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TOT 231 1989 Autumn A J5 1x2 in MxN board

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March 12, 2021
combinatoricscombinatorial geometryTiling

Problem Statement

A rectangular M×NM \times N board is divided into 1×1 \times cells. There are also many domino pieces of size 1×21 \times 2. These pieces are placed on a board so that each piece occupies two cells. The board is not entirely covered, but it is impossible to move the domino pieces (the board has a frame, so that the pieces cannot stick out of it). Prove that the number of uncovered cells is (a) less than 14MN\frac14 MN,
(b) less than 15MN\frac15 MN.
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