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Lets count: Domino tilings Christopher R. H. Hanusa Queens College, CUNY 2 n 3 n n n Aztec Domino Tilings Today well discuss domino tilings, where: Our board is made up of squares. Our dominoes have no spots and all look

  1. 2 × n 3 × n n × n Aztec Proof that f 2 n = ( f n ) 2 + ( f n − 1 ) 2 Proof. How many ways are there to tile a 2 × (2 n ) board? Answer 1. Duh, f 2 n . Answer 2. Ask whether there is a break in the middle of the tiling: Either there is... Or there isn’t... f n f n f n − 1 f n − 1 �� � � Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 5 / 19

  2. 2 × n 3 × n n × n Aztec Proof that f 2 n = ( f n ) 2 + ( f n − 1 ) 2 Proof. How many ways are there to tile a 2 × (2 n ) board? Answer 1. Duh, f 2 n . Answer 2. Ask whether there is a break in the middle of the tiling: Either there is... Or there isn’t... f n f n f n − 1 f n − 1 �� � � For a total of ( f n ) 2 + ( f n − 1 ) 2 tilings. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 5 / 19

  3. 2 × n 3 × n n × n Aztec Proof that f 2 n = ( f n ) 2 + ( f n − 1 ) 2 Proof. How many ways are there to tile a 2 × (2 n ) board? Answer 1. Duh, f 2 n . Answer 2. Ask whether there is a break in the middle of the tiling: Either there is... Or there isn’t... f n f n f n − 1 f n − 1 �� � � For a total of ( f n ) 2 + ( f n − 1 ) 2 tilings. We counted f 2 n in two different ways, so they must be equal. � Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 5 / 19

  4. 2 × n 3 × n n × n Aztec Proof that f 2 n = ( f n ) 2 + ( f n − 1 ) 2 Proof. How many ways are there to tile a 2 × (2 n ) board? Answer 1. Duh, f 2 n . Answer 2. Ask whether there is a break in the middle of the tiling: Either there is... Or there isn’t... f n f n f n − 1 f n − 1 �� � � For a total of ( f n ) 2 + ( f n − 1 ) 2 tilings. We counted f 2 n in two different ways, so they must be equal. � Further reading: Arthur T. Benjamin and Jennifer J. Quinn Proofs that Really Count, MAA Press, 2003. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 5 / 19

  5. 2 × n 3 × n n × n Aztec 3 × n board Question. How many tilings are there on a 3 × n board? Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 6 / 19

  6. 2 × n 3 × n n × n Aztec 3 × n board Question. How many tilings are there on a 3 × n board? Definition. Let t n = # of ways to tile a 3 × n board. t 0 = 1 t 1 = t 2 = t 3 = t 4 = t 5 = t 6 = t 7 = Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 6 / 19

  7. 2 × n 3 × n n × n Aztec 3 × n board Question. How many tilings are there on a 3 × n board? Definition. Let t n = # of ways to tile a 3 × n board. t 0 = 1 None t 1 = 0 t 2 = t 3 = t 4 = t 5 = t 6 = t 7 = Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 6 / 19

  8. 2 × n 3 × n n × n Aztec 3 × n board Question. How many tilings are there on a 3 × n board? Definition. Let t n = # of ways to tile a 3 × n board. t 0 = 1 None None None None t 1 = 0 t 2 = t 3 = 0 t 4 = t 5 = 0 t 6 = t 7 = 0 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 6 / 19

  9. 2 × n 3 × n n × n Aztec 3 × n board Question. How many tilings are there on a 3 × n board? Definition. Let t n = # of ways to tile a 3 × n board. t 0 = 1 None None None None t 1 = 0 t 2 = 3 t 3 = 0 t 4 = t 5 = 0 t 6 = t 7 = 0 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 6 / 19

  10. 2 × n 3 × n n × n Aztec 3 × n board Question. How many tilings are there on a 3 × n board? Definition. Let t n = # of ways to tile a 3 × n board. t 0 = 1 None None None None t 1 = 0 t 2 = 3 t 3 = 0 t 4 = 11 t 5 = 0 t 6 = 41 t 7 = 0 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 6 / 19

  11. 2 × n 3 × n n × n Aztec Hunting sequences Question. How many tilings are there on a 3 × n board? ◮ Our Sequence: 1, 3, 11, 41, . . . Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 7 / 19

  12. 2 × n 3 × n n × n Aztec Hunting sequences Question. How many tilings are there on a 3 × n board? ◮ Our Sequence: 1, 3, 11, 41, . . . Go to the Online Encyclopedia of Integer Sequences (OEIS). http://oeis.org/ ◮ (Search) Information on a sequence ◮ Formula ◮ Other interpretations ◮ References ◮ (Browse) Learn new math ◮ (Contribute) Submit your own! Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 7 / 19

  13. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  14. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Question. How can we count these tilings intelligently? Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  15. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Question. How can we count these tilings intelligently? Answer. Use the transfer matrix method. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  16. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Question. How can we count these tilings intelligently? Answer. Use the transfer matrix method. ◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  17. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Question. How can we count these tilings intelligently? Answer. Use the transfer matrix method. ◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  18. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Question. How can we count these tilings intelligently? Answer. Use the transfer matrix method. ◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  19. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Question. How can we count these tilings intelligently? Answer. Use the transfer matrix method. ◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  20. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Question. How can we count these tilings intelligently? Answer. Use the transfer matrix method. ◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  21. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Question. How can we count these tilings intelligently? Answer. Use the transfer matrix method. ◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  22. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Question. How can we count these tilings intelligently? Answer. Use the transfer matrix method. ◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  23. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Question. How can we count these tilings intelligently? Answer. Use the transfer matrix method. ◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  24. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Question. How can we count these tilings intelligently? Answer. Use the transfer matrix method. ◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  25. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Question. How can we count these tilings intelligently? Answer. Use the transfer matrix method. ◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  26. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Question. How can we count these tilings intelligently? Answer. Use the transfer matrix method. ◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  27. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Question. How can we count these tilings intelligently? Answer. Use the transfer matrix method. ◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  28. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Question. How can we count these tilings intelligently? Answer. Use the transfer matrix method. ◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  29. 2 × n 3 × n n × n Aztec The transfer matrix method Question. How many tilings are there on a 3 × n board? Question. How can we count these tilings intelligently? Answer. Use the transfer matrix method. ◮ Like a finite state machine. ◮ Build the tiling dynamically one column at a time. ◮ A “state” corresponds to which squares are free in a column. ◮ Filling the free squares “transitions” to the next state. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 8 / 19

  30. 2 × n 3 × n n × n Aztec The transfer matrix for the 3 × n board For 3 × n tilings, the possible states are: And the possible transitions are: Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 9 / 19

  31. 2 × n 3 × n n × n Aztec The transfer matrix for the 3 × n board For 3 × n tilings, the possible states are: And the possible transitions are: Use a matrix to keep track of how many transitions there are. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 9 / 19

  32. 2 × n 3 × n n × n Aztec The transfer matrix for the 3 × n board For 3 × n tilings, the possible states are: And the possible transitions are: FROM:     TO:  = A      Use a matrix to keep track of how many transitions there are. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 9 / 19

  33. 2 × n 3 × n n × n Aztec The transfer matrix for the 3 × n board For 3 × n tilings, the possible states are: And the possible transitions are: FROM:   2   TO:  = A      1 Use a matrix to keep track of how many transitions there are. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 9 / 19

  34. 2 × n 3 × n n × n Aztec The transfer matrix for the 3 × n board For 3 × n tilings, the possible states are: And the possible transitions are: FROM: 1   2   TO:  = A    1   1 Use a matrix to keep track of how many transitions there are. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 9 / 19

  35. 2 × n 3 × n n × n Aztec The transfer matrix for the 3 × n board For 3 × n tilings, the possible states are: And the possible transitions are: FROM: 1   2 1   TO:  = A    1   1 Use a matrix to keep track of how many transitions there are. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 9 / 19

  36. 2 × n 3 × n n × n Aztec The transfer matrix for the 3 × n board For 3 × n tilings, the possible states are: And the possible transitions are: FROM: 1 1   2 1   TO:  = A    1   1 Use a matrix to keep track of how many transitions there are. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 9 / 19

  37. 2 × n 3 × n n × n Aztec The transfer matrix for the 3 × n board For 3 × n tilings, the possible states are: And the possible transitions are: FROM: 0 1 0 1   2 0 1 0   TO:  = A    0 1 0 0   1 0 0 0 Use a matrix to keep track of how many transitions there are. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 9 / 19

  38. 2 × n 3 × n n × n Aztec The power of the transfer matrix Multiply by A . This shows that one step after : t 1 = 0  0 1 0 1   1   0  t 2 = 3 2 0 1 0 0 2        =       t 3 = 0 0 1 0 0 0 0      t 4 = 11 1 0 0 0 0 1 t 5 = t 6 = t 7 = t 8 = t 9 = t 10 = Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

  39. 2 × n 3 × n n × n Aztec The power of the transfer matrix Multiply by A . This shows that one step after : t 1 = 0  0 1 0 1   1   0  t 2 = 3 2 0 1 0 0 2        =       t 3 = 0 0 1 0 0 0 0      t 4 = 11 1 0 0 0 0 1 t 5 = Let’s do it again! t 6 = t 7 = t 8 = t 9 = t 10 = Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

  40. 2 × n 3 × n n × n Aztec The power of the transfer matrix Multiply by A . This shows that two steps after : t 1 = 0 2   0 1 0 1  1   3  t 2 = 3 2 0 1 0 0 0        = t 3 = 0       0 1 0 0 0 2      t 4 = 11 1 0 0 0 0 0 t 5 = t 6 = t 7 = t 8 = t 9 = t 10 = Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

  41. 2 × n 3 × n n × n Aztec The power of the transfer matrix Multiply by A . This shows that two steps after : t 1 = 0 2   0 1 0 1  1   3  t 2 = 3 2 0 1 0 0 0        = t 3 = 0       0 1 0 0 0 2      t 4 = 11 1 0 0 0 0 0 t 5 = Let’s do it again! t 6 = t 7 = t 8 = t 9 = t 10 = Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

  42. 2 × n 3 × n n × n Aztec The power of the transfer matrix Multiply by A . This shows that three steps after : t 1 = 0 3   0 1 0 1  1   0  t 2 = 3 2 0 1 0 0 8        = t 3 = 0       0 1 0 0 0 0      t 4 = 11 1 0 0 0 0 3 t 5 = t 6 = t 7 = t 8 = t 9 = t 10 = Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

  43. 2 × n 3 × n n × n Aztec The power of the transfer matrix Multiply by A . This shows that three steps after : t 1 = 0 3   0 1 0 1  1   0  t 2 = 3 2 0 1 0 0 8        = t 3 = 0       0 1 0 0 0 0      t 4 = 11 1 0 0 0 0 3 t 5 = Let’s do it again! t 6 = t 7 = t 8 = t 9 = t 10 = Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

  44. 2 × n 3 × n n × n Aztec The power of the transfer matrix Multiply by A . This shows that four steps after : t 1 = 0 4   0 1 0 1  1   11  t 2 = 3 2 0 1 0 0 0        = t 3 = 0       0 1 0 0 0 8      t 4 = 11 1 0 0 0 0 0 t 5 = t 6 = t 7 = t 8 = t 9 = t 10 = Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

  45. 2 × n 3 × n n × n Aztec The power of the transfer matrix Multiply by A . This shows that four steps after : t 1 = 0 4   0 1 0 1  1   11  t 2 = 3 2 0 1 0 0 0        = t 3 = 0       0 1 0 0 0 8      t 4 = 11 1 0 0 0 0 0 t 5 = t 6 = ◮ A complete tiling of 3 × n ↔ ends in t 7 = t 8 = t 9 = t 10 = Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

  46. 2 × n 3 × n n × n Aztec The power of the transfer matrix Multiply by A . This shows that four steps after : t 1 = 0 4   0 1 0 1  1   11  t 2 = 3 2 0 1 0 0 0        = t 3 = 0       0 1 0 0 0 8      t 4 = 11 1 0 0 0 0 0 t 5 = t 6 = ◮ A complete tiling of 3 × n ↔ ends in � 1 t 7 = � ↔ first entry of A n 0 # of tilings of 3 × n ◮ t 8 = 0 0 t 9 = t 10 = Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

  47. 2 × n 3 × n n × n Aztec The power of the transfer matrix Multiply by A . This shows that four steps after : t 1 = 0 4   0 1 0 1  1   11  t 2 = 3 2 0 1 0 0 0        = t 3 = 0       0 1 0 0 0 8      t 4 = 11 1 0 0 0 0 0 t 5 = 0 t 6 = 153 ◮ A complete tiling of 3 × n ↔ ends in � 1 t 7 = 0 � ↔ first entry of A n 0 # of tilings of 3 × n ◮ t 8 = 571 0 0 t 9 = 0 We can calculate values. t 10 = 2131 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

  48. 2 × n 3 × n n × n Aztec The power of the transfer matrix Multiply by A . This shows that four steps after : t 1 = 0 4   0 1 0 1  1   11  t 2 = 3 2 0 1 0 0 0        = t 3 = 0       0 1 0 0 0 8      t 4 = 11 1 0 0 0 0 0 t 5 = 0 t 6 = 153 ◮ A complete tiling of 3 × n ↔ ends in � 1 t 7 = 0 � ↔ first entry of A n 0 # of tilings of 3 × n ◮ t 8 = 571 0 0 t 9 = 0 We can calculate values. Is there a formula? t 10 = 2131 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 10 / 19

  49. 2 × n 3 × n n × n Aztec A formula for t n Solve by diagonalizing A : A = P − 1 DP Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 11 / 19

  50. 2 × n 3 × n n × n Aztec A formula for t n Solve by diagonalizing A : A = P − 1 DP A n = P − 1 DP P − 1 DP P − 1 DP � �� � � � · · · Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 11 / 19

  51. 2 × n 3 × n n × n Aztec A formula for t n Solve by diagonalizing A : A = P − 1 DP A n = P − 1 D � � D DP · · · Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 11 / 19

  52. 2 × n 3 × n n × n Aztec A formula for t n Solve by diagonalizing A : A = P − 1 DP A n = P − 1 DD · · · DP Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 11 / 19

  53. 2 × n 3 × n n × n Aztec A formula for t n Solve by diagonalizing A : A = P − 1 DP A n = P − 1 D n P Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 11 / 19

  54. 2 × n 3 × n n × n Aztec A formula for t n Solve by diagonalizing A : − √ √   2+ 3 0 0 0 A = P − 1 DP √ √ 0 2+ 3 0 0 D =  √  √ A n = P − 1 D n P   0 0 − 2 − 3 0 0 √ √ 0 0 2 − 3 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 11 / 19

  55. 2 × n 3 × n n × n Aztec A formula for t n Solve by diagonalizing A : − √ √   2+ 3 0 0 0 A = P − 1 DP √ √ 0 2+ 3 0 0 D =  √  √ A n = P − 1 D n P   0 0 − 2 − 3 0 0 √ √ 0 0 2 − 3 We conclude: � 2 n +1 � 2 n +1 �� �� 1 √ + 1 √ t 2 n = 2 − 3 2 + 3 √ √ 6 6 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 11 / 19

  56. 2 × n 3 × n n × n Aztec A formula for t n Solve by diagonalizing A : − √ √   2+ 3 0 0 0 A = P − 1 DP √ √ 0 2+ 3 0 0 D =  √  √ A n = P − 1 D n P   0 0 − 2 − 3 0 0 √ √ 0 0 2 − 3 We conclude: � 2 n +1 � 2 n +1 �� �� 1 √ + 1 √ t 2 n = 2 − 3 2 + 3 √ √ 6 6 ◮ Method works for rectangular boards of fixed width Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 11 / 19

  57. 2 × n 3 × n n × n Aztec On a chessboard Back to our original question: How many domino tilings are there on an 8 × 8 board? Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

  58. 2 × n 3 × n n × n Aztec On a chessboard Back to our original question: How many domino tilings are there on an 8 × 8 board? Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

  59. 2 × n 3 × n n × n Aztec On a chessboard Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than: 10 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

  60. 2 × n 3 × n n × n Aztec On a chessboard Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than: 100 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

  61. 2 × n 3 × n n × n Aztec On a chessboard Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than: 100 2 × 8: f 8 = 34 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

  62. 2 × n 3 × n n × n Aztec On a chessboard Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than: 1,000 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

  63. 2 × n 3 × n n × n Aztec On a chessboard Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than: 1,000 3 × 8: t 8 = 571 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

  64. 2 × n 3 × n n × n Aztec On a chessboard Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than: 10,000 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

  65. 2 × n 3 × n n × n Aztec On a chessboard Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than: 100,000 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

  66. 2 × n 3 × n n × n Aztec On a chessboard Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than: 1,000,000 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

  67. 2 × n 3 × n n × n Aztec On a chessboard Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than: 10,000,000 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

  68. 2 × n 3 × n n × n Aztec On a chessboard Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than: 100,000,000 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

  69. 2 × n 3 × n n × n Aztec On a chessboard Back to our original question: How many domino tilings are there on an 8 × 8 board? How many people think there are more than: 1,000,000,000 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

  70. 2 × n 3 × n n × n Aztec On a chessboard Back to our original question: How many domino tilings are there on an 8 × 8 board? The TRUE number is: Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

  71. 2 × n 3 × n n × n Aztec On a chessboard Back to our original question: How many domino tilings are there on an 8 × 8 board? The TRUE number is: 12,988,816 Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

  72. 2 × n 3 × n n × n Aztec On a chessboard Back to our original question: How many domino tilings are there on an 8 × 8 board? The TRUE number is: 12,988,816 How to determine? Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 12 / 19

  73. 2 × n 3 × n n × n Aztec A Chessboard Graph A graph is a collection of vertices and edges . A perfect matching is a selection of edges which pairs all vertices. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 13 / 19

  74. 2 × n 3 × n n × n Aztec A Chessboard Graph A graph is a collection of vertices and edges . A perfect matching is a selection of edges which pairs all vertices. Create a graph from the chessboard: Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 13 / 19

  75. 2 × n 3 × n n × n Aztec A Chessboard Graph A graph is a collection of vertices and edges . A perfect matching is a selection of edges which pairs all vertices. Create a graph from the chessboard: A tiling of the chessboard ← → A perfect matching of the graph. Let’s count: Domino tilings Manhattan College ΠME & TΣK Induction Christopher R. H. Hanusa Queens College, CUNY April 14, 2011 13 / 19

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