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Use a proof by exhaustion to show that a tiling using
Chapter 8, Problem 45E(choose chapter or problem)
Use a proof by exhaustion to show that a tiling using dominoes of a 4 × 4 checkerboard with opposite corners removed does not exist. [Hint: First show that you can assume that the squares in the upper left and lower right comers are removed. Number the squares of the original checkerboard from 1 to 16, starting in the first row, moving right in this row, then starring in the leftmost square in the second row and moving right, and so on. Remove squares 1 and 16 To begin the proof, note that square 2 is covered either by a domino laid horizontally, which covers squares 2 and 3, or vertically, which covers squares 2 and 6. Consider each of these cases separately, and work through all the subcases that arise.]
Questions & Answers
QUESTION:
Use a proof by exhaustion to show that a tiling using dominoes of a 4 × 4 checkerboard with opposite corners removed does not exist. [Hint: First show that you can assume that the squares in the upper left and lower right comers are removed. Number the squares of the original checkerboard from 1 to 16, starting in the first row, moving right in this row, then starring in the leftmost square in the second row and moving right, and so on. Remove squares 1 and 16 To begin the proof, note that square 2 is covered either by a domino laid horizontally, which covers squares 2 and 3, or vertically, which covers squares 2 and 6. Consider each of these cases separately, and work through all the subcases that arise.]
ANSWER:SolutionStep 1Let us assume that a checkerboard will have 4 rows and 4 column. Now, we will spin the checkerboard if required to make then delete the square from 1 to 16.Square 2 must be protected by the domino.First caseIf the domino is place to wrapped square 2 and 6 then the following arrangement of the domino will be 5 - 9 , 10 - 11 and 13 - 14. So square 15 is not covered by any of the square.