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

Batch NFS

• D. J. Bernstein

University of Illinois at Chicago & Technische Universiteit Eindhoven Tanja Lange Technische Universiteit Eindhoven In this talk log ▲ means (1 + ♦(1))(log ◆)1❂3(log log ◆)2❂3. ▲ is often written “▲◆(1❂3)” or “▲◆(1❂3)1+♦(1)”. Exponents of ▲ in this talk are limited to 106Z. Rigorously proven? Ha ha ha. 2003 Shamir–Tromer, 2003 Lenstra–Tromer–Shamir– Kortsmit–Dodson–Hughes– Leyland, 2005 Geiselmann– Shamir–Steinwandt–Tromer, 2005 Franke–Kleinjung–Paar–Pelzl– Priplata–Stahlke, etc.: RSA-1024 is breakable in a year by an attack machine costing ❁ 109 dollars.

Batch NFS

• D. J. Bernstein

University of Illinois at Chicago & Technische Universiteit Eindhoven Tanja Lange Technische Universiteit Eindhoven In this talk log ▲ means (1 + ♦(1))(log ◆)1❂3(log log ◆)2❂3. ▲ is often written “▲◆(1❂3)” or “▲◆(1❂3)1+♦(1)”. Exponents of ▲ in this talk are limited to 106Z. Rigorously proven? Ha ha ha. 2003 Shamir–Tromer, 2003 Lenstra–Tromer–Shamir– Kortsmit–Dodson–Hughes– Leyland, 2005 Geiselmann– Shamir–Steinwandt–Tromer, 2005 Franke–Kleinjung–Paar–Pelzl– Priplata–Stahlke, etc.: RSA-1024 is breakable in a year by an attack machine costing ❁ 109 dollars. So the Internet switched to RSA-2048, and we no longer care about RSA-1024 security, right?

Batch NFS

• D. J. Bernstein

University of Illinois at Chicago & Technische Universiteit Eindhoven Tanja Lange Technische Universiteit Eindhoven In this talk log ▲ means (1 + ♦(1))(log ◆)1❂3(log log ◆)2❂3. ▲ is often written “▲◆(1❂3)” or “▲◆(1❂3)1+♦(1)”. Exponents of ▲ in this talk are limited to 106Z. Rigorously proven? Ha ha ha. 2003 Shamir–Tromer, 2003 Lenstra–Tromer–Shamir– Kortsmit–Dodson–Hughes– Leyland, 2005 Geiselmann– Shamir–Steinwandt–Tromer, 2005 Franke–Kleinjung–Paar–Pelzl– Priplata–Stahlke, etc.: RSA-1024 is breakable in a year by an attack machine costing ❁ 109 dollars. So the Internet switched to RSA-2048, and we no longer care about RSA-1024 security, right? Wrong!

NFS Bernstein University of Illinois at Chicago & echnische Universiteit Eindhoven Lange echnische Universiteit Eindhoven talk log ▲ means ♦(1))(log ◆)1❂3(log log ◆)2❂3. ▲

• ften written

▲◆ ❂3)” or “▲◆(1❂3)1+♦(1)”.

• nents of ▲ in this talk

limited to 106Z. rously proven? Ha ha ha. 2003 Shamir–Tromer, 2003 Lenstra–Tromer–Shamir– Kortsmit–Dodson–Hughes– Leyland, 2005 Geiselmann– Shamir–Steinwandt–Tromer, 2005 Franke–Kleinjung–Paar–Pelzl– Priplata–Stahlke, etc.: RSA-1024 is breakable in a year by an attack machine costing ❁ 109 dollars. So the Internet switched to RSA-2048, and we no longer care about RSA-1024 security, right? Wrong! Example: dnssec-deployment.org is signed

Illinois at Chicago & Universiteit Eindhoven Universiteit Eindhoven ▲ means ♦ ◆)1❂3(log log ◆)2❂3. ▲ written ▲◆ ❂ ▲◆(1❂3)1+♦(1)”. ▲ in this talk

6Z.

roven? Ha ha ha. 2003 Shamir–Tromer, 2003 Lenstra–Tromer–Shamir– Kortsmit–Dodson–Hughes– Leyland, 2005 Geiselmann– Shamir–Steinwandt–Tromer, 2005 Franke–Kleinjung–Paar–Pelzl– Priplata–Stahlke, etc.: RSA-1024 is breakable in a year by an attack machine costing ❁ 109 dollars. So the Internet switched to RSA-2048, and we no longer care about RSA-1024 security, right? Wrong! Example: The IP addr dnssec-deployment.org is signed by an RSA-1024

Chicago & Eindhoven Eindhoven ▲ ♦ ◆

❂

log ◆)2❂3. ▲ ▲◆ ❂ ▲◆ ❂

♦(1)”.

▲ talk

• ha.

2003 Shamir–Tromer, 2003 Lenstra–Tromer–Shamir– Kortsmit–Dodson–Hughes– Leyland, 2005 Geiselmann– Shamir–Steinwandt–Tromer, 2005 Franke–Kleinjung–Paar–Pelzl– Priplata–Stahlke, etc.: RSA-1024 is breakable in a year by an attack machine costing ❁ 109 dollars. So the Internet switched to RSA-2048, and we no longer care about RSA-1024 security, right? Wrong! Example: The IP address of dnssec-deployment.org is signed by an RSA-1024 key

2003 Shamir–Tromer, 2003 Lenstra–Tromer–Shamir– Kortsmit–Dodson–Hughes– Leyland, 2005 Geiselmann– Shamir–Steinwandt–Tromer, 2005 Franke–Kleinjung–Paar–Pelzl– Priplata–Stahlke, etc.: RSA-1024 is breakable in a year by an attack machine costing ❁ 109 dollars. So the Internet switched to RSA-2048, and we no longer care about RSA-1024 security, right? Wrong! Example: The IP address of dnssec-deployment.org is signed by an RSA-1024 key

2003 Shamir–Tromer, 2003 Lenstra–Tromer–Shamir– Kortsmit–Dodson–Hughes– Leyland, 2005 Geiselmann– Shamir–Steinwandt–Tromer, 2005 Franke–Kleinjung–Paar–Pelzl– Priplata–Stahlke, etc.: RSA-1024 is breakable in a year by an attack machine costing ❁ 109 dollars. So the Internet switched to RSA-2048, and we no longer care about RSA-1024 security, right? Wrong! Example: The IP address of dnssec-deployment.org is signed by an RSA-1024 key signed by an RSA-2048 key

2003 Shamir–Tromer, 2003 Lenstra–Tromer–Shamir– Kortsmit–Dodson–Hughes– Leyland, 2005 Geiselmann– Shamir–Steinwandt–Tromer, 2005 Franke–Kleinjung–Paar–Pelzl– Priplata–Stahlke, etc.: RSA-1024 is breakable in a year by an attack machine costing ❁ 109 dollars. So the Internet switched to RSA-2048, and we no longer care about RSA-1024 security, right? Wrong! Example: The IP address of dnssec-deployment.org is signed by an RSA-1024 key signed by an RSA-2048 key signed by org’s RSA-1024 key

2003 Shamir–Tromer, 2003 Lenstra–Tromer–Shamir– Kortsmit–Dodson–Hughes– Leyland, 2005 Geiselmann– Shamir–Steinwandt–Tromer, 2005 Franke–Kleinjung–Paar–Pelzl– Priplata–Stahlke, etc.: RSA-1024 is breakable in a year by an attack machine costing ❁ 109 dollars. So the Internet switched to RSA-2048, and we no longer care about RSA-1024 security, right? Wrong! Example: The IP address of dnssec-deployment.org is signed by an RSA-1024 key signed by an RSA-2048 key signed by org’s RSA-1024 key signed by an RSA-2048 key

2003 Shamir–Tromer, 2003 Lenstra–Tromer–Shamir– Kortsmit–Dodson–Hughes– Leyland, 2005 Geiselmann– Shamir–Steinwandt–Tromer, 2005 Franke–Kleinjung–Paar–Pelzl– Priplata–Stahlke, etc.: RSA-1024 is breakable in a year by an attack machine costing ❁ 109 dollars. So the Internet switched to RSA-2048, and we no longer care about RSA-1024 security, right? Wrong! Example: The IP address of dnssec-deployment.org is signed by an RSA-1024 key signed by an RSA-2048 key signed by org’s RSA-1024 key signed by an RSA-2048 key signed by a root RSA-1024 key

2003 Shamir–Tromer, 2003 Lenstra–Tromer–Shamir– Kortsmit–Dodson–Hughes– Leyland, 2005 Geiselmann– Shamir–Steinwandt–Tromer, 2005 Franke–Kleinjung–Paar–Pelzl– Priplata–Stahlke, etc.: RSA-1024 is breakable in a year by an attack machine costing ❁ 109 dollars. So the Internet switched to RSA-2048, and we no longer care about RSA-1024 security, right? Wrong! Example: The IP address of dnssec-deployment.org is signed by an RSA-1024 key signed by an RSA-2048 key signed by org’s RSA-1024 key signed by an RSA-2048 key signed by a root RSA-1024 key signed by an RSA-2048 key.

2003 Shamir–Tromer, 2003 Lenstra–Tromer–Shamir– Kortsmit–Dodson–Hughes– Leyland, 2005 Geiselmann– Shamir–Steinwandt–Tromer, 2005 Franke–Kleinjung–Paar–Pelzl– Priplata–Stahlke, etc.: RSA-1024 is breakable in a year by an attack machine costing ❁ 109 dollars. So the Internet switched to RSA-2048, and we no longer care about RSA-1024 security, right? Wrong! Example: The IP address of dnssec-deployment.org is signed by an RSA-1024 key signed by an RSA-2048 key signed by org’s RSA-1024 key signed by an RSA-2048 key signed by a root RSA-1024 key signed by an RSA-2048 key. Most “DNSSEC” signatures follow a similar pattern.

2003 Shamir–Tromer, 2003 Lenstra–Tromer–Shamir– Kortsmit–Dodson–Hughes– Leyland, 2005 Geiselmann– Shamir–Steinwandt–Tromer, 2005 Franke–Kleinjung–Paar–Pelzl– Priplata–Stahlke, etc.: RSA-1024 is breakable in a year by an attack machine costing ❁ 109 dollars. So the Internet switched to RSA-2048, and we no longer care about RSA-1024 security, right? Wrong! Example: The IP address of dnssec-deployment.org is signed by an RSA-1024 key signed by an RSA-2048 key signed by org’s RSA-1024 key signed by an RSA-2048 key signed by a root RSA-1024 key signed by an RSA-2048 key. Most “DNSSEC” signatures 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.

Shamir–Tromer, 2003 Lenstra–Tromer–Shamir– rtsmit–Dodson–Hughes– Leyland, 2005 Geiselmann– Shamir–Steinwandt–Tromer, 2005 e–Kleinjung–Paar–Pelzl– Priplata–Stahlke, etc.: RSA-1024 reakable in a year by an attack machine costing ❁ 109 dollars. Internet switched to RSA-2048, and we no longer care RSA-1024 security, right? rong! Example: The IP address of dnssec-deployment.org is signed by an RSA-1024 key signed by an RSA-2048 key signed by org’s RSA-1024 key signed by an RSA-2048 key signed by a root RSA-1024 key signed by an RSA-2048 key. Most “DNSSEC” signatures 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. Users seem

• 1. “The

more than

• 2. “The
• ff-the-shelf;

attackers

• 3. For signatures:

switch keys the attack

romer, 2003 romer–Shamir– dson–Hughes– Geiselmann– andt–Tromer, 2005 ung–Paar–Pelzl– e, etc.: RSA-1024 year by an attack ❁ 109 dollars. switched to we no longer care security, right? Example: The IP address of dnssec-deployment.org is signed by an RSA-1024 key signed by an RSA-2048 key signed by org’s RSA-1024 key signed by an RSA-2048 key signed by a root RSA-1024 key signed by an RSA-2048 key. Most “DNSSEC” signatures 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. Users seem unconcerned:

• 1. “The attack machine

more than this RSA

• 2. “The attack machine
• ff-the-shelf; it’s only

attackers building

• 3. For signatures:

switch keys every month, the attack machine

2003 dson–Hughes– Geiselmann– romer, 2005 elzl– RSA-1024 an attack ❁ dollars. to longer care right? Example: The IP address of dnssec-deployment.org is signed by an RSA-1024 key signed by an RSA-2048 key signed by org’s RSA-1024 key signed by an RSA-2048 key signed by a root RSA-1024 key signed by an RSA-2048 key. Most “DNSSEC” signatures 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. Users seem unconcerned:

• 1. “The attack machine costs

more than this RSA key is w

• 2. “The attack machine isn’t
• ff-the-shelf; it’s only for

attackers building ASICs.”

• 3. For signatures: “We

switch keys every month, and the attack machine takes a y

Example: The IP address of dnssec-deployment.org is signed by an RSA-1024 key signed by an RSA-2048 key signed by org’s RSA-1024 key signed by an RSA-2048 key signed by a root RSA-1024 key signed by an RSA-2048 key. Most “DNSSEC” signatures 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. Users seem unconcerned:

• 1. “The attack machine costs

more than this RSA key is worth.”

• 2. “The attack machine isn’t
• ff-the-shelf; it’s only for

attackers building ASICs.”

• 3. For signatures: “We

switch keys every month, and the attack machine takes a year.”

Example: The IP address of dnssec-deployment.org is signed by an RSA-1024 key signed by an RSA-2048 key signed by org’s RSA-1024 key signed by an RSA-2048 key signed by a root RSA-1024 key signed by an RSA-2048 key. Most “DNSSEC” signatures 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. Users seem unconcerned:

• 1. “The attack machine costs

more than this RSA key is worth.”

• 2. “The attack machine isn’t
• ff-the-shelf; it’s only for

attackers building ASICs.”

• 3. For signatures: “We

switch keys every month, and the attack machine takes a year.” Real quote: “DNSSEC signing keys should be large enough to avoid all known cryptographic attacks during the effectivity period of the key.”

Example: The IP address of dnssec-deployment.org signed by an RSA-1024 key by an RSA-2048 key by org’s RSA-1024 key by an RSA-2048 key by a root RSA-1024 key by an RSA-2048 key. “DNSSEC” signatures a similar pattern. Another example: SSL has used millions of RSA-1024 keys. Imagine that an attacker has rded tons of SSL traffic. Users seem unconcerned:

• 1. “The attack machine costs

more than this RSA key is worth.”

• 2. “The attack machine isn’t
• ff-the-shelf; it’s only for

attackers building ASICs.”

• 3. For signatures: “We

switch keys every month, and the attack machine takes a year.” Real quote: “DNSSEC signing keys should be large enough to avoid all known cryptographic attacks during the effectivity period of the key.” Continuation despite huge broken a fact, the estimated

• f a 700-bit

breaking would need amounts power in be detected single key estimated safely use least the

address of dnssec-deployment.org RSA-1024 key RSA-2048 key RSA-1024 key RSA-2048 key RSA-1024 key RSA-2048 key. “DNSSEC” signatures pattern. example: SSL has used RSA-1024 keys. attacker has SSL traffic. Users seem unconcerned:

• 1. “The attack machine costs

more than this RSA key is worth.”

• 2. “The attack machine isn’t
• ff-the-shelf; it’s only for

attackers building ASICs.”

• 3. For signatures: “We

switch keys every month, and the attack machine takes a year.” Real quote: “DNSSEC signing keys should be large enough to avoid all known cryptographic attacks during the effectivity period of the key.” Continuation of quote: despite huge efforts, broken a regular 1024-bit fact, the best completed estimated to be the

• f a 700-bit key. An

breaking a 1024-bit would need to exp amounts of network power in a way that be detected in order single key. Because estimated that most safely use 1024-bit least the next ten

• f

key ey key ey RSA-1024 key ey. signatures as used RSA-1024 keys. has traffic. Users seem unconcerned:

• 1. “The attack machine costs

more than this RSA key is worth.”

• 2. “The attack machine isn’t
• ff-the-shelf; it’s only for

attackers building ASICs.”

• 3. For signatures: “We

switch keys every month, and the attack machine takes a year.” Real quote: “DNSSEC signing keys should be large enough to avoid all known cryptographic attacks during the effectivity period of the key.” Continuation of quote: “To despite huge efforts, no one broken a regular 1024-bit key; fact, the best completed attack estimated to be the equivalent

• f a 700-bit key. An attacker

breaking a 1024-bit signing k would need to expend phenomenal amounts of networked computing power in a way that would not be detected in order to break single key. Because of this, it estimated that most zones c safely use 1024-bit keys for at least the next ten years.”

Users seem unconcerned:

• 1. “The attack machine costs

more than this RSA key is worth.”

• 2. “The attack machine isn’t
• ff-the-shelf; it’s only for

attackers building ASICs.”

• 3. For signatures: “We

switch keys every month, and the attack machine takes a year.” Real quote: “DNSSEC signing keys should be large enough to avoid all known cryptographic attacks during the effectivity period of the key.” Continuation of quote: “To date, despite huge efforts, no one has broken a regular 1024-bit key; in fact, the best completed attack is estimated to be the equivalent

• f a 700-bit key. An attacker

breaking a 1024-bit signing key would need to expend phenomenal amounts of networked computing power in a way that would not be detected in order to break a single key. Because of this, it is estimated that most zones can safely use 1024-bit keys for at least the next ten years.”

seem unconcerned: “The attack machine costs than this RSA key is worth.” “The attack machine isn’t

• ff-the-shelf; it’s only for

ers building ASICs.” signatures: “We keys every month, and attack machine takes a year.” quote: “DNSSEC signing should be large enough to all known cryptographic attacks during the effectivity

• f the key.”

Continuation of quote: “To date, despite huge efforts, no one has broken a regular 1024-bit key; in fact, the best completed attack is estimated to be the equivalent

• f a 700-bit key. An attacker

breaking a 1024-bit signing key would need to expend phenomenal amounts of networked computing power in a way that would not be detected in order to break a single key. Because of this, it is estimated that most zones can safely use 1024-bit keys for at least the next ten years.” Goal of ou analyze the specifically ratio, of “Many”: “Price-perfo area-time “RAM” bit integers accessing realistic; ❆❚ “Asymptotic”: suppress speedups

unconcerned: machine costs RSA key is worth.” machine isn’t

• nly for

ing ASICs.” signatures: “We every month, and machine takes a year.” “DNSSEC signing large enough to cryptographic the effectivity ey.” Continuation of quote: “To date, despite huge efforts, no one has broken a regular 1024-bit key; in fact, the best completed attack is estimated to be the equivalent

• f a 700-bit key. An attacker

breaking a 1024-bit signing key would need to expend phenomenal amounts of networked computing power in a way that would not be detected in order to break a single key. Because of this, it is estimated that most zones can safely use 1024-bit keys for at least the next ten years.” Goal of our paper: analyze the asymptotic specifically price-perfo ratio, of breaking many “Many”: e.g. millions. “Price-performance area-time product “RAM” metric (adding bit integers has same accessing array of realistic; “❆❚” metric “Asymptotic”: We suppress polynomial speedups are superp

costs worth.” isn’t and a year.” signing

• ugh to

cryptographic effectivity Continuation of quote: “To date, despite huge efforts, no one has broken a regular 1024-bit key; in fact, the best completed attack is estimated to be the equivalent

• f a 700-bit key. An attacker

breaking a 1024-bit signing key would need to expend phenomenal amounts of networked computing power in a way that would not be detected in order to break a single key. Because of this, it is estimated that most zones can safely use 1024-bit keys for at least the next ten years.” Goal of our paper: analyze the asymptotic cost, specifically price-performance ratio, of breaking many RSA “Many”: e.g. millions. “Price-performance ratio”: area-time product for chips. “RAM” metric (adding two bit integers has same cost as accessing array of size 264) is realistic; “❆❚” metric is realistic. “Asymptotic”: We systematically suppress polynomial factors. speedups are superpolynomial.

Continuation of quote: “To date, despite huge efforts, no one has broken a regular 1024-bit key; in fact, the best completed attack is estimated to be the equivalent

• f a 700-bit key. An attacker

breaking a 1024-bit signing key would need to expend phenomenal amounts of networked computing power in a way that would not be detected in order to break a single key. Because of this, it is estimated that most zones can safely use 1024-bit keys for at least the next ten years.” Goal of our paper: analyze the asymptotic cost, specifically price-performance ratio, of breaking many RSA keys. “Many”: e.g. millions. “Price-performance ratio”: area-time product for chips. “RAM” metric (adding two 64- bit integers has same cost as accessing array of size 264) is not realistic; “❆❚” metric is realistic. “Asymptotic”: We systematically suppress polynomial factors. Our speedups are superpolynomial.

Continuation of quote: “To date, despite huge efforts, no one has a regular 1024-bit key; in the best completed attack is estimated to be the equivalent 700-bit key. An attacker reaking a 1024-bit signing key need to expend phenomenal amounts of networked computing in a way that would not detected in order to break a

• key. Because of this, it is

estimated that most zones can use 1024-bit keys for at the next ten years.” Goal of our paper: analyze the asymptotic cost, specifically price-performance ratio, of breaking many RSA keys. “Many”: e.g. millions. “Price-performance ratio”: area-time product for chips. “RAM” metric (adding two 64- bit integers has same cost as accessing array of size 264) is not realistic; “❆❚” metric is realistic. “Asymptotic”: We systematically suppress polynomial factors. Our speedups are superpolynomial. Best result time ▲1✿185632 using chip ▲ ✿ ❆❚ is ▲1✿ Our main a batch of ▲ ✿ time ▲1✿022400 using chip ▲ ✿ ❆❚ per k ▲ ✿ This pap at ▲♦(1), speedup Results a guess from

quote: “To date, efforts, no one has 1024-bit key; in completed attack is the equivalent An attacker 1024-bit signing key expend phenomenal

• rked computing

that would not rder to break a Because of this, it is most zones can 1024-bit keys for at ten years.” Goal of our paper: analyze the asymptotic cost, specifically price-performance ratio, of breaking many RSA keys. “Many”: e.g. millions. “Price-performance ratio”: area-time product for chips. “RAM” metric (adding two 64- bit integers has same cost as accessing array of size 264) is not realistic; “❆❚” metric is realistic. “Asymptotic”: We systematically suppress polynomial factors. Our speedups are superpolynomial. Best result known time ▲1✿185632 using chip area ▲0✿ ❆❚ is ▲1✿976052. Our main result fo a batch of ▲0✿5 keys: time ▲1✿022400 using chip area ▲1✿ ❆❚ per key is ▲1✿704000 This paper also looks at ▲♦(1), analyzing speedup from early-ab Results are not what guess from 1982 P

• date,
• ne has

key; in attack is equivalent attacker signing key phenomenal computing not reak a this, it is can r at Goal of our paper: analyze the asymptotic cost, specifically price-performance ratio, of breaking many RSA keys. “Many”: e.g. millions. “Price-performance ratio”: area-time product for chips. “RAM” metric (adding two 64- bit integers has same cost as accessing array of size 264) is not realistic; “❆❚” metric is realistic. “Asymptotic”: We systematically suppress polynomial factors. Our speedups are superpolynomial. Best result known for one key time ▲1✿185632 using chip area ▲0✿790420; ❆❚ is ▲1✿976052. Our main result for a batch of ▲0✿5 keys: time ▲1✿022400 using chip area ▲1✿181600; ❆❚ per key is ▲1✿704000. This paper also looks more closely at ▲♦(1), analyzing asymptotic speedup from early-abort ECM. Results are not what one would guess from 1982 Pomerance.

Goal of our paper: analyze the asymptotic cost, specifically price-performance ratio, of breaking many RSA keys. “Many”: e.g. millions. “Price-performance ratio”: area-time product for chips. “RAM” metric (adding two 64- bit integers has same cost as accessing array of size 264) is not realistic; “❆❚” metric is realistic. “Asymptotic”: We systematically suppress polynomial factors. Our speedups are superpolynomial. Best result known for one key time ▲1✿185632 using chip area ▲0✿790420; ❆❚ is ▲1✿976052. Our main result for a batch of ▲0✿5 keys: time ▲1✿022400 using chip area ▲1✿181600; ❆❚ per key is ▲1✿704000. This paper also looks more closely at ▲♦(1), analyzing asymptotic speedup from early-abort ECM. Results are not what one would guess from 1982 Pomerance.

• f our paper:

analyze the asymptotic cost, ecifically price-performance

• f breaking many RSA keys.

“Many”: e.g. millions. “Price-performance ratio”: ime product for chips. “RAM” metric (adding two 64- integers has same cost as accessing array of size 264) is not realistic; “❆❚” metric is realistic. “Asymptotic”: We systematically ress polynomial factors. Our eedups are superpolynomial. Best result known for one key time ▲1✿185632 using chip area ▲0✿790420; ❆❚ is ▲1✿976052. Our main result for a batch of ▲0✿5 keys: time ▲1✿022400 using chip area ▲1✿181600; ❆❚ per key is ▲1✿704000. This paper also looks more closely at ▲♦(1), analyzing asymptotic speedup from early-abort ECM. Results are not what one would guess from 1982 Pomerance. Asymptotic

• 1. Attack

is reduced, can targe

• 2. Prima

memory for off-the-shelf

• 3. Attack

(and can breaking

er: asymptotic cost, rice-performance reaking many RSA keys. millions. rmance ratio”: duct for chips. (adding two 64- same cost as

• f size 264) is not

❆❚ metric is realistic. e systematically

• lynomial factors. Our

erpolynomial. Best result known for one key time ▲1✿185632 using chip area ▲0✿790420; ❆❚ is ▲1✿976052. Our main result for a batch of ▲0✿5 keys: time ▲1✿022400 using chip area ▲1✿181600; ❆❚ per key is ▲1✿704000. This paper also looks more closely at ▲♦(1), analyzing asymptotic speedup from early-abort ECM. Results are not what one would guess from 1982 Pomerance. Asymptotic consequ

• 1. Attack cost per

is reduced, so attack can target lower-value

• 2. Primary bottleneck

memory factorization—w for off-the-shelf graphics

• 3. Attack time is reduced

(and can be reduced breaking key rotation.

cost, rmance RSA keys. ratio”: chips.

• 64-

as ) is not ❆❚ realistic. systematically

• rs. Our
• lynomial.

Best result known for one key time ▲1✿185632 using chip area ▲0✿790420; ❆❚ is ▲1✿976052. Our main result for a batch of ▲0✿5 keys: time ▲1✿022400 using chip area ▲1✿181600; ❆❚ per key is ▲1✿704000. This paper also looks more closely at ▲♦(1), analyzing asymptotic speedup from early-abort ECM. Results are not what one would guess from 1982 Pomerance. Asymptotic consequences:

• 1. Attack cost per key

is reduced, so attacker can target lower-value keys.

• 2. Primary bottleneck is low-

memory factorization—well suited for off-the-shelf graphics cards.

• 3. Attack time is reduced

(and can be reduced more), breaking key rotation.

Best result known for one key time ▲1✿185632 using chip area ▲0✿790420; ❆❚ is ▲1✿976052. Our main result for a batch of ▲0✿5 keys: time ▲1✿022400 using chip area ▲1✿181600; ❆❚ per key is ▲1✿704000. This paper also looks more closely at ▲♦(1), analyzing asymptotic speedup from early-abort ECM. Results are not what one would guess from 1982 Pomerance. Asymptotic consequences:

• 1. Attack cost per key

is reduced, so attacker can target lower-value keys.

• 2. Primary bottleneck is low-

memory factorization—well suited for off-the-shelf graphics cards.

• 3. Attack time is reduced

(and can be reduced more), breaking key rotation.

Best result known for one key time ▲1✿185632 using chip area ▲0✿790420; ❆❚ is ▲1✿976052. Our main result for a batch of ▲0✿5 keys: time ▲1✿022400 using chip area ▲1✿181600; ❆❚ per key is ▲1✿704000. This paper also looks more closely at ▲♦(1), analyzing asymptotic speedup from early-abort ECM. Results are not what one would guess from 1982 Pomerance. Asymptotic consequences:

• 1. Attack cost per key

is reduced, so attacker can target lower-value keys.

• 2. Primary bottleneck is low-

memory factorization—well suited for off-the-shelf graphics cards.

• 3. Attack time is reduced

(and can be reduced more), breaking key rotation. “Do the asymptotics really kick in before 1024 bits?” — Maybe not, but no basis for confidence.

result known for one key ▲1✿185632 chip area ▲0✿790420; ❆❚ ▲1✿976052. main result for batch of ▲0✿5 keys: ▲1✿022400 chip area ▲1✿181600; ❆❚ er key is ▲1✿704000. paper also looks more closely ▲♦(1), analyzing asymptotic eedup from early-abort ECM. Results are not what one would from 1982 Pomerance. Asymptotic consequences:

• 1. Attack cost per key

is reduced, so attacker can target lower-value keys.

• 2. Primary bottleneck is low-

memory factorization—well suited for off-the-shelf graphics cards.

• 3. Attack time is reduced

(and can be reduced more), breaking key rotation. “Do the asymptotics really kick in before 1024 bits?” — Maybe not, but no basis for confidence. Eratosthenes Sieving small ✐ ❃ using prim ❀ ❀ ❀

1 2 2 3 3 4 2 2 5 6 2 3 7 8 2 2 2 9 3 3 10 2 11 12 2 2 3 13 14 2 15 3 16 2 2 2 2 17 18 2 3 3 19 20 2 2

etc.

wn for one key ▲ ✿ ▲0✿790420; ❆❚ ▲ ✿ for ▲ ✿ keys: ▲ ✿ ▲1✿181600; ❆❚ ▲ ✿704000. looks more closely ▲♦ analyzing asymptotic rly-abort ECM. what one would Pomerance. Asymptotic consequences:

• 1. Attack cost per key

is reduced, so attacker can target lower-value keys.

• 2. Primary bottleneck is low-

memory factorization—well suited for off-the-shelf graphics cards.

• 3. Attack time is reduced

(and can be reduced more), breaking key rotation. “Do the asymptotics really kick in before 1024 bits?” — Maybe not, but no basis for confidence. Eratosthenes for smo Sieving small integers ✐ ❃ using primes 2❀ 3❀ 5❀

1 2 2 3 3 4 2 2 5 5 6 2 3 7 7 8 2 2 2 9 3 3 10 2 5 11 12 2 2 3 13 14 2 7 15 3 5 16 2 2 2 2 17 18 2 3 3 19 20 2 2 5

etc.

key ▲ ✿ ▲ ✿ ❆❚ ▲ ✿ ▲ ✿ ▲ ✿ ▲ ✿ ❆❚ ▲ ✿ re closely ▲♦ ptotic ECM. would

• merance.

Asymptotic consequences:

• 1. Attack cost per key

is reduced, so attacker can target lower-value keys.

• 2. Primary bottleneck is low-

memory factorization—well suited for off-the-shelf graphics cards.

• 3. Attack time is reduced

(and can be reduced more), breaking key rotation. “Do the asymptotics really kick in before 1024 bits?” — Maybe not, but no basis for confidence. Eratosthenes for smoothness Sieving small integers ✐ ❃ 0 using primes 2❀ 3❀ 5❀ 7:

1 2 2 3 3 4 2 2 5 5 6 2 3 7 7 8 2 2 2 9 3 3 10 2 5 11 12 2 2 3 13 14 2 7 15 3 5 16 2 2 2 2 17 18 2 3 3 19 20 2 2 5

etc.

Asymptotic consequences:

• 1. Attack cost per key

is reduced, so attacker can target lower-value keys.

• 2. Primary bottleneck is low-

memory factorization—well suited for off-the-shelf graphics cards.

• 3. Attack time is reduced

(and can be reduced more), breaking key rotation. “Do the asymptotics really kick in before 1024 bits?” — Maybe not, but no basis for confidence. Eratosthenes for smoothness Sieving small integers ✐ ❃ 0 using primes 2❀ 3❀ 5❀ 7:

1 2 2 3 3 4 2 2 5 5 6 2 3 7 7 8 2 2 2 9 3 3 10 2 5 11 12 2 2 3 13 14 2 7 15 3 5 16 2 2 2 2 17 18 2 3 3 19 20 2 2 5

etc.

Asymptotic consequences: ttack cost per key reduced, so attacker rget lower-value keys. Primary bottleneck is low- ry factorization—well suited

• the-shelf graphics cards.

ttack time is reduced can be reduced more), reaking key rotation. the asymptotics really kick in 1024 bits?” — Maybe not, basis for confidence. Eratosthenes for smoothness Sieving small integers ✐ ❃ 0 using primes 2❀ 3❀ 5❀ 7:

1 2 2 3 3 4 2 2 5 5 6 2 3 7 7 8 2 2 2 9 3 3 10 2 5 11 12 2 2 3 13 14 2 7 15 3 5 16 2 2 2 2 17 18 2 3 3 19 20 2 2 5

etc. The Q sieve Sieving ✐ ✐ ✐ using prime ❀ ❀ ❀

1 2 2 3 3 4 2 2 5 6 2 3 7 8 2 2 2 9 3 3 10 2 11 12 2 2 3 13 14 2 15 3 16 2 2 2 2 17 18 2 3 3 19 20 2 2

etc.

consequences: er key attacker er-value keys.

• ttleneck is low-

rization—well suited graphics cards. is reduced reduced more), rotation. asymptotics really kick in bits?” — Maybe not, confidence. Eratosthenes for smoothness Sieving small integers ✐ ❃ 0 using primes 2❀ 3❀ 5❀ 7:

1 2 2 3 3 4 2 2 5 5 6 2 3 7 7 8 2 2 2 9 3 3 10 2 5 11 12 2 2 3 13 14 2 7 15 3 5 16 2 2 2 2 17 18 2 3 3 19 20 2 2 5

etc. The Q sieve Sieving ✐ and 611 + ✐ ✐ using primes 2❀ 3❀ 5❀

1 2 2 3 3 4 2 2 5 5 6 2 3 7 7 8 2 2 2 9 3 3 10 2 5 11 12 2 2 3 13 14 2 7 15 3 5 16 2 2 2 2 17 18 2 3 3 19 20 2 2 5 612 2 613 614 2 615 616 2 617 618 2 619 620 2 621 622 2 623 624 2 625 626 2 627 628 2 629 630 2 631

etc.

eys. low- ell suited cards. re), kick in ybe not, confidence. Eratosthenes for smoothness Sieving small integers ✐ ❃ 0 using primes 2❀ 3❀ 5❀ 7:

1 2 2 3 3 4 2 2 5 5 6 2 3 7 7 8 2 2 2 9 3 3 10 2 5 11 12 2 2 3 13 14 2 7 15 3 5 16 2 2 2 2 17 18 2 3 3 19 20 2 2 5

etc. The Q sieve Sieving ✐ and 611 + ✐ for sm ✐ using primes 2❀ 3❀ 5❀ 7:

1 2 2 3 3 4 2 2 5 5 6 2 3 7 7 8 2 2 2 9 3 3 10 2 5 11 12 2 2 3 13 14 2 7 15 3 5 16 2 2 2 2 17 18 2 3 3 19 20 2 2 5 612 2 2 3 3 613 614 2 615 3 5 616 2 2 2 617 618 2 3 619 620 2 2 5 621 3 3 3 622 2 623 624 2 2 2 2 3 625 5 626 2 627 3 628 2 2 629 630 2 3 3 5 631

etc.

Eratosthenes for smoothness Sieving small integers ✐ ❃ 0 using primes 2❀ 3❀ 5❀ 7:

1 2 2 3 3 4 2 2 5 5 6 2 3 7 7 8 2 2 2 9 3 3 10 2 5 11 12 2 2 3 13 14 2 7 15 3 5 16 2 2 2 2 17 18 2 3 3 19 20 2 2 5

etc. The Q sieve Sieving ✐ and 611 + ✐ for small ✐ using primes 2❀ 3❀ 5❀ 7:

1 2 2 3 3 4 2 2 5 5 6 2 3 7 7 8 2 2 2 9 3 3 10 2 5 11 12 2 2 3 13 14 2 7 15 3 5 16 2 2 2 2 17 18 2 3 3 19 20 2 2 5 612 2 2 3 3 613 614 2 615 3 5 616 2 2 2 7 617 618 2 3 619 620 2 2 5 621 3 3 3 622 2 623 7 624 2 2 2 2 3 625 5 5 5 5 626 2 627 3 628 2 2 629 630 2 3 3 5 7 631

etc.

Eratosthenes for smoothness Sieving small integers ✐ ❃ 0 primes 2❀ 3❀ 5❀ 7:

3 5 3 7 3 3 5 3 7 3 5 3 3 5

The Q sieve Sieving ✐ and 611 + ✐ for small ✐ using primes 2❀ 3❀ 5❀ 7:

1 2 2 3 3 4 2 2 5 5 6 2 3 7 7 8 2 2 2 9 3 3 10 2 5 11 12 2 2 3 13 14 2 7 15 3 5 16 2 2 2 2 17 18 2 3 3 19 20 2 2 5 612 2 2 3 3 613 614 2 615 3 5 616 2 2 2 7 617 618 2 3 619 620 2 2 5 621 3 3 3 622 2 623 7 624 2 2 2 2 3 625 5 5 5 5 626 2 627 3 628 2 2 629 630 2 3 3 5 7 631

etc. Have complet the congruences ✐ ✑ ✐ for some ✐ 14 ✁ 625 64 ✁ 675 75 ✁ 686 14 ✁ 64 ✁ 75 ✁ ✁ ✁ = 283458 gcd ✟ 611❀ ✁ ✁

• ✠

= 47. 611 = 47 ✁

smoothness integers ✐ ❃ 0 ❀ ❀ 5❀ 7: The Q sieve Sieving ✐ and 611 + ✐ for small ✐ using primes 2❀ 3❀ 5❀ 7:

1 2 2 3 3 4 2 2 5 5 6 2 3 7 7 8 2 2 2 9 3 3 10 2 5 11 12 2 2 3 13 14 2 7 15 3 5 16 2 2 2 2 17 18 2 3 3 19 20 2 2 5 612 2 2 3 3 613 614 2 615 3 5 616 2 2 2 7 617 618 2 3 619 620 2 2 5 621 3 3 3 622 2 623 7 624 2 2 2 2 3 625 5 5 5 5 626 2 627 3 628 2 2 629 630 2 3 3 5 7 631

etc. Have complete facto the congruences ✐ ✑ ✐ for some ✐’s. 14 ✁ 625 = 2130547 64 ✁ 675 = 2633527 75 ✁ 686 = 2131527 14 ✁ 64 ✁ 75 ✁ 625 ✁ 675 ✁ = 28345874 = (243 gcd ✟ 611❀ 14 ✁ 64 ✁ 75 ✠ = 47. 611 = 47 ✁ 13.

• thness

✐ ❃ 0 ❀ ❀ ❀ The Q sieve Sieving ✐ and 611 + ✐ for small ✐ using primes 2❀ 3❀ 5❀ 7:

1 2 2 3 3 4 2 2 5 5 6 2 3 7 7 8 2 2 2 9 3 3 10 2 5 11 12 2 2 3 13 14 2 7 15 3 5 16 2 2 2 2 17 18 2 3 3 19 20 2 2 5 612 2 2 3 3 613 614 2 615 3 5 616 2 2 2 7 617 618 2 3 619 620 2 2 5 621 3 3 3 622 2 623 7 624 2 2 2 2 3 625 5 5 5 5 626 2 627 3 628 2 2 629 630 2 3 3 5 7 631

etc. Have complete factorization the congruences ✐ ✑ 611 + ✐ for some ✐’s. 14 ✁ 625 = 21305471. 64 ✁ 675 = 26335270. 75 ✁ 686 = 21315273. 14 ✁ 64 ✁ 75 ✁ 625 ✁ 675 ✁ 686 = 28345874 = (24325472)2. gcd ✟ 611❀ 14 ✁ 64 ✁ 75 24325 ✠ = 47. 611 = 47 ✁ 13.

The Q sieve Sieving ✐ and 611 + ✐ for small ✐ using primes 2❀ 3❀ 5❀ 7:

1 2 2 3 3 4 2 2 5 5 6 2 3 7 7 8 2 2 2 9 3 3 10 2 5 11 12 2 2 3 13 14 2 7 15 3 5 16 2 2 2 2 17 18 2 3 3 19 20 2 2 5 612 2 2 3 3 613 614 2 615 3 5 616 2 2 2 7 617 618 2 3 619 620 2 2 5 621 3 3 3 622 2 623 7 624 2 2 2 2 3 625 5 5 5 5 626 2 627 3 628 2 2 629 630 2 3 3 5 7 631

etc. Have complete factorization of the congruences ✐ ✑ 611 + ✐ for some ✐’s. 14 ✁ 625 = 21305471. 64 ✁ 675 = 26335270. 75 ✁ 686 = 21315273. 14 ✁ 64 ✁ 75 ✁ 625 ✁ 675 ✁ 686 = 28345874 = (24325472)2. gcd ✟ 611❀ 14 ✁ 64 ✁ 75 24325472✠ = 47. 611 = 47 ✁ 13.

sieve Sieving ✐ and 611 + ✐ for small ✐ primes 2❀ 3❀ 5❀ 7:

3 5 3 7 3 3 5 3 7 3 5 3 3 5 612 2 2 3 3 613 614 2 615 3 5 616 2 2 2 7 617 618 2 3 619 620 2 2 5 621 3 3 3 622 2 623 7 624 2 2 2 2 3 625 5 5 5 5 626 2 627 3 628 2 2 629 630 2 3 3 5 7 631

Have complete factorization of the congruences ✐ ✑ 611 + ✐ for some ✐’s. 14 ✁ 625 = 21305471. 64 ✁ 675 = 26335270. 75 ✁ 686 = 21315273. 14 ✁ 64 ✁ 75 ✁ 625 ✁ 675 ✁ 686 = 28345874 = (24325472)2. gcd ✟ 611❀ 14 ✁ 64 ✁ 75 24325472✠ = 47. 611 = 47 ✁ 13. The numb Generalize ✐ ✑ ✐ ◆ ◆ ✦ ❛ ✑ ❛ ❜◆ ◆ ✦ ❛❜♠ ✑ ❛❜☛ ♠☛ for root ☛ ✷

• f nonzero

For any ♠ ☛ so that facto ♠ ☛ produces ◆ Optimal ♠ (✖+♦(1))(log ◆

❂

◆

❂

✐ 611 + ✐ for small ✐ ❀ ❀ 5❀ 7:

2 2 3 3 2 3 5 2 2 2 7 2 3 2 2 5 3 3 3 2 7 2 2 2 2 3 5 5 5 5 2 3 2 2 2 3 3 5 7

Have complete factorization of the congruences ✐ ✑ 611 + ✐ for some ✐’s. 14 ✁ 625 = 21305471. 64 ✁ 675 = 26335270. 75 ✁ 686 = 21315273. 14 ✁ 64 ✁ 75 ✁ 625 ✁ 675 ✁ 686 = 28345874 = (24325472)2. gcd ✟ 611❀ 14 ✁ 64 ✁ 75 24325472✠ = 47. 611 = 47 ✁ 13. The number-field sieve Generalize ✐ ✑ ✐ + ◆ ◆ ✦ ❛ ✑ ❛ + ❜◆ (mo ◆ ✦ ❛❜♠ ✑ ❛❜☛ ♠☛ for root ☛ ✷ C

• f nonzero integer

For any ♠ can find ☛ so that factoring ♠ ☛ produces factorization ◆ Optimal choice of ♠ (✖+♦(1))(log ◆)2❂ ◆

❂

✐ ✐ small ✐ ❀ ❀ ❀

5 7 5 7 5 5 5 5 5 7

Have complete factorization of the congruences ✐ ✑ 611 + ✐ for some ✐’s. 14 ✁ 625 = 21305471. 64 ✁ 675 = 26335270. 75 ✁ 686 = 21315273. 14 ✁ 64 ✁ 75 ✁ 625 ✁ 675 ✁ 686 = 28345874 = (24325472)2. gcd ✟ 611❀ 14 ✁ 64 ✁ 75 24325472✠ = 47. 611 = 47 ✁ 13. The number-field sieve Generalize ✐ ✑ ✐ + ◆ (mod ◆ ✦ ❛ ✑ ❛ + ❜◆ (mod ◆) ✦ ❛❜♠ ✑ ❛❜☛ (mod ♠☛ for root ☛ ✷ C

• f nonzero integer poly.

For any ♠ can find ☛ so that factoring ♠ ☛ produces factorization of ◆. Optimal choice of log ♠ is (✖+♦(1))(log ◆)2❂3(log log ◆

❂

Have complete factorization of the congruences ✐ ✑ 611 + ✐ for some ✐’s. 14 ✁ 625 = 21305471. 64 ✁ 675 = 26335270. 75 ✁ 686 = 21315273. 14 ✁ 64 ✁ 75 ✁ 625 ✁ 675 ✁ 686 = 28345874 = (24325472)2. gcd ✟ 611❀ 14 ✁ 64 ✁ 75 24325472✠ = 47. 611 = 47 ✁ 13. The number-field sieve Generalize ✐ ✑ ✐ + ◆ (mod ◆) ✦ ❛ ✑ ❛ + ❜◆ (mod ◆) ✦ ❛❜♠ ✑ ❛❜☛ (mod ♠☛) for root ☛ ✷ C

• f nonzero integer poly.

For any ♠ can find ☛ so that factoring ♠ ☛ produces factorization of ◆. Optimal choice of log ♠ is (✖+♦(1))(log ◆)2❂3(log log ◆)1❂3.

complete factorization of congruences ✐ ✑ 611 + ✐

• me ✐’s.

✁ 625 = 21305471. ✁ 675 = 26335270. ✁ 686 = 21315273. ✁ ✁ 75 ✁ 625 ✁ 675 ✁ 686 5874 = (24325472)2. ✟ 611❀ 14 ✁ 64 ✁ 75 24325472✠ 47 ✁ 13. The number-field sieve Generalize ✐ ✑ ✐ + ◆ (mod ◆) ✦ ❛ ✑ ❛ + ❜◆ (mod ◆) ✦ ❛❜♠ ✑ ❛❜☛ (mod ♠☛) for root ☛ ✷ C

• f nonzero integer poly.

For any ♠ can find ☛ so that factoring ♠ ☛ produces factorization of ◆. Optimal choice of log ♠ is (✖+♦(1))(log ◆)2❂3(log log ◆)1❂3. RAM cost 1993 Buhler–Lenstra–P Smoothness ▲ ✿ Sieve ▲1✿ ❛❀ ❜ Find ▲0✿961500 with ❛ ❜♠ ❛ ❜☛ Total RAM ▲ ✿ 1993 Copp Total RAM ▲ ✿ using multiple (Multiple don’t seem with ❆❚

factorization of ✐ ✑ 611 + ✐ ✐ ✁

471.

✁

270.

✁

273.

✁ ✁ ✁ ✁ 675 ✁ 686 (24325472)2. ✟ ❀ ✁ ✁ 75 24325472✠ ✁ The number-field sieve Generalize ✐ ✑ ✐ + ◆ (mod ◆) ✦ ❛ ✑ ❛ + ❜◆ (mod ◆) ✦ ❛❜♠ ✑ ❛❜☛ (mod ♠☛) for root ☛ ✷ C

• f nonzero integer poly.

For any ♠ can find ☛ so that factoring ♠ ☛ produces factorization of ◆. Optimal choice of log ♠ is (✖+♦(1))(log ◆)2❂3(log log ◆)1❂3. RAM cost analysis 1993 Buhler–Lenstra–P Smoothness bound ▲ ✿ Sieve ▲1✿923000 pairs ❛❀ ❜ Find ▲0✿961500 pairs with ❛ ❜♠ and ❛ ❜☛ Total RAM time ▲ ✿ 1993 Coppersmith: Total RAM time ▲ ✿ using multiple numb (Multiple number don’t seem to combine with ❆❚, factory, et

rization of ✐ ✑ ✐ ✐ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ . ✟ ❀ ✁ ✁

• 25472✠

✁ The number-field sieve Generalize ✐ ✑ ✐ + ◆ (mod ◆) ✦ ❛ ✑ ❛ + ❜◆ (mod ◆) ✦ ❛❜♠ ✑ ❛❜☛ (mod ♠☛) for root ☛ ✷ C

• f nonzero integer poly.

For any ♠ can find ☛ so that factoring ♠ ☛ produces factorization of ◆. Optimal choice of log ♠ is (✖+♦(1))(log ◆)2❂3(log log ◆)1❂3. RAM cost analysis 1993 Buhler–Lenstra–Pomera Smoothness bound ▲0✿961500 Sieve ▲1✿923000 pairs (❛❀ ❜). Find ▲0✿961500 pairs with ❛ ❜♠ and ❛ ❜☛ smo Total RAM time ▲1✿923000. 1993 Coppersmith: Total RAM time ▲1✿901884 using multiple number fields. (Multiple number fields don’t seem to combine well with ❆❚, factory, et al.)

The number-field sieve Generalize ✐ ✑ ✐ + ◆ (mod ◆) ✦ ❛ ✑ ❛ + ❜◆ (mod ◆) ✦ ❛❜♠ ✑ ❛❜☛ (mod ♠☛) for root ☛ ✷ C

• f nonzero integer poly.

For any ♠ can find ☛ so that factoring ♠ ☛ produces factorization of ◆. Optimal choice of log ♠ is (✖+♦(1))(log ◆)2❂3(log log ◆)1❂3. RAM cost analysis 1993 Buhler–Lenstra–Pomerance: Smoothness bound ▲0✿961500. Sieve ▲1✿923000 pairs (❛❀ ❜). Find ▲0✿961500 pairs with ❛ ❜♠ and ❛ ❜☛ smooth. Total RAM time ▲1✿923000. 1993 Coppersmith: Total RAM time ▲1✿901884 using multiple number fields. (Multiple number fields don’t seem to combine well with ❆❚, factory, et al.)

number-field sieve Generalize ✐ ✑ ✐ + ◆ (mod ◆) ✦ ❛ ✑ ❛ + ❜◆ (mod ◆) ✦ ❛❜♠ ✑ ❛❜☛ (mod ♠☛)

• t ☛ ✷ C

nonzero integer poly. any ♠ can find ☛ that factoring ♠ ☛ duces factorization of ◆. Optimal choice of log ♠ is ✖ ♦(1))(log ◆)2❂3(log log ◆)1❂3. RAM cost analysis 1993 Buhler–Lenstra–Pomerance: Smoothness bound ▲0✿961500. Sieve ▲1✿923000 pairs (❛❀ ❜). Find ▲0✿961500 pairs with ❛ ❜♠ and ❛ ❜☛ smooth. Total RAM time ▲1✿923000. 1993 Coppersmith: Total RAM time ▲1✿901884 using multiple number fields. (Multiple number fields don’t seem to combine well with ❆❚, factory, et al.) ❆❚ cost Sieving is in realistic ❆❚ cost ▲ ✿

er-field sieve ✐ ✑ ✐ + ◆ (mod ◆) ✦ ❛ ✑ ❛ ❜◆ (mod ◆) ✦ ❛❜♠ ✑ ❛❜☛ (mod ♠☛) ☛ ✷ integer poly. ♠ find ☛ ♠ ☛ zation of ◆.

• f log ♠ is

✖ ♦ ◆)2❂3(log log ◆)1❂3. RAM cost analysis 1993 Buhler–Lenstra–Pomerance: Smoothness bound ▲0✿961500. Sieve ▲1✿923000 pairs (❛❀ ❜). Find ▲0✿961500 pairs with ❛ ❜♠ and ❛ ❜☛ smooth. Total RAM time ▲1✿923000. 1993 Coppersmith: Total RAM time ▲1✿901884 using multiple number fields. (Multiple number fields don’t seem to combine well with ❆❚, factory, et al.) ❆❚ cost analysis Sieving is a disaster in realistic cost metric. ❆❚ cost ▲2✿403750.

✐ ✑ ✐ ◆ (mod ◆) ✦ ❛ ✑ ❛ ❜◆ ◆ ✦ ❛❜♠ ✑ ❛❜☛ ♠☛) ☛ ✷ ♠ ☛ ♠ ☛ ◆. ♠ ✖ ♦ ◆

❂

log ◆)1❂3. RAM cost analysis 1993 Buhler–Lenstra–Pomerance: Smoothness bound ▲0✿961500. Sieve ▲1✿923000 pairs (❛❀ ❜). Find ▲0✿961500 pairs with ❛ ❜♠ and ❛ ❜☛ smooth. Total RAM time ▲1✿923000. 1993 Coppersmith: Total RAM time ▲1✿901884 using multiple number fields. (Multiple number fields don’t seem to combine well with ❆❚, factory, et al.) ❆❚ cost analysis Sieving is a disaster in realistic cost metric. ❆❚ cost ▲2✿403750.

RAM cost analysis 1993 Buhler–Lenstra–Pomerance: Smoothness bound ▲0✿961500. Sieve ▲1✿923000 pairs (❛❀ ❜). Find ▲0✿961500 pairs with ❛ ❜♠ and ❛ ❜☛ smooth. Total RAM time ▲1✿923000. 1993 Coppersmith: Total RAM time ▲1✿901884 using multiple number fields. (Multiple number fields don’t seem to combine well with ❆❚, factory, et al.) ❆❚ cost analysis Sieving is a disaster in realistic cost metric. ❆❚ cost ▲2✿403750.

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

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

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

cost analysis Buhler–Lenstra–Pomerance:

• thness bound ▲0✿961500.

▲1✿923000 pairs (❛❀ ❜). ▲0✿961500 pairs ❛ ❜♠ and ❛ ❜☛ smooth. RAM time ▲1✿923000. Coppersmith: RAM time ▲1✿901884 multiple number fields. (Multiple number fields seem to combine well ❆❚, factory, et al.) ❆❚ cost analysis Sieving is a disaster in realistic cost metric. ❆❚ cost ▲2✿403750. Fix: find smooth using ECM. ❆❚ cost ▲1✿923000. Linear algebra is also a disaster. ❆❚ cost ▲2✿403750. Semi-fix: Reduce smoothness bounds to rebalance. ❆❚ cost ▲1✿976052. (2001 Bernstein) The facto 1993 Copp There exists that facto with same ◆ in RAM ▲ ✿ Smoothness ▲ ✿ Smaller than so need mo ❛❀ ❜ Algorithm ❛❀ ❜ such that ❛ ❜♠ Note: one ♠ ◆ Algorithm whether ❛ ❜☛◆

analysis Buhler–Lenstra–Pomerance:

• und ▲0✿961500.

▲ ✿ pairs (❛❀ ❜). ▲ ✿ pairs ❛ ❜♠ and ❛ ❜☛ smooth. ▲1✿923000. ersmith: ▲1✿901884 number fields. er fields combine well ❆❚ , et al.) ❆❚ cost analysis Sieving is a disaster in realistic cost metric. ❆❚ cost ▲2✿403750. Fix: find smooth using ECM. ❆❚ cost ▲1✿923000. Linear algebra is also a disaster. ❆❚ cost ▲2✿403750. Semi-fix: Reduce smoothness bounds to rebalance. ❆❚ cost ▲1✿976052. (2001 Bernstein) The factorization facto 1993 Coppersmith: There exists an algo that factors any integer with same #bits as ◆ in RAM time ▲1✿638587 Smoothness bound ▲ ✿ Smaller than before, so need more (❛❀ ❜ Algorithm knows all ❛❀ ❜ such that ❛ ❜♠ Note: one ♠ works ◆ Algorithm uses ECM whether ❛ ❜☛◆ is

• merance:

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

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

❆❚ cost analysis Sieving is a disaster realistic cost metric. ❆❚ cost ▲2✿403750. find smooth using ECM. ❆❚ cost ▲1✿923000. algebra is also a disaster. ❆❚ cost ▲2✿403750. Semi-fix: Reduce smoothness

• unds to rebalance.

❆❚ cost ▲1✿976052. Bernstein) The factorization factory 1993 Coppersmith: There exists an algorithm that factors any integer with same #bits as ◆ in RAM time ▲1✿638587. Smoothness bound ▲0✿819290. Smaller than before, so need more (❛❀ ❜). Algorithm knows all (❛❀ ❜) such that ❛ ❜♠ is smooth. Note: one ♠ works for all ◆. Algorithm uses ECM to check whether ❛ ❜☛◆ is smooth. Finding is slower Need to ❛❀ ❜ such that ❛ ❜♠ RAM time ▲ ✿

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

❆❚ ❆❚ ▲ ✿ ECM. ❆❚ ▲ ✿ disaster. ❆❚ ▲ ✿

• thness

❆❚ ▲ ✿ The factorization factory 1993 Coppersmith: There exists an algorithm that factors any integer with same #bits as ◆ in RAM time ▲1✿638587. Smoothness bound ▲0✿819290. Smaller than before, so need more (❛❀ ❜). Algorithm knows all (❛❀ ❜) such that ❛ ❜♠ is smooth. Note: one ♠ works for all ◆. Algorithm uses ECM to check whether ❛ ❜☛◆ is smooth. Finding this algorithm is slower than running it. Need to precompute all (❛❀ ❜ such that ❛ ❜♠ is smooth. RAM time ▲2✿006853.

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

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

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

factorization factory Coppersmith: exists an algorithm factors any integer same #bits as ◆ time ▲1✿638587.

• thness bound ▲0✿819290.

Smaller than before, need more (❛❀ ❜). rithm knows all (❛❀ ❜) that ❛ ❜♠ is smooth.

• ne ♠ works for all ◆.

rithm uses ECM to check whether ❛ ❜☛◆ is smooth. Finding this algorithm is slower than running it. Need to precompute all (❛❀ ❜) such that ❛ ❜♠ is smooth. RAM time ▲2✿006853. Standard conversion of precomputation into batching: if there are enough targets, more than ▲0✿368266, then precomputation cost becomes negligible. The big problem: Coppersmith’s algorithm has size ▲1✿638587. Huge ❆❚ cost; useless in reality. Batch NFS Goal: Optimize ❆❚

• 1. Generate ❛❀ ❜

Test ❛ ❜♠

• 2. Make

◆ close to ❛❀ ❜ When smo ❛ ❜♠ test each ❛ ❜☛◆

• 3. After

reorganize: ◆ relevant ❛❀ ❜

• 4. Linear

ization factory ersmith: algorithm integer as ◆ ▲ ✿638587.

• und ▲0✿819290.

efore, ❛❀ ❜). ws all (❛❀ ❜) ❛ ❜♠ is smooth. ♠ rks for all ◆. ECM to check ❛ ❜☛◆ is smooth. Finding this algorithm is slower than running it. Need to precompute all (❛❀ ❜) such that ❛ ❜♠ is smooth. RAM time ▲2✿006853. Standard conversion of precomputation into batching: if there are enough targets, more than ▲0✿368266, then precomputation cost becomes negligible. The big problem: Coppersmith’s algorithm has size ▲1✿638587. Huge ❆❚ cost; useless in reality. Batch NFS Goal: Optimize ❆❚

• 1. Generate (❛❀ ❜)

Test ❛ ❜♠ for smo

• 2. Make many copies

◆ close to each (❛❀ ❜) When smooth ❛ ❜♠ test each ❛ ❜☛◆

• 3. After all smooths

reorganize: for each ◆ relevant (❛❀ ❜) close

• 4. Linear algebra.

◆ ▲ ✿ ▲ ✿819290. ❛❀ ❜ ❛❀ ❜ ❛ ❜♠

• th.

♠ ◆. check ❛ ❜☛◆

• th.

Finding this algorithm is slower than running it. Need to precompute all (❛❀ ❜) such that ❛ ❜♠ is smooth. RAM time ▲2✿006853. Standard conversion of precomputation into batching: if there are enough targets, more than ▲0✿368266, then precomputation cost becomes negligible. The big problem: Coppersmith’s algorithm has size ▲1✿638587. Huge ❆❚ cost; useless in reality. Batch NFS Goal: Optimize ❆❚ asymptotics.

• 1. Generate (❛❀ ❜) in parallel.

Test ❛ ❜♠ for smoothness.

• 2. Make many copies of eac

◆ close to each (❛❀ ❜) generato When smooth ❛ ❜♠ is found, test each ❛ ❜☛◆ for smoothness.

• 3. After all smooths are found,

reorganize: for each ◆, bring relevant (❛❀ ❜) close together.

• 4. Linear algebra.

Finding this algorithm is slower than running it. Need to precompute all (❛❀ ❜) such that ❛ ❜♠ is smooth. RAM time ▲2✿006853. Standard conversion of precomputation into batching: if there are enough targets, more than ▲0✿368266, then precomputation cost becomes negligible. The big problem: Coppersmith’s algorithm has size ▲1✿638587. Huge ❆❚ cost; useless in reality. Batch NFS Goal: Optimize ❆❚ asymptotics.

• 1. Generate (❛❀ ❜) in parallel.

Test ❛ ❜♠ for smoothness.

• 2. Make many copies of each ◆,

close to each (❛❀ ❜) generator. When smooth ❛ ❜♠ is found, test each ❛ ❜☛◆ for smoothness.

• 3. After all smooths are found,

reorganize: for each ◆, bring relevant (❛❀ ❜) close together.

• 4. Linear algebra.

Finding this algorithm er than running it. to precompute all (❛❀ ❜) that ❛ ❜♠ is smooth. time ▲2✿006853. Standard conversion of recomputation into batching: there are enough targets, than ▲0✿368266, recomputation cost ecomes negligible. big problem: Coppersmith’s rithm has size ▲1✿638587. ❆❚ cost; useless in reality. Batch NFS Goal: Optimize ❆❚ asymptotics.

• 1. Generate (❛❀ ❜) in parallel.

Test ❛ ❜♠ for smoothness.

• 2. Make many copies of each ◆,

close to each (❛❀ ❜) generator. When smooth ❛ ❜♠ is found, test each ❛ ❜☛◆ for smoothness.

• 3. After all smooths are found,

reorganize: for each ◆, bring relevant (❛❀ ❜) close together.

• 4. Linear algebra.

Generate (❛❀ ❜). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Is ❛ ❜♠ ❛ ❜♠ ❛ ❜♠ ❛ ❜♠ smooth? If so, store. Repeat. Generate (❛❀ ❜). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Is ❛ ❜♠ ❛ ❜♠ ❛ ❜♠ ❛ ❜♠ smooth? If so, store. Repeat. Generate (❛❀ ❜). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Is ❛ ❜♠ ❛ ❜♠ ❛ ❜♠ ❛ ❜♠ smooth? If so, store. Repeat. Generate (❛❀ ❜). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Is ❛ ❜♠ ❛ ❜♠ ❛ ❜♠ ❛ ❜♠ smooth? If so, store. Repeat.

algorithm running it. recompute all (❛❀ ❜) ❛ ❜♠ is smooth. ▲ ✿006853. conversion of into batching: enough targets, ▲ ✿368266, computation cost negligible. roblem: Coppersmith’s size ▲1✿638587. ❆❚ useless in reality. Batch NFS Goal: Optimize ❆❚ asymptotics.

• 1. Generate (❛❀ ❜) in parallel.

Test ❛ ❜♠ for smoothness.

• 2. Make many copies of each ◆,

close to each (❛❀ ❜) generator. When smooth ❛ ❜♠ is found, test each ❛ ❜☛◆ for smoothness.

• 3. After all smooths are found,

reorganize: for each ◆, bring relevant (❛❀ ❜) close together.

• 4. Linear algebra.

Generate (❛❀ ❜). Generate (❛❀ ❜). Generate ❛❀ ❜ ❛❀ ❜ Is ❛ ❜♠ Is ❛ ❜♠ ❛ ❜♠ ❛ ❜♠ smooth? smooth? If so, store. If so, store. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate ❛❀ ❜ ❛❀ ❜ Is ❛ ❜♠ Is ❛ ❜♠ ❛ ❜♠ ❛ ❜♠ smooth? smooth? If so, store. If so, store. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate ❛❀ ❜ ❛❀ ❜ Is ❛ ❜♠ Is ❛ ❜♠ ❛ ❜♠ ❛ ❜♠ smooth? smooth? If so, store. If so, store. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate ❛❀ ❜ ❛❀ ❜ Is ❛ ❜♠ Is ❛ ❜♠ ❛ ❜♠ ❛ ❜♠ smooth? smooth? If so, store. If so, store. Repeat. Repeat.

❛❀ ❜) ❛ ❜♠

• th.

▲ ✿ batching: rgets, ▲ ✿ rsmith’s ▲ ✿638587. ❆❚ reality. Batch NFS Goal: Optimize ❆❚ asymptotics.

• 1. Generate (❛❀ ❜) in parallel.

Test ❛ ❜♠ for smoothness.

• 2. Make many copies of each ◆,

close to each (❛❀ ❜) generator. When smooth ❛ ❜♠ is found, test each ❛ ❜☛◆ for smoothness.

• 3. After all smooths are found,

reorganize: for each ◆, bring relevant (❛❀ ❜) close together.

• 4. Linear algebra.

Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate ❛❀ ❜ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? If so, store. If so, store. If so, store. If Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate ❛❀ ❜ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? If so, store. If so, store. If so, store. If Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate ❛❀ ❜ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? If so, store. If so, store. If so, store. If Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate ❛❀ ❜ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? If so, store. If so, store. If so, store. If Repeat. Repeat. Repeat.

Batch NFS Goal: Optimize ❆❚ asymptotics.

• 1. Generate (❛❀ ❜) in parallel.

Test ❛ ❜♠ for smoothness.

• 2. Make many copies of each ◆,

close to each (❛❀ ❜) generator. When smooth ❛ ❜♠ is found, test each ❛ ❜☛◆ for smoothness.

• 3. After all smooths are found,

reorganize: for each ◆, bring relevant (❛❀ ❜) close together.

• 4. Linear algebra.

Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat.

NFS Optimize ❆❚ asymptotics. Generate (❛❀ ❜) in parallel. ❛ ❜♠ for smoothness. Make many copies of each ◆, to each (❛❀ ❜) generator. smooth ❛ ❜♠ is found, each ❛ ❜☛◆ for smoothness. After all smooths are found, rganize: for each ◆, bring relevant (❛❀ ❜) close together. Linear algebra.

Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Is ❛ ❜☛1 ❛ ❜☛ ❛ ❜☛ ❛ ❜☛ smooth? If so, store. Send (❛❀ ❜). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ right. Repeat. Is ❛ ❜☛5 ❛ ❜☛ ❛ ❜☛ ❛ ❜☛ smooth? If so, store. Send (❛❀ ❜). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ up. Repeat. Is ❛ ❜☛9 ❛ ❜☛ ❛ ❜☛ ❛ ❜☛ smooth? If so, store. Send (❛❀ ❜). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ right. Repeat. Is ❛ ❜☛13 ❛ ❜☛ ❛ ❜☛ ❛ ❜☛ smooth? If so, store. Send (❛❀ ❜). ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ up. Repeat.

❆❚ asymptotics. ❛❀ ❜) in parallel. ❛ ❜♠ smoothness. copies of each ◆, ❛❀ ❜) generator. ❛ ❜♠ is found, ❛ ❜☛◆ for smoothness.

• ths are found,

each ◆, bring ❛❀ ❜ close together. ra.

Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Is ❛ ❜☛1 Is ❛ ❜☛2 ❛ ❜☛ ❛ ❜☛ smooth? smooth? If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). ❛❀ ❜ ❛❀ ❜ right. Repeat. right. Repeat. right. Is ❛ ❜☛5 Is ❛ ❜☛6 ❛ ❜☛ ❛ ❜☛ smooth? smooth? If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). ❛❀ ❜ ❛❀ ❜ up. Repeat. left. Repeat. Is ❛ ❜☛9 Is ❛ ❜☛10 ❛ ❜☛ ❛ ❜☛ smooth? smooth? If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). ❛❀ ❜ ❛❀ ❜ right. Repeat. right. Repeat. right. Is ❛ ❜☛13 Is ❛ ❜☛14 ❛ ❜☛ ❛ ❜☛ smooth? smooth? If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). ❛❀ ❜ ❛❀ ❜ up. Repeat. left. Repeat.

❆❚ asymptotics. ❛❀ ❜ rallel. ❛ ❜♠

• thness.

ach ◆, ❛❀ ❜ generator. ❛ ❜♠ found, ❛ ❜☛◆

• thness.

found, ◆ ring ❛❀ ❜ together.

Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Is ❛ ❜☛1 Is ❛ ❜☛2 Is ❛ ❜☛3 Is ❛ ❜☛ smooth? smooth? smooth? If so, store. If so, store. If so, store. If Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send ❛❀ ❜ right. Repeat. right. Repeat. right.

• Repeat. down.

Is ❛ ❜☛5 Is ❛ ❜☛6 Is ❛ ❜☛7 Is ❛ ❜☛ smooth? smooth? smooth? If so, store. If so, store. If so, store. If Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send ❛❀ ❜ up. Repeat. left. Repeat. left. Repeat. left. Is ❛ ❜☛9 Is ❛ ❜☛10 Is ❛ ❜☛11 Is ❛ ❜☛ smooth? smooth? smooth? If so, store. If so, store. If so, store. If Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send ❛❀ ❜ right. Repeat. right. Repeat. right.

• Repeat. down.

Is ❛ ❜☛13 Is ❛ ❜☛14 Is ❛ ❜☛15 Is ❛ ❜☛ smooth? smooth? smooth? If so, store. If so, store. If so, store. If Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send ❛❀ ❜ up. Repeat. left. Repeat. left. Repeat. left.

Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Repeat. Is ❛ ❜☛1 Is ❛ ❜☛2 Is ❛ ❜☛3 Is ❛ ❜☛4 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). right. Repeat. right. Repeat. right.

• Repeat. down.

Repeat. Is ❛ ❜☛5 Is ❛ ❜☛6 Is ❛ ❜☛7 Is ❛ ❜☛8 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). up. Repeat. left. Repeat. left. Repeat. left. Repeat. Is ❛ ❜☛9 Is ❛ ❜☛10 Is ❛ ❜☛11 Is ❛ ❜☛12 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). right. Repeat. right. Repeat. right.

• Repeat. down.

Repeat. Is ❛ ❜☛13 Is ❛ ❜☛14 Is ❛ ❜☛15 Is ❛ ❜☛16 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). up. Repeat. left. Repeat. left. Repeat. left. Repeat.

❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? . If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. ❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? . If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. ❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? . If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. ❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). Generate (❛❀ ❜). ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? smooth? . If so, store. If so, store. If so, store. Repeat. Repeat. Repeat. Is ❛ ❜☛1 Is ❛ ❜☛2 Is ❛ ❜☛3 Is ❛ ❜☛4 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). right. Repeat. right. Repeat. right.

• Repeat. down.

Repeat. Is ❛ ❜☛5 Is ❛ ❜☛6 Is ❛ ❜☛7 Is ❛ ❜☛8 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). up. Repeat. left. Repeat. left. Repeat. left. Repeat. Is ❛ ❜☛9 Is ❛ ❜☛10 Is ❛ ❜☛11 Is ❛ ❜☛12 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). right. Repeat. right. Repeat. right.

• Repeat. down.

Repeat. Is ❛ ❜☛13 Is ❛ ❜☛14 Is ❛ ❜☛15 Is ❛ ❜☛16 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). up. Repeat. left. Repeat. left. Repeat. left. Repeat. ◆1❀ ◆2❀ ◆3❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆

❛❀ ❜ ❛❀ ❜ Generate (❛❀ ❜). Generate (❛❀ ❜). ❛ ❜♠ ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? If so, store. If so, store. Repeat. Repeat. ❛❀ ❜ ❛❀ ❜ Generate (❛❀ ❜). Generate (❛❀ ❜). ❛ ❜♠ ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? If so, store. If so, store. Repeat. Repeat. ❛❀ ❜ ❛❀ ❜ Generate (❛❀ ❜). Generate (❛❀ ❜). ❛ ❜♠ ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? If so, store. If so, store. Repeat. Repeat. ❛❀ ❜ ❛❀ ❜ Generate (❛❀ ❜). Generate (❛❀ ❜). ❛ ❜♠ ❛ ❜♠ Is ❛ ❜♠ Is ❛ ❜♠ smooth? smooth? If so, store. If so, store. Repeat. Repeat. Is ❛ ❜☛1 Is ❛ ❜☛2 Is ❛ ❜☛3 Is ❛ ❜☛4 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). right. Repeat. right. Repeat. right.

• Repeat. down.

Repeat. Is ❛ ❜☛5 Is ❛ ❜☛6 Is ❛ ❜☛7 Is ❛ ❜☛8 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). up. Repeat. left. Repeat. left. Repeat. left. Repeat. Is ❛ ❜☛9 Is ❛ ❜☛10 Is ❛ ❜☛11 Is ❛ ❜☛12 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). right. Repeat. right. Repeat. right.

• Repeat. down.

Repeat. Is ❛ ❜☛13 Is ❛ ❜☛14 Is ❛ ❜☛15 Is ❛ ❜☛16 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). up. Repeat. left. Repeat. left. Repeat. left. Repeat. ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆

❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Generate (❛❀ ❜). ❛ ❜♠ ❛ ❜♠ ❛ ❜♠ Is ❛ ❜♠ smooth? If so, store. Repeat. ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Generate (❛❀ ❜). ❛ ❜♠ ❛ ❜♠ ❛ ❜♠ Is ❛ ❜♠ smooth? If so, store. Repeat. ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Generate (❛❀ ❜). ❛ ❜♠ ❛ ❜♠ ❛ ❜♠ Is ❛ ❜♠ smooth? If so, store. Repeat. ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Generate (❛❀ ❜). ❛ ❜♠ ❛ ❜♠ ❛ ❜♠ Is ❛ ❜♠ smooth? If so, store. Repeat. Is ❛ ❜☛1 Is ❛ ❜☛2 Is ❛ ❜☛3 Is ❛ ❜☛4 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). right. Repeat. right. Repeat. right.

• Repeat. down.

Repeat. Is ❛ ❜☛5 Is ❛ ❜☛6 Is ❛ ❜☛7 Is ❛ ❜☛8 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). up. Repeat. left. Repeat. left. Repeat. left. Repeat. Is ❛ ❜☛9 Is ❛ ❜☛10 Is ❛ ❜☛11 Is ❛ ❜☛12 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). right. Repeat. right. Repeat. right.

• Repeat. down.

Repeat. Is ❛ ❜☛13 Is ❛ ❜☛14 Is ❛ ❜☛15 Is ❛ ❜☛16 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). up. Repeat. left. Repeat. left. Repeat. left. Repeat. ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆

Is ❛ ❜☛1 Is ❛ ❜☛2 Is ❛ ❜☛3 Is ❛ ❜☛4 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). right. Repeat. right. Repeat. right.

• Repeat. down.

Repeat. Is ❛ ❜☛5 Is ❛ ❜☛6 Is ❛ ❜☛7 Is ❛ ❜☛8 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). up. Repeat. left. Repeat. left. Repeat. left. Repeat. Is ❛ ❜☛9 Is ❛ ❜☛10 Is ❛ ❜☛11 Is ❛ ❜☛12 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). right. Repeat. right. Repeat. right.

• Repeat. down.

Repeat. Is ❛ ❜☛13 Is ❛ ❜☛14 Is ❛ ❜☛15 Is ❛ ❜☛16 smooth? smooth? smooth? smooth? If so, store. If so, store. If so, store. If so, store. Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). up. Repeat. left. Repeat. left. Repeat. left. Repeat. ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆

❛ ❜☛ Is ❛ ❜☛2 Is ❛ ❜☛3 Is ❛ ❜☛4 smooth? smooth? smooth? . If so, store. If so, store. If so, store. ❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). eat. right. Repeat. right.

• Repeat. down.

Repeat. ❛ ❜☛ Is ❛ ❜☛6 Is ❛ ❜☛7 Is ❛ ❜☛8 smooth? smooth? smooth? . If so, store. If so, store. If so, store. ❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). eat. left. Repeat. left. Repeat. left. Repeat. ❛ ❜☛ Is ❛ ❜☛10 Is ❛ ❜☛11 Is ❛ ❜☛12 smooth? smooth? smooth? . If so, store. If so, store. If so, store. ❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). eat. right. Repeat. right.

• Repeat. down.

Repeat. ❛ ❜☛ Is ❛ ❜☛14 Is ❛ ❜☛15 Is ❛ ❜☛16 smooth? smooth? smooth? . If so, store. If so, store. If so, store. ❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). Send (❛❀ ❜). eat. left. Repeat. left. Repeat. left. Repeat. ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆

❛ ❜☛ ❛ ❜☛ Is ❛ ❜☛3 Is ❛ ❜☛4 smooth? smooth? If so, store. If so, store. ❛❀ ❜ ❛❀ ❜ Send (❛❀ ❜). Send (❛❀ ❜). right.

• Repeat. down.

Repeat. ❛ ❜☛ ❛ ❜☛ Is ❛ ❜☛7 Is ❛ ❜☛8 smooth? smooth? If so, store. If so, store. ❛❀ ❜ ❛❀ ❜ Send (❛❀ ❜). Send (❛❀ ❜). left. Repeat. left. Repeat. ❛ ❜☛ ❛ ❜☛ Is ❛ ❜☛11 Is ❛ ❜☛12 smooth? smooth? If so, store. If so, store. ❛❀ ❜ ❛❀ ❜ Send (❛❀ ❜). Send (❛❀ ❜). right.

• Repeat. down.

Repeat. ❛ ❜☛ ❛ ❜☛ Is ❛ ❜☛15 Is ❛ ❜☛16 smooth? smooth? If so, store. If so, store. ❛❀ ❜ ❛❀ ❜ Send (❛❀ ❜). Send (❛❀ ❜). left. Repeat. left. Repeat. ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆

❛ ❜☛ ❛ ❜☛ ❛ ❜☛ Is ❛ ❜☛4 smooth? If so, store. ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Send (❛❀ ❜). down. Repeat. ❛ ❜☛ ❛ ❜☛ ❛ ❜☛ Is ❛ ❜☛8 smooth? If so, store. ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Send (❛❀ ❜). left. Repeat. ❛ ❜☛ ❛ ❜☛ ❛ ❜☛ Is ❛ ❜☛12 smooth? If so, store. ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Send (❛❀ ❜). down. Repeat. ❛ ❜☛ ❛ ❜☛ ❛ ❜☛ Is ❛ ❜☛16 smooth? If so, store. ❛❀ ❜ ❛❀ ❜ ❛❀ ❜ Send (❛❀ ❜). left. Repeat. ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16

◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆1❀ ◆2❀ ◆3❀ ◆4 ◆1❀ ◆2❀ ◆3❀ ◆4 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆5❀ ◆6❀ ◆7❀ ◆8 ◆5❀ ◆6❀ ◆7❀ ◆8 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆9❀ ◆10❀ ◆11❀ ◆12 ◆9❀ ◆10❀ ◆11❀ ◆12 ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆ ❀ ◆ ❀ ◆ ❀ ◆ ◆13❀ ◆14❀ ◆15❀ ◆16 ◆13❀ ◆14❀ ◆15❀ ◆16

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

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

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

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

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

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

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

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