β—† ) 2 ❂ 3 (log log β—† ) 1 ❂ 3 . (Multiple 47 ✁ 13. don’t seem with β†βš

  40. factorization of The number-field sieve RAM cost analysis ✐ βœ‘ 611 + ✐ Generalize ✐ βœ‘ ✐ + β—† (mod β—† ) 1993 Buhler–Lenstra–P ✐ Smoothness bound β–² ✿ ✦ ❛ βœ‘ ❛ + βœβ—† (mod β—† ) Sieve β–² 1 ✿ 923000 pairs ❛❀ ❜ 4 7 1 . ✦ ❛ οΏ½ βœβ™  βœ‘ ❛ οΏ½ βœβ˜› (mod β™  οΏ½ β˜› ) ✁ Find β–² 0 ✿ 961500 pairs 2 7 0 . for root β˜› ✷ C ✁ 2 7 3 . of nonzero integer poly. with ❛ οΏ½ βœβ™  and ❛ οΏ½ βœβ˜› ✁ Total RAM time β–² ✿ ✁ 675 ✁ 686 For any β™  can find β˜› ✁ ✁ ✁ (2 4 3 2 5 4 7 2 ) 2 . so that factoring β™  οΏ½ β˜› 1993 Coppersmith: Total RAM time β–² ✿ produces factorization of β—† . ✁ 75 οΏ½ 2 4 3 2 5 4 7 2 ✠ ✟ ❀ ✁ using multiple numb Optimal choice of log β™  is ( βœ– + ♦ (1))(log β—† ) 2 ❂ 3 (log log β—† ) 1 ❂ 3 . (Multiple number ✁ don’t seem to combine with β†βš , factory, et

  41. rization of The number-field sieve RAM cost analysis ✐ βœ‘ ✐ Generalize ✐ βœ‘ ✐ + β—† (mod β—† ) 1993 Buhler–Lenstra–Pomera ✐ Smoothness bound β–² 0 ✿ 961500 ✦ ❛ βœ‘ ❛ + βœβ—† (mod β—† ) Sieve β–² 1 ✿ 923000 pairs ( ❛❀ ❜ ). ✦ ❛ οΏ½ βœβ™  βœ‘ ❛ οΏ½ βœβ˜› (mod β™  οΏ½ β˜› ) ✁ Find β–² 0 ✿ 961500 pairs for root β˜› ✷ C ✁ of nonzero integer poly. with ❛ οΏ½ βœβ™  and ❛ οΏ½ βœβ˜› smo ✁ Total RAM time β–² 1 ✿ 923000 . For any β™  can find β˜› ✁ ✁ ✁ ✁ ✁ . so that factoring β™  οΏ½ β˜› 1993 Coppersmith: Total RAM time β–² 1 ✿ 901884 produces factorization of β—† . 2 5 4 7 2 ✠ ✟ ❀ ✁ ✁ οΏ½ using multiple number fields. Optimal choice of log β™  is ( βœ– + ♦ (1))(log β—† ) 2 ❂ 3 (log log β—† ) 1 ❂ 3 . (Multiple number fields ✁ don’t seem to combine well with β†βš , factory, et al.)

  42. The number-field sieve RAM cost analysis Generalize ✐ βœ‘ ✐ + β—† (mod β—† ) 1993 Buhler–Lenstra–Pomerance: Smoothness bound β–² 0 ✿ 961500 . ✦ ❛ βœ‘ ❛ + βœβ—† (mod β—† ) Sieve β–² 1 ✿ 923000 pairs ( ❛❀ ❜ ). ✦ ❛ οΏ½ βœβ™  βœ‘ ❛ οΏ½ βœβ˜› (mod β™  οΏ½ β˜› ) Find β–² 0 ✿ 961500 pairs for root β˜› ✷ C of nonzero integer poly. with ❛ οΏ½ βœβ™  and ❛ οΏ½ βœβ˜› smooth. Total RAM time β–² 1 ✿ 923000 . For any β™  can find β˜› so that factoring β™  οΏ½ β˜› 1993 Coppersmith: Total RAM time β–² 1 ✿ 901884 produces factorization of β—† . using multiple number fields. Optimal choice of log β™  is ( βœ– + ♦ (1))(log β—† ) 2 ❂ 3 (log log β—† ) 1 ❂ 3 . (Multiple number fields don’t seem to combine well with β†βš , factory, et al.)

  43. number-field sieve RAM cost analysis β†βš cost Generalize ✐ βœ‘ ✐ + β—† (mod β—† ) 1993 Buhler–Lenstra–Pomerance: Sieving is Smoothness bound β–² 0 ✿ 961500 . ✦ ❛ βœ‘ ❛ + βœβ—† (mod β—† ) in realistic Sieve β–² 1 ✿ 923000 pairs ( ❛❀ ❜ ). β†βš cost β–² ✿ ✦ ❛ οΏ½ βœβ™  βœ‘ ❛ οΏ½ βœβ˜› (mod β™  οΏ½ β˜› ) Find β–² 0 ✿ 961500 pairs ot β˜› ✷ C nonzero integer poly. with ❛ οΏ½ βœβ™  and ❛ οΏ½ βœβ˜› smooth. Total RAM time β–² 1 ✿ 923000 . any β™  can find β˜› that factoring β™  οΏ½ β˜› 1993 Coppersmith: Total RAM time β–² 1 ✿ 901884 duces factorization of β—† . using multiple number fields. Optimal choice of log β™  is ♦ (1))(log β—† ) 2 ❂ 3 (log log β—† ) 1 ❂ 3 . βœ– (Multiple number fields don’t seem to combine well with β†βš , factory, et al.)

  44. er-field sieve RAM cost analysis β†βš cost analysis ✐ βœ‘ ✐ + β—† (mod β—† ) 1993 Buhler–Lenstra–Pomerance: Sieving is a disaster Smoothness bound β–² 0 ✿ 961500 . ✦ ❛ βœ‘ ❛ βœβ—† (mod β—† ) in realistic cost metric. Sieve β–² 1 ✿ 923000 pairs ( ❛❀ ❜ ). β†βš cost β–² 2 ✿ 403750 . ✦ ❛ οΏ½ βœβ™  βœ‘ ❛ οΏ½ βœβ˜› (mod β™  οΏ½ β˜› ) Find β–² 0 ✿ 961500 pairs β˜› ✷ integer poly. with ❛ οΏ½ βœβ™  and ❛ οΏ½ βœβ˜› smooth. Total RAM time β–² 1 ✿ 923000 . β™  find β˜› β™  οΏ½ β˜› 1993 Coppersmith: Total RAM time β–² 1 ✿ 901884 zation of β—† . using multiple number fields. of log β™  is β—† ) 2 ❂ 3 (log log β—† ) 1 ❂ 3 . βœ– ♦ (Multiple number fields don’t seem to combine well with β†βš , factory, et al.)

  45. RAM cost analysis β†βš cost analysis ✐ βœ‘ ✐ β—† (mod β—† ) 1993 Buhler–Lenstra–Pomerance: Sieving is a disaster Smoothness bound β–² 0 ✿ 961500 . ✦ ❛ βœ‘ ❛ βœβ—† β—† in realistic cost metric. Sieve β–² 1 ✿ 923000 pairs ( ❛❀ ❜ ). β†βš cost β–² 2 ✿ 403750 . ✦ ❛ οΏ½ βœβ™  βœ‘ ❛ οΏ½ βœβ˜› β™  οΏ½ β˜› ) Find β–² 0 ✿ 961500 pairs β˜› ✷ with ❛ οΏ½ βœβ™  and ❛ οΏ½ βœβ˜› smooth. Total RAM time β–² 1 ✿ 923000 . β™  β˜› β™  οΏ½ β˜› 1993 Coppersmith: Total RAM time β–² 1 ✿ 901884 β—† . using multiple number fields. β™  ❂ log β—† ) 1 ❂ 3 . βœ– ♦ β—† (Multiple number fields don’t seem to combine well with β†βš , factory, et al.)

  46. RAM cost analysis β†βš cost analysis 1993 Buhler–Lenstra–Pomerance: Sieving is a disaster Smoothness bound β–² 0 ✿ 961500 . in realistic cost metric. Sieve β–² 1 ✿ 923000 pairs ( ❛❀ ❜ ). β†βš cost β–² 2 ✿ 403750 . Find β–² 0 ✿ 961500 pairs with ❛ οΏ½ βœβ™  and ❛ οΏ½ βœβ˜› smooth. Total RAM time β–² 1 ✿ 923000 . 1993 Coppersmith: Total RAM time β–² 1 ✿ 901884 using multiple number fields. (Multiple number fields don’t seem to combine well with β†βš , factory, et al.)

  47. RAM cost analysis β†βš cost analysis 1993 Buhler–Lenstra–Pomerance: Sieving is a disaster Smoothness bound β–² 0 ✿ 961500 . in realistic cost metric. Sieve β–² 1 ✿ 923000 pairs ( ❛❀ ❜ ). β†βš cost β–² 2 ✿ 403750 . Find β–² 0 ✿ 961500 pairs Fix: find smooth using ECM. with ❛ οΏ½ βœβ™  and ❛ οΏ½ βœβ˜› smooth. β†βš cost β–² 1 ✿ 923000 . Total RAM time β–² 1 ✿ 923000 . 1993 Coppersmith: Total RAM time β–² 1 ✿ 901884 using multiple number fields. (Multiple number fields don’t seem to combine well with β†βš , factory, et al.)

  48. RAM cost analysis β†βš cost analysis 1993 Buhler–Lenstra–Pomerance: Sieving is a disaster Smoothness bound β–² 0 ✿ 961500 . in realistic cost metric. Sieve β–² 1 ✿ 923000 pairs ( ❛❀ ❜ ). β†βš cost β–² 2 ✿ 403750 . Find β–² 0 ✿ 961500 pairs Fix: find smooth using ECM. with ❛ οΏ½ βœβ™  and ❛ οΏ½ βœβ˜› smooth. β†βš cost β–² 1 ✿ 923000 . Total RAM time β–² 1 ✿ 923000 . Linear algebra is also a disaster. 1993 Coppersmith: β†βš cost β–² 2 ✿ 403750 . Total RAM time β–² 1 ✿ 901884 using multiple number fields. (Multiple number fields don’t seem to combine well with β†βš , factory, et al.)

  49. RAM cost analysis β†βš cost analysis 1993 Buhler–Lenstra–Pomerance: Sieving is a disaster Smoothness bound β–² 0 ✿ 961500 . in realistic cost metric. Sieve β–² 1 ✿ 923000 pairs ( ❛❀ ❜ ). β†βš cost β–² 2 ✿ 403750 . Find β–² 0 ✿ 961500 pairs Fix: find smooth using ECM. with ❛ οΏ½ βœβ™  and ❛ οΏ½ βœβ˜› smooth. β†βš cost β–² 1 ✿ 923000 . Total RAM time β–² 1 ✿ 923000 . Linear algebra is also a disaster. 1993 Coppersmith: β†βš cost β–² 2 ✿ 403750 . Total RAM time β–² 1 ✿ 901884 Semi-fix: Reduce smoothness using multiple number fields. bounds to rebalance. (Multiple number fields β†βš cost β–² 1 ✿ 976052 . don’t seem to combine well (2001 Bernstein) with β†βš , factory, et al.)

  50. cost analysis β†βš cost analysis The facto Buhler–Lenstra–Pomerance: Sieving is a disaster 1993 Copp othness bound β–² 0 ✿ 961500 . in realistic cost metric. There exists β–² 1 ✿ 923000 pairs ( ❛❀ ❜ ). β†βš cost β–² 2 ✿ 403750 . that facto β–² 0 ✿ 961500 pairs with same β—† Fix: find smooth using ECM. β–² ✿ ❛ οΏ½ βœβ™  and ❛ οΏ½ βœβ˜› smooth. in RAM β†βš cost β–² 1 ✿ 923000 . RAM time β–² 1 ✿ 923000 . β–² ✿ Smoothness Linear algebra is also a disaster. Coppersmith: Smaller than β†βš cost β–² 2 ✿ 403750 . RAM time β–² 1 ✿ 901884 so need mo ❛❀ ❜ Semi-fix: Reduce smoothness multiple number fields. Algorithm ❛❀ ❜ bounds to rebalance. (Multiple number fields such that ❛ οΏ½ βœβ™  β†βš cost β–² 1 ✿ 976052 . seem to combine well Note: one β™  β—† (2001 Bernstein) β†βš , factory, et al.) Algorithm whether ❛ οΏ½ βœβ˜› β—†

  51. analysis β†βš cost analysis The factorization facto Buhler–Lenstra–Pomerance: Sieving is a disaster 1993 Coppersmith: ound β–² 0 ✿ 961500 . in realistic cost metric. There exists an algo β–² ✿ β†βš cost β–² 2 ✿ 403750 . pairs ( ❛❀ ❜ ). that factors any integer β–² ✿ pairs with same #bits as β—† Fix: find smooth using ECM. in RAM time β–² 1 ✿ 638587 ❛ οΏ½ βœβ™  and ❛ οΏ½ βœβ˜› smooth. β†βš cost β–² 1 ✿ 923000 . β–² 1 ✿ 923000 . Smoothness bound β–² ✿ Linear algebra is also a disaster. ersmith: Smaller than before, β†βš cost β–² 2 ✿ 403750 . β–² 1 ✿ 901884 so need more ( ❛❀ ❜ Semi-fix: Reduce smoothness number fields. Algorithm knows all ❛❀ ❜ bounds to rebalance. er fields such that ❛ οΏ½ βœβ™  β†βš cost β–² 1 ✿ 976052 . combine well Note: one β™  works β—† (2001 Bernstein) β†βš , et al.) Algorithm uses ECM whether ❛ οΏ½ βœβ˜› β—† is

  52. β†βš cost analysis The factorization factory omerance: Sieving is a disaster 1993 Coppersmith: β–² ✿ 961500 . in realistic cost metric. There exists an algorithm β–² ✿ β†βš cost β–² 2 ✿ 403750 . ❛❀ ❜ ). that factors any integer β–² ✿ with same #bits as β—† Fix: find smooth using ECM. in RAM time β–² 1 ✿ 638587 . ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ˜› smooth. β†βš cost β–² 1 ✿ 923000 . β–² ✿ . Smoothness bound β–² 0 ✿ 819290 Linear algebra is also a disaster. Smaller than before, β†βš cost β–² 2 ✿ 403750 . β–² ✿ so need more ( ❛❀ ❜ ). Semi-fix: Reduce smoothness fields. Algorithm knows all ( ❛❀ ❜ ) bounds to rebalance. such that ❛ οΏ½ βœβ™  is smooth. β†βš cost β–² 1 ✿ 976052 . ell Note: one β™  works for all β—† (2001 Bernstein) β†βš Algorithm uses ECM to check whether ❛ οΏ½ βœβ˜› β—† is smooth.

  53. β†βš cost analysis The factorization factory Sieving is a disaster 1993 Coppersmith: in realistic cost metric. There exists an algorithm β†βš cost β–² 2 ✿ 403750 . that factors any integer with same #bits as β—† Fix: find smooth using ECM. in RAM time β–² 1 ✿ 638587 . β†βš cost β–² 1 ✿ 923000 . Smoothness bound β–² 0 ✿ 819290 . Linear algebra is also a disaster. Smaller than before, β†βš cost β–² 2 ✿ 403750 . so need more ( ❛❀ ❜ ). Semi-fix: Reduce smoothness Algorithm knows all ( ❛❀ ❜ ) bounds to rebalance. such that ❛ οΏ½ βœβ™  is smooth. β†βš cost β–² 1 ✿ 976052 . Note: one β™  works for all β—† . (2001 Bernstein) Algorithm uses ECM to check whether ❛ οΏ½ βœβ˜› β—† is smooth.

  54. β†βš cost analysis The factorization factory Finding is slower Sieving is a disaster 1993 Coppersmith: Need to ❛❀ ❜ realistic cost metric. There exists an algorithm such that ❛ οΏ½ βœβ™  β†βš cost β–² 2 ✿ 403750 . that factors any integer RAM time β–² ✿ with same #bits as β—† find smooth using ECM. in RAM time β–² 1 ✿ 638587 . β†βš cost β–² 1 ✿ 923000 . Smoothness bound β–² 0 ✿ 819290 . algebra is also a disaster. Smaller than before, β†βš cost β–² 2 ✿ 403750 . so need more ( ❛❀ ❜ ). Semi-fix: Reduce smoothness Algorithm knows all ( ❛❀ ❜ ) ounds to rebalance. such that ❛ οΏ½ βœβ™  is smooth. β†βš cost β–² 1 ✿ 976052 . Note: one β™  works for all β—† . Bernstein) Algorithm uses ECM to check whether ❛ οΏ½ βœβ˜› β—† is smooth.

  55. β†βš The factorization factory Finding this algorithm is slower than running disaster 1993 Coppersmith: Need to precompute ❛❀ ❜ metric. There exists an algorithm such that ❛ οΏ½ βœβ™  β–² ✿ 403750 . β†βš that factors any integer RAM time β–² 2 ✿ 006853 with same #bits as β—† using ECM. in RAM time β–² 1 ✿ 638587 . β–² ✿ 923000 . β†βš Smoothness bound β–² 0 ✿ 819290 . also a disaster. Smaller than before, β–² ✿ 403750 . β†βš so need more ( ❛❀ ❜ ). Reduce smoothness Algorithm knows all ( ❛❀ ❜ ) rebalance. such that ❛ οΏ½ βœβ™  is smooth. β–² ✿ 976052 . β†βš Note: one β™  works for all β—† . Bernstein) Algorithm uses ECM to check whether ❛ οΏ½ βœβ˜› β—† is smooth.

  56. β†βš The factorization factory Finding this algorithm is slower than running it. 1993 Coppersmith: Need to precompute all ( ❛❀ ❜ There exists an algorithm such that ❛ οΏ½ βœβ™  is smooth. β–² ✿ β†βš that factors any integer RAM time β–² 2 ✿ 006853 . with same #bits as β—† ECM. in RAM time β–² 1 ✿ 638587 . β–² ✿ β†βš Smoothness bound β–² 0 ✿ 819290 . disaster. Smaller than before, β–² ✿ β†βš so need more ( ❛❀ ❜ ). othness Algorithm knows all ( ❛❀ ❜ ) such that ❛ οΏ½ βœβ™  is smooth. β–² ✿ β†βš Note: one β™  works for all β—† . Algorithm uses ECM to check whether ❛ οΏ½ βœβ˜› β—† is smooth.

  57. The factorization factory Finding this algorithm is slower than running it. 1993 Coppersmith: Need to precompute all ( ❛❀ ❜ ) There exists an algorithm such that ❛ οΏ½ βœβ™  is smooth. that factors any integer RAM time β–² 2 ✿ 006853 . with same #bits as β—† in RAM time β–² 1 ✿ 638587 . Smoothness bound β–² 0 ✿ 819290 . Smaller than before, so need more ( ❛❀ ❜ ). Algorithm knows all ( ❛❀ ❜ ) such that ❛ οΏ½ βœβ™  is smooth. Note: one β™  works for all β—† . Algorithm uses ECM to check whether ❛ οΏ½ βœβ˜› β—† is smooth.

  58. The factorization factory Finding this algorithm is slower than running it. 1993 Coppersmith: Need to precompute all ( ❛❀ ❜ ) There exists an algorithm such that ❛ οΏ½ βœβ™  is smooth. that factors any integer RAM time β–² 2 ✿ 006853 . with same #bits as β—† in RAM time β–² 1 ✿ 638587 . Standard conversion of precomputation into batching: Smoothness bound β–² 0 ✿ 819290 . if there are enough targets, Smaller than before, more than β–² 0 ✿ 368266 , so need more ( ❛❀ ❜ ). then precomputation cost Algorithm knows all ( ❛❀ ❜ ) becomes negligible. such that ❛ οΏ½ βœβ™  is smooth. Note: one β™  works for all β—† . Algorithm uses ECM to check whether ❛ οΏ½ βœβ˜› β—† is smooth.

  59. The factorization factory Finding this algorithm is slower than running it. 1993 Coppersmith: Need to precompute all ( ❛❀ ❜ ) There exists an algorithm such that ❛ οΏ½ βœβ™  is smooth. that factors any integer RAM time β–² 2 ✿ 006853 . with same #bits as β—† in RAM time β–² 1 ✿ 638587 . Standard conversion of precomputation into batching: Smoothness bound β–² 0 ✿ 819290 . if there are enough targets, Smaller than before, more than β–² 0 ✿ 368266 , so need more ( ❛❀ ❜ ). then precomputation cost Algorithm knows all ( ❛❀ ❜ ) becomes negligible. such that ❛ οΏ½ βœβ™  is smooth. The big problem: Coppersmith’s Note: one β™  works for all β—† . algorithm has size β–² 1 ✿ 638587 . Algorithm uses ECM to check Huge β†βš cost; useless in reality. whether ❛ οΏ½ βœβ˜› β—† is smooth.

  60. factorization factory Finding this algorithm Batch NFS is slower than running it. Coppersmith: Goal: Optimize β†βš Need to precompute all ( ❛❀ ❜ ) exists an algorithm 1. Generate ❛❀ ❜ such that ❛ οΏ½ βœβ™  is smooth. factors any integer RAM time β–² 2 ✿ 006853 . Test ❛ οΏ½ βœβ™  same #bits as β—† time β–² 1 ✿ 638587 . 2. Make β—† Standard conversion of close to ❛❀ ❜ precomputation into batching: othness bound β–² 0 ✿ 819290 . When smo ❛ οΏ½ βœβ™  if there are enough targets, Smaller than before, more than β–² 0 ✿ 368266 , test each ❛ οΏ½ βœβ˜› β—† need more ( ❛❀ ❜ ). then precomputation cost 3. After rithm knows all ( ❛❀ ❜ ) becomes negligible. reorganize: β—† that ❛ οΏ½ βœβ™  is smooth. relevant ❛❀ ❜ The big problem: Coppersmith’s one β™  works for all β—† . algorithm has size β–² 1 ✿ 638587 . 4. Linear rithm uses ECM to check Huge β†βš cost; useless in reality. whether ❛ οΏ½ βœβ˜› β—† is smooth.

  61. ization factory Finding this algorithm Batch NFS is slower than running it. ersmith: Goal: Optimize β†βš Need to precompute all ( ❛❀ ❜ ) algorithm 1. Generate ( ❛❀ ❜ ) such that ❛ οΏ½ βœβ™  is smooth. integer RAM time β–² 2 ✿ 006853 . Test ❛ οΏ½ βœβ™  for smo as β—† β–² ✿ 638587 . 2. Make many copies β—† Standard conversion of close to each ( ❛❀ ❜ ) precomputation into batching: ound β–² 0 ✿ 819290 . When smooth ❛ οΏ½ βœβ™  if there are enough targets, efore, more than β–² 0 ✿ 368266 , test each ❛ οΏ½ βœβ˜› β—† ❛❀ ❜ ). then precomputation cost 3. After all smooths ws all ( ❛❀ ❜ ) becomes negligible. reorganize: for each β—† ❛ οΏ½ βœβ™  is smooth. relevant ( ❛❀ ❜ ) close The big problem: Coppersmith’s β™  rks for all β—† . algorithm has size β–² 1 ✿ 638587 . 4. Linear algebra. ECM to check Huge β†βš cost; useless in reality. ❛ οΏ½ βœβ˜› β—† is smooth.

  62. Finding this algorithm Batch NFS is slower than running it. Goal: Optimize β†βš asymptotics. Need to precompute all ( ❛❀ ❜ ) 1. Generate ( ❛❀ ❜ ) in parallel. such that ❛ οΏ½ βœβ™  is smooth. RAM time β–² 2 ✿ 006853 . Test ❛ οΏ½ βœβ™  for smoothness. β—† β–² ✿ 2. Make many copies of eac β—† Standard conversion of close to each ( ❛❀ ❜ ) generato precomputation into batching: β–² ✿ 819290 . When smooth ❛ οΏ½ βœβ™  is found, if there are enough targets, more than β–² 0 ✿ 368266 , test each ❛ οΏ½ βœβ˜› β—† for smoothness. ❛❀ ❜ then precomputation cost 3. After all smooths are found, ❛❀ ❜ becomes negligible. reorganize: for each β—† , bring ❛ οΏ½ βœβ™  oth. relevant ( ❛❀ ❜ ) close together. The big problem: Coppersmith’s β™  β—† . algorithm has size β–² 1 ✿ 638587 . 4. Linear algebra. check Huge β†βš cost; useless in reality. ❛ οΏ½ βœβ˜› β—† oth.

  63. Finding this algorithm Batch NFS is slower than running it. Goal: Optimize β†βš asymptotics. Need to precompute all ( ❛❀ ❜ ) 1. Generate ( ❛❀ ❜ ) in parallel. such that ❛ οΏ½ βœβ™  is smooth. RAM time β–² 2 ✿ 006853 . Test ❛ οΏ½ βœβ™  for smoothness. 2. Make many copies of each β—† , Standard conversion of close to each ( ❛❀ ❜ ) generator. precomputation into batching: When smooth ❛ οΏ½ βœβ™  is found, if there are enough targets, more than β–² 0 ✿ 368266 , test each ❛ οΏ½ βœβ˜› β—† for smoothness. then precomputation cost 3. After all smooths are found, becomes negligible. reorganize: for each β—† , bring relevant ( ❛❀ ❜ ) close together. The big problem: Coppersmith’s algorithm has size β–² 1 ✿ 638587 . 4. Linear algebra. Huge β†βš cost; useless in reality.

  64. Finding this algorithm Batch NFS Generate ( ❛❀ ❜ ). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ er than running it. Is ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  Goal: Optimize β†βš asymptotics. smooth? to precompute all ( ❛❀ ❜ ) If so, store. 1. Generate ( ❛❀ ❜ ) in parallel. that ❛ οΏ½ βœβ™  is smooth. Repeat. time β–² 2 ✿ 006853 . Test ❛ οΏ½ βœβ™  for smoothness. Generate ( ❛❀ ❜ ). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Is ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  smooth? 2. Make many copies of each β—† , Standard conversion of If so, store. close to each ( ❛❀ ❜ ) generator. recomputation into batching: Repeat. Generate ( ❛❀ ❜ ). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ When smooth ❛ οΏ½ βœβ™  is found, there are enough targets, Is ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  than β–² 0 ✿ 368266 , test each ❛ οΏ½ βœβ˜› β—† for smoothness. smooth? If so, store. recomputation cost Repeat. 3. After all smooths are found, Generate ( ❛❀ ❜ ). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ecomes negligible. reorganize: for each β—† , bring Is ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  smooth? relevant ( ❛❀ ❜ ) close together. big problem: Coppersmith’s If so, store. rithm has size β–² 1 ✿ 638587 . Repeat. 4. Linear algebra. β†βš cost; useless in reality.

  65. algorithm Batch NFS Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ❛❀ ❜ ❛❀ ❜ running it. Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  Goal: Optimize β†βš asymptotics. smooth? smooth? recompute all ( ❛❀ ❜ ) If so, store. If so, store. 1. Generate ( ❛❀ ❜ ) in parallel. ❛ οΏ½ βœβ™  is smooth. Repeat. Repeat. β–² ✿ 006853 . Test ❛ οΏ½ βœβ™  for smoothness. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ❛❀ ❜ ❛❀ ❜ Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  smooth? smooth? 2. Make many copies of each β—† , conversion of If so, store. If so, store. close to each ( ❛❀ ❜ ) generator. into batching: Repeat. Repeat. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ❛❀ ❜ ❛❀ ❜ When smooth ❛ οΏ½ βœβ™  is found, enough targets, Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  β–² ✿ 368266 , test each ❛ οΏ½ βœβ˜› β—† for smoothness. smooth? smooth? If so, store. If so, store. computation cost Repeat. Repeat. 3. After all smooths are found, Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ❛❀ ❜ ❛❀ ❜ negligible. reorganize: for each β—† , bring Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  smooth? smooth? relevant ( ❛❀ ❜ ) close together. roblem: Coppersmith’s If so, store. If so, store. size β–² 1 ✿ 638587 . Repeat. Repeat. 4. Linear algebra. β†βš useless in reality.

  66. Batch NFS Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ❛❀ ❜ Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Goal: Optimize β†βš asymptotics. smooth? smooth? smooth? ❛❀ ❜ ) If so, store. If so, store. If so, store. If 1. Generate ( ❛❀ ❜ ) in parallel. ❛ οΏ½ βœβ™  oth. Repeat. Repeat. Repeat. β–² ✿ Test ❛ οΏ½ βœβ™  for smoothness. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ❛❀ ❜ Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? 2. Make many copies of each β—† , If so, store. If so, store. If so, store. If close to each ( ❛❀ ❜ ) generator. batching: Repeat. Repeat. Repeat. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ❛❀ ❜ When smooth ❛ οΏ½ βœβ™  is found, rgets, Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  β–² ✿ test each ❛ οΏ½ βœβ˜› β—† for smoothness. smooth? smooth? smooth? If so, store. If so, store. If so, store. If Repeat. Repeat. Repeat. 3. After all smooths are found, Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ❛❀ ❜ reorganize: for each β—† , bring Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? relevant ( ❛❀ ❜ ) close together. rsmith’s If so, store. If so, store. If so, store. If β–² ✿ 638587 . Repeat. Repeat. Repeat. 4. Linear algebra. β†βš reality.

  67. Batch NFS Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Goal: Optimize β†βš asymptotics. smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. 1. Generate ( ❛❀ ❜ ) in parallel. Repeat. Repeat. Repeat. Repeat. Test ❛ οΏ½ βœβ™  for smoothness. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? 2. Make many copies of each β—† , If so, store. If so, store. If so, store. If so, store. close to each ( ❛❀ ❜ ) generator. Repeat. Repeat. Repeat. Repeat. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). When smooth ❛ οΏ½ βœβ™  is found, Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  test each ❛ οΏ½ βœβ˜› β—† for smoothness. smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. 3. After all smooths are found, Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). reorganize: for each β—† , bring Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? relevant ( ❛❀ ❜ ) close together. If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. 4. Linear algebra.

  68. NFS Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 1 ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? Optimize β†βš asymptotics. smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Generate ( ❛❀ ❜ ) in parallel. Repeat. Repeat. Repeat. Repeat. right. Repeat. ❛ οΏ½ βœβ™  for smoothness. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 5 ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? smooth? If so, store. Make many copies of each β—† , If so, store. If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ to each ( ❛❀ ❜ ) generator. Repeat. Repeat. Repeat. Repeat. up. Repeat. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 9 ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› smooth ❛ οΏ½ βœβ™  is found, Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? each ❛ οΏ½ βœβ˜› β—† for smoothness. smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Repeat. Repeat. Repeat. Repeat. right. Repeat. After all smooths are found, Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 13 ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› rganize: for each β—† , bring Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? smooth? If so, store. relevant ( ❛❀ ❜ ) close together. If so, store. If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Repeat. Repeat. Repeat. Repeat. up. Repeat. Linear algebra.

  69. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 1 Is ❛ οΏ½ βœβ˜› 2 ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? β†βš asymptotics. smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ) in parallel. Repeat. Repeat. Repeat. Repeat. right. Repeat. right. Repeat. right. smoothness. ❛ οΏ½ βœβ™  Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 5 Is ❛ οΏ½ βœβ˜› 6 ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? smooth? smooth? If so, store. If so, store. copies of each β—† , If so, store. If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ) generator. Repeat. Repeat. Repeat. Repeat. up. Repeat. left. Repeat. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 9 Is ❛ οΏ½ βœβ˜› 10 ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ™  is found, Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? ❛ οΏ½ βœβ˜› β—† for smoothness. smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). ❛❀ ❜ ❛❀ ❜ Repeat. Repeat. Repeat. Repeat. right. Repeat. right. Repeat. right. oths are found, Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 13 Is ❛ οΏ½ βœβ˜› 14 ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› each β—† , bring Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? smooth? smooth? If so, store. If so, store. ❛❀ ❜ close together. If so, store. If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). ❛❀ ❜ ❛❀ ❜ Repeat. Repeat. Repeat. Repeat. up. Repeat. left. Repeat. ra.

  70. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 1 Is ❛ οΏ½ βœβ˜› 2 Is ❛ οΏ½ βœβ˜› 3 Is ❛ οΏ½ βœβ˜› Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? β†βš asymptotics. smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If If so, store. If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ❛❀ ❜ ❛❀ ❜ rallel. Repeat. Repeat. Repeat. Repeat. right. Repeat. right. Repeat. right. Repeat. down. othness. ❛ οΏ½ βœβ™  Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 5 Is ❛ οΏ½ βœβ˜› 6 Is ❛ οΏ½ βœβ˜› 7 Is ❛ οΏ½ βœβ˜› Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If ach β—† , If so, store. If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ❛❀ ❜ ❛❀ ❜ generator. Repeat. Repeat. Repeat. Repeat. up. Repeat. left. Repeat. left. Repeat. left. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 9 Is ❛ οΏ½ βœβ˜› 10 Is ❛ οΏ½ βœβ˜› 11 Is ❛ οΏ½ βœβ˜› found, ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? ❛ οΏ½ βœβ˜› β—† othness. smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If If so, store. If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ❛❀ ❜ Repeat. Repeat. Repeat. Repeat. right. Repeat. right. Repeat. right. Repeat. down. found, Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 13 Is ❛ οΏ½ βœβ˜› 14 Is ❛ οΏ½ βœβ˜› 15 Is ❛ οΏ½ βœβ˜› ring β—† Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If ❛❀ ❜ together. If so, store. If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ❛❀ ❜ Repeat. Repeat. Repeat. Repeat. up. Repeat. left. Repeat. left. Repeat. left.

  71. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 1 Is ❛ οΏ½ βœβ˜› 2 Is ❛ οΏ½ βœβ˜› 3 Is ❛ οΏ½ βœβ˜› 4 Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. If so, store. If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Repeat. Repeat. Repeat. Repeat. right. Repeat. right. Repeat. right. Repeat. down. Repeat. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 5 Is ❛ οΏ½ βœβ˜› 6 Is ❛ οΏ½ βœβ˜› 7 Is ❛ οΏ½ βœβ˜› 8 Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. If so, store. If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Repeat. Repeat. Repeat. Repeat. up. Repeat. left. Repeat. left. Repeat. left. Repeat. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 9 Is ❛ οΏ½ βœβ˜› 10 Is ❛ οΏ½ βœβ˜› 11 Is ❛ οΏ½ βœβ˜› 12 Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. If so, store. If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Repeat. Repeat. Repeat. Repeat. right. Repeat. right. Repeat. right. Repeat. down. Repeat. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 13 Is ❛ οΏ½ βœβ˜› 14 Is ❛ οΏ½ βœβ˜› 15 Is ❛ οΏ½ βœβ˜› 16 Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. If so, store. If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Repeat. Repeat. Repeat. Repeat. up. Repeat. left. Repeat. left. Repeat. left. Repeat.

  72. ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 1 Is ❛ οΏ½ βœβ˜› 2 Is ❛ οΏ½ βœβ˜› 3 Is ❛ οΏ½ βœβ˜› 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† . If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Repeat. Repeat. Repeat. right. Repeat. right. Repeat. right. Repeat. down. Repeat. β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 5 Is ❛ οΏ½ βœβ˜› 6 Is ❛ οΏ½ βœβ˜› 7 Is ❛ οΏ½ βœβ˜› 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? ❛ οΏ½ βœβ™  β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† . If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Repeat. Repeat. Repeat. up. Repeat. left. Repeat. left. Repeat. left. Repeat. β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 9 Is ❛ οΏ½ βœβ˜› 10 Is ❛ οΏ½ βœβ˜› 11 Is ❛ οΏ½ βœβ˜› 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? ❛ οΏ½ βœβ™  β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† . If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Repeat. Repeat. Repeat. right. Repeat. right. Repeat. right. Repeat. down. Repeat. β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 13 Is ❛ οΏ½ βœβ˜› 14 Is ❛ οΏ½ βœβ˜› 15 Is ❛ οΏ½ βœβ˜› 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❛ οΏ½ βœβ™  smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† . If so, store. If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Repeat. Repeat. Repeat. up. Repeat. left. Repeat. left. Repeat. left. Repeat. β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—†

  73. Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 1 Is ❛ οΏ½ βœβ˜› 2 Is ❛ οΏ½ βœβ˜› 3 Is ❛ οΏ½ βœβ˜› 4 ❛❀ ❜ ❛❀ ❜ β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? If so, store. If so, store. If so, store. If so, store. β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Repeat. Repeat. right. Repeat. right. Repeat. right. Repeat. down. Repeat. β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 5 Is ❛ οΏ½ βœβ˜› 6 Is ❛ οΏ½ βœβ˜› 7 Is ❛ οΏ½ βœβ˜› 8 ❛❀ ❜ ❛❀ ❜ β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? If so, store. If so, store. If so, store. If so, store. β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Repeat. Repeat. up. Repeat. left. Repeat. left. Repeat. left. Repeat. β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 9 Is ❛ οΏ½ βœβ˜› 10 Is ❛ οΏ½ βœβ˜› 11 Is ❛ οΏ½ βœβ˜› 12 ❛❀ ❜ ❛❀ ❜ β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? If so, store. If so, store. If so, store. If so, store. β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Repeat. Repeat. right. Repeat. right. Repeat. right. Repeat. down. Repeat. β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Generate ( ❛❀ ❜ ). Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 13 Is ❛ οΏ½ βœβ˜› 14 Is ❛ οΏ½ βœβ˜› 15 Is ❛ οΏ½ βœβ˜› 16 ❛❀ ❜ ❛❀ ❜ β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  smooth? smooth? If so, store. If so, store. If so, store. If so, store. β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† If so, store. If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Repeat. Repeat. up. Repeat. left. Repeat. left. Repeat. left. Repeat. β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—†

  74. Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 1 Is ❛ οΏ½ βœβ˜› 2 Is ❛ οΏ½ βœβ˜› 3 Is ❛ οΏ½ βœβ˜› 4 ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? If so, store. If so, store. If so, store. If so, store. β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Repeat. right. Repeat. right. Repeat. right. Repeat. down. Repeat. β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 5 Is ❛ οΏ½ βœβ˜› 6 Is ❛ οΏ½ βœβ˜› 7 Is ❛ οΏ½ βœβ˜› 8 ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? If so, store. If so, store. If so, store. If so, store. β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Repeat. up. Repeat. left. Repeat. left. Repeat. left. Repeat. β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 9 Is ❛ οΏ½ βœβ˜› 10 Is ❛ οΏ½ βœβ˜› 11 Is ❛ οΏ½ βœβ˜› 12 ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? If so, store. If so, store. If so, store. If so, store. β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Repeat. right. Repeat. right. Repeat. right. Repeat. down. Repeat. β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Generate ( ❛❀ ❜ ). Is ❛ οΏ½ βœβ˜› 13 Is ❛ οΏ½ βœβ˜› 14 Is ❛ οΏ½ βœβ˜› 15 Is ❛ οΏ½ βœβ˜› 16 ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ™  smooth? smooth? smooth? smooth? β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  ❛ οΏ½ βœβ™  smooth? If so, store. If so, store. If so, store. If so, store. β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† If so, store. Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Repeat. up. Repeat. left. Repeat. left. Repeat. left. Repeat. β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—†

  75. Is ❛ οΏ½ βœβ˜› 1 Is ❛ οΏ½ βœβ˜› 2 Is ❛ οΏ½ βœβ˜› 3 Is ❛ οΏ½ βœβ˜› 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? smooth? smooth? β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† If so, store. If so, store. If so, store. If so, store. β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† right. Repeat. right. Repeat. right. Repeat. down. Repeat. β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ˜› 5 Is ❛ οΏ½ βœβ˜› 6 Is ❛ οΏ½ βœβ˜› 7 Is ❛ οΏ½ βœβ˜› 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? smooth? smooth? β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† If so, store. If so, store. If so, store. If so, store. β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† up. Repeat. left. Repeat. left. Repeat. left. Repeat. β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ˜› 9 Is ❛ οΏ½ βœβ˜› 10 Is ❛ οΏ½ βœβ˜› 11 Is ❛ οΏ½ βœβ˜› 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? smooth? smooth? β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† If so, store. If so, store. If so, store. If so, store. β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† right. Repeat. right. Repeat. right. Repeat. down. Repeat. β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ˜› 13 Is ❛ οΏ½ βœβ˜› 14 Is ❛ οΏ½ βœβ˜› 15 Is ❛ οΏ½ βœβ˜› 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? smooth? smooth? β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† If so, store. If so, store. If so, store. If so, store. β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† up. Repeat. left. Repeat. left. Repeat. left. Repeat. β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—†

  76. Is ❛ οΏ½ βœβ˜› 2 Is ❛ οΏ½ βœβ˜› 3 Is ❛ οΏ½ βœβ˜› 4 ❛ οΏ½ βœβ˜› β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? smooth? β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† . If so, store. If so, store. If so, store. β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† eat. right. Repeat. right. Repeat. down. Repeat. β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ˜› 6 Is ❛ οΏ½ βœβ˜› 7 Is ❛ οΏ½ βœβ˜› 8 ❛ οΏ½ βœβ˜› β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? smooth? β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† . If so, store. If so, store. If so, store. β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† eat. left. Repeat. left. Repeat. left. Repeat. β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ˜› 10 Is ❛ οΏ½ βœβ˜› 11 Is ❛ οΏ½ βœβ˜› 12 ❛ οΏ½ βœβ˜› β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? smooth? β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† . If so, store. If so, store. If so, store. β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† eat. right. Repeat. right. Repeat. down. Repeat. β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ˜› 14 Is ❛ οΏ½ βœβ˜› 15 Is ❛ οΏ½ βœβ˜› 16 ❛ οΏ½ βœβ˜› β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? smooth? β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† . If so, store. If so, store. If so, store. β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† eat. left. Repeat. left. Repeat. left. Repeat. β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—†

  77. Is ❛ οΏ½ βœβ˜› 3 Is ❛ οΏ½ βœβ˜› 4 ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† If so, store. If so, store. β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). ❛❀ ❜ ❛❀ ❜ β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† ❀ β—† ❀ β—† ❀ β—† right. Repeat. down. Repeat. β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ˜› 7 Is ❛ οΏ½ βœβ˜› 8 ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† smooth? smooth? β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† ❀ β—† ❀ β—† If so, store. If so, store. β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† ❀ β—† ❀ β—† ❀ β—† Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ❛❀ ❜ left. Repeat. left. Repeat. β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ˜› 11 Is ❛ οΏ½ βœβ˜› 12 ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† If so, store. If so, store. β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). ❛❀ ❜ ❛❀ ❜ β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† right. Repeat. down. Repeat. β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ˜› 15 Is ❛ οΏ½ βœβ˜› 16 ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† smooth? smooth? β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† If so, store. If so, store. β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Send ( ❛❀ ❜ ). Send ( ❛❀ ❜ ). β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ❛❀ ❜ left. Repeat. left. Repeat. β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—†

  78. Is ❛ οΏ½ βœβ˜› 4 ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 smooth? β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 If so, store. β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† Send ( ❛❀ ❜ ). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† down. Repeat. β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ˜› 8 ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† smooth? β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† If so, store. β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† Send ( ❛❀ ❜ ). β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ left. Repeat. β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ˜› 12 ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† smooth? β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† If so, store. β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† Send ( ❛❀ ❜ ). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† down. Repeat. β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† Is ❛ οΏ½ βœβ˜› 16 ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› ❛ οΏ½ βœβ˜› β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† smooth? β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† If so, store. β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† Send ( ❛❀ ❜ ). β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ left. Repeat. β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—†

  79. β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—†

  80. Linear algeb β—† ❀ β—† ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† β—† β—† β—† using congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ β—† ❀ β—† β—† ❀ β—† ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† 15 ❀ β—† 16 ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† Linear algeb β—† β—† β—† β—† using congruences β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 11 ❀ β—† 12 ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† 15 ❀ β—† 16 ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Linear algeb β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 11 ❀ β—† 12 ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† β—† β—† β—† using congruences β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† 15 ❀ β—† 16 ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† β—† ❀ β—† ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Linear algebra β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† 15 ❀ β—† 16 ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† β—† β—† β—† ❀ β—† ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† using congruences β—† ❀ β—† ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† β—† ❀ β—† ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† β—† ❀ β—† ❀ β—† ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜

  81. Linear algebra for β—† 1 Linea β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† β—† β—† using congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† ❀ β—† ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 Linear algebra for β—† 5 Linea β—† β—† β—† using congruences β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† ❀ β—† ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Linear algebra for β—† 9 Linea β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† β—† using congruences β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† ❀ β—† ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Linear algebra for β—† 13 Linea β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† ❀ β—† ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 using congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜

  82. Linear algebra for β—† 1 Linear algebra for β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† using congruences using congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† Linear algebra for β—† 5 Linear algebra for β—† β—† β—† using congruences using congruences β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Linear algebra for β—† 9 Linear algebra for β—† β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† using congruences using congruences β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Linear algebra for β—† 13 Linear algebra for β—† β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 using congruences using congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† β—† ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜

  83. Linear algebra for β—† 1 Linear algebra for β—† 2 Linear algeb β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† β—† using congruences using congruences using congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 Linear algebra for β—† 5 Linear algebra for β—† 6 Linear algeb β—† β—† using congruences using congruences using congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Linear algebra for β—† 9 Linear algebra for β—† 10 Linear algeb β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† β—† using congruences using congruences using congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Linear algebra for β—† 13 Linear algebra for β—† 14 Linear algeb β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 using congruences using congruences using congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜

  84. Linear algebra for β—† 1 Linear algebra for β—† 2 Linear algeb algebra for β—† 3 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 β—† β—† β—† β—† β—† using congruences using congruences using congruences congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 Linear algebra for β—† 5 β—† Linear algebra for β—† 6 β—† Linear algeb algebra for β—† 7 β—† β—† β—† using congruences using congruences using congruences congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Linear algebra for β—† 9 Linear algebra for β—† 10 Linear algeb algebra for β—† 11 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 β—† β—† β—† β—† β—† using congruences using congruences using congruences congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Linear algebra for β—† 13 Linear algebra for β—† 14 Linear algeb algebra for β—† 15 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 β—† β—† β—† β—† β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 1 ❀ β—† 2 ❀ β—† 3 ❀ β—† 4 using congruences using congruences using congruences congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 5 ❀ β—† 6 ❀ β—† 7 ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 9 ❀ β—† 10 ❀ β—† 11 ❀ β—† 12 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† 13 ❀ β—† 14 ❀ β—† 15 ❀ β—† 16 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜

  85. Linear algebra for β—† 1 Linear algebra for β—† 2 Linear algeb algebra for β—† 3 Linear algebra fo β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 4 β—† β—† β—† β—† β—† using congruences using congruences using congruences congruences using congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 12 ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 16 ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 4 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 8 Linear algebra for β—† 5 β—† Linear algebra for β—† 6 β—† Linear algeb algebra for β—† 7 β—† Linear algebra fo β—† β—† using congruences using congruences using congruences congruences using congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 12 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 16 ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 4 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Linear algebra for β—† 9 Linear algebra for β—† 10 Linear algeb algebra for β—† 11 Linear algebra for β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 12 β—† β—† β—† β—† using congruences using congruences using congruences congruences using congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 4 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 12 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Linear algebra for β—† 13 Linear algebra for β—† 14 Linear algeb algebra for β—† 15 Linear algebra for β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 16 β—† β—† β—† β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 4 using congruences using congruences using congruences congruences using congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 8 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 12 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 16 ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜

  86. Linear algebra for β—† 1 Linear algebra for β—† 2 Linear algeb algebra for β—† 3 Linear algebra for β—† 4 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† β—† β—† using congruences using congruences using congruences congruences using congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† Linear algebra for β—† 5 β—† Linear algebra for β—† 6 β—† Linear algeb algebra for β—† 7 β—† Linear algebra for β—† 8 β—† using congruences using congruences using congruences congruences using congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Linear algebra for β—† 9 Linear algebra for β—† 10 Linear algeb algebra for β—† 11 Linear algebra for β—† 12 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† β—† β—† β—† using congruences using congruences using congruences congruences using congruences β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ( ❛❀ ❜ ) ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Linear algebra for β—† 13 Linear algebra for β—† 14 Linear algeb algebra for β—† 15 Linear algebra for β—† 16 β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† β—† ❀ β—† ❀ β—† ❀ β—† 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