Problem
COM-B1-M07-P014 Where the Single Cell May Stand
#14
★★★☆☆ Level 3 of 5
An \(8\times8\) board is to be covered by \(21\) straight \(1\times3\) trominoes and one single cell. Prove that the single cell cannot be \((1,1)\) or \((8,8)\).
Count colors under the coloring \(i+j\pmod3\).
Under the coloring \((i,j)\mapsto i+j\pmod3\), every tromino covers one cell of each color. Thus \(21\) trominoes cover \(21\) cells of each color. On the \(8\times8\) board, colors \(0,1,2\) occur \(22,21,21\) times. Therefore the single cell must have color \(0\). But \((1,1)\) has color \(2\), and \((8,8)\) has color \(16\equiv1\pmod3\). Neither can be the single cell.
A good necessary-condition problem without promising existence.