Problem
COM-B1-M07-P019 Uncovered Cell on \(7\times7\)
#19
★★★☆☆ Level 3 of 5
A \(7\times7\) board is covered by \(16\) straight \(1\times3\) trominoes and one monomino. Prove that the monomino can stand only on a cell with \(i+j\equiv2\pmod3\).
Count colors under the coloring by \(i+j\pmod3\).
Under coloring by \(i+j\pmod3\), every straight tromino covers one cell of each color. Thus \(16\) trominoes cover \(16\) cells of each color. On the \(7\times7\) board the color counts are \(16,16,17\), with the extra color being \(2\). Therefore the monomino must stand on a cell of color \(2\), meaning \(i+j\equiv2\pmod3\).
Make clear that this is a necessary condition.