Question

1. A chess board has its top left and bottom right corner squares cut out leaving 62 squares. We have a supply of dominoes, each of which will cover 2 adjacent squares. Is there a way to exactly cover an entire board? What difference does cutting out the 2 opposing corners make ? Could we just simply try and see if we can do this?

a) If a rectangular board is completely covered with tetrominoes show that at least one side
must be of even length.

b) If a rectangular board can be completely covered by T tetrominoes show that the
number of squares on the board must be a multiple of 8.

Answer 1

Hello,
          Please find the answer attached as under. Please give a thumbs up rating if you find the answer useful! Have a rocking day ahead!

Note that the opposite diagonal corners of a chess board are of the same color. Hence, if these corners were black in color, there would now be 32 white squares and 30 black squares. A rectangular domino, if placed on the chess-board, will have to exactly cover one black and one white square. Thus, the dominos can cover the chess board only if the number of black squares=number of white squares.

Hence not possible!

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