Problem
COM-B1-M07-P022 One Monomino Among Tetrominoes
#22
★★★★☆ Level 4 of 5
A \(9\times9\) board is covered by \(20\) straight \(1\times4\) tetrominoes and one monomino. Prove that the monomino lies on a cell with \(i+j\equiv2\pmod4\).
Count the four colors on the \(9\times9\) board.
Color cell \((i,j)\) by \(i+j\pmod4\). Each straight tetromino covers one cell of each color, so \(20\) tetrominoes cover \(20\) cells of each color. On the \(9\times9\) board, the counts of colors \(0,1,2,3\) are \(20,20,21,20\). Hence the single monomino must cover the extra cell of color \(2\).
Trains deriving a condition on an exceptional cell under four-coloring.