06 Induction and Recusion 2019 2.pdf - Induction and RecursionPart 2 1 Covering a Board with L-shaped Trominoes Prove that for any integer ≥ 1 if one

# 06 Induction and Recusion 2019 2.pdf - Induction and...

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Induction and RecursionPart 2 1
Covering a Board with L-shaped Trominoes Prove that for any integer if one corner square is removed from a checkerboard, the remaining squares can be completely covered by L-shaped trominoes. ࠵?   ≥ 1 2 ࠵?   ×   2 ࠵?
Proof by induction on n Basis case: Show that a checkerboard minus a corner square can be covered by a trominoe. Inductive Hypothesis: Assume that a checkerboard minus a corner can be completely covered by l-shaped trominoes. 2 1   ×   2 1 2 ࠵?   ×   2 ࠵?
Inductive Step: Show that a checkerboard minus a corner can be completely covered by trominoes. Consider an checkerboard minus a corner. 2 ࠵? +1   ×   2 ࠵? +1 2 ࠵? +1   ×   2 ࠵? +1
Inductive Step: Show that a checkerboard minus a corner can be completely covered by trominoes. Consider an checkerboard minus a corner. Divide into four quadrants…use I.H. on each quadrant…. 2 ࠵? +1   ×   2 ࠵? +1 2 ࠵? +1   ×   2 ࠵? +1
Inductive Step: Show that a checkerboard minus a corner can be completely covered by trominoes. Consider an checkerboard minus a corner. use I.H. 4 times, different corners 2 ࠵? +1   ×   2 ࠵? +1 2 ࠵? +1   ×   2 ࠵? +1
Inductive Step: Show that a checkerboard minus a corner can be completely covered by trominoes. Consider an checkerboard minus a corner. 4 times plus one 2 ࠵? +1   ×   2 ࠵? +1 2 ࠵? +1   ×   2 ࠵? +1
Combinations Define as the number of sets of size that can be chosen from a set of size . Like the sum of the first numbers, the argument seems better than the proof. Let’s recall the argument for a formula for . ( n k ) k n n ( n k )
Combinations - the argument If we are looking to pick a subset of elements from a set of size , we have choices to write down a first element, choices for the second element, and so on down to choices for the last element, giving us permutations. But, we are talking combinations. (i.e., choosing a set of committee members, not ways of arranging of them around a table). Any set of members can be written in how many different ways? k n n n 1 n ( k 1) n ( n 1) ( n 2) ( n ( k 1)) k
Combinations - the argument If we are looking to pick a subset of elements from a set of size , we have choices to write down a first element, choices for the second element, and so on down to choices for the last element, giving us permutations. But, we are talking combinations. (i.e., choosing a set of committee members, not ways of arranging of them around a table). Any set of members can be written in how many different ways? k n n n 1 n ( k 1) n ( n 1) ( n 2) ( n ( k 1)) k We can write any of members in the first position, any of the remaining in the second position, and so on, giving us k k 1 k ( k 1) ( k 2) 3 2 1
Combinations - the argument If we are looking to pick a subset of elements from a set of size , we have choices to write down a first element, choices for the second element, and so on down to choices for the last element, giving us permutations. But, we are talking combinations. (i.e., choosing a set

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