Straight trominoes and the one uncovered square

Question

An 8 by 8 board has 64 squares. You have 21 straight trominoes (1 by 3 tiles, placed horizontally or vertically) and one monomino (a 1 by 1 tile), which together cover exactly 64 squares.

On which squares can the monomino be placed so that the rest of the board can be covered by the trominoes? Find all of them and prove no others work.

Accepted Answer

Only four squares: in chess notation c3, c6, f3, f6. First colouring. Number rows and columns from 0 to 7 and give the square in row r, column c the colour (r + c) mod 3: diagonal stripes. Any straight tromino, horizontal or vertical, covers three consecutive values of r + c, so it covers one square of each colour. Count the colours on the board: colour 0 appears 21 times, colour 1 appears 22 times, colour 2 appears 21 times (you can check by counting along rows). Twenty-one trominoes use 21 of each colour, so the monomino must sit on a square of the colour that has 22: r + c leaves remainder…

Straight trominoes and the one uncovered square

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