hw3 - SOLUTIONS FOR HOMEWORK 3 2.40. (a) Suppose, for sake...

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SOLUTIONS FOR HOMEWORK 3 2.40. (a) Suppose, for sake of contradiction, that the chessboard with corners removed (CWCR, for short) is covered by dominoes. The number of dominoes required equals 62 / 2 = 31 (each domino covers 2 squares, out of the 62 available). Moreover, each domino covers one black square, and one white square, hence the CWCR must contain 31 squares of each color. However, the original chessboard had 8 × 8 / 2 = 32 squares of each type, and the two squares removed are of the same color (white on the picture in your textbook). Thus, CWCR has 32 black and 30 white squares left. This is a contradiction. Therefore, our assumption (that dominoes can cover CWCR) is false. (b) Now we consider a “crippled chessboard”, with four squares removed (two in each of the two opposite corners). This new board contains 60 squares – 30 white and 30 black. Suppose, for the sake of contradiction, that this “crippled chessboard” is covered by “T-shapes”. As each of the T-shapes covers four squares, the total number of them equals 60
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This note was uploaded on 01/27/2010 for the course MATH 347 taught by Professor ? during the Fall '09 term at University of Illinois at Urbana–Champaign.

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hw3 - SOLUTIONS FOR HOMEWORK 3 2.40. (a) Suppose, for sake...

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