6. Now let D represent the number of domino tilings of a 3 x n checkerboard....
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6. Now let D₁ represent the number of domino tilings of a 3 x n checkerboard. It is considerably trickier to come up with a recurrence relation in this case, though the ideas involved are similar. (a) List (by drawing them) three possibilities for how the rightmost column of a 3 x n board can be covered. (b) Let C₁, be the number of a domino tilings of a 3 x n board with one missing corner. By considering the options you gave in part (a), explain why Dn = Dn-2 +2Cn-1. (c) Now start with a board missing a corner (say the top right corner). Again consider the possibilities for how to cover the rightmost column, and use your analysis to show that Cn Dn-1 + Cn-2- - = (d) Combine the results of (b) and (c) to show that Dn 4Dn-2 - Dn-4. (e) Find initial conditions Do and D₂. What is D, for any odd value of n? 6. Now let D₁ represent the number of domino tilings of a 3 x n checkerboard. It is considerably trickier to come up with a recurrence relation in this case, though the ideas involved are similar. (a) List (by drawing them) three possibilities for how the rightmost column of a 3 x n board can be covered. (b) Let C₁, be the number of a domino tilings of a 3 x n board with one missing corner. By considering the options you gave in part (a), explain why Dn = Dn-2 +2Cn-1. (c) Now start with a board missing a corner (say the top right corner). Again consider the possibilities for how to cover the rightmost column, and use your analysis to show that Cn Dn-1 + Cn-2- - = (d) Combine the results of (b) and (c) to show that Dn 4Dn-2 - Dn-4. (e) Find initial conditions Do and D₂. What is D, for any odd value of n?
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Related Book For
Introduction to Probability and Statistics
ISBN: 978-1133103752
14th edition
Authors: William Mendenhall, Robert Beaver, Barbara Beaver
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