Tiling a 10 × 10 board with 1 × 4 tiles

Idea. Choose a colouring for which every tile covers one square of each colour.

Colouring. Give square $(i,j)$ the colour $(i+j)\bmod4$. A horizontal or vertical tile covers four consecutive residues modulo $4$: all different.

Consequence. A tiling by $25$ tiles would cover exactly $25$ squares of each colour.

Counting. Counting the squares of each colour on the $10\times10$ board gives colour $0$: $25$; colour $1$: $24$; colour $2$: $25$; colour $3$: $26$. Since colour $3$ has $26$ squares, a tiling is impossible.

Answer. No.

Remark. The argument adapts to every $n\times n$ board: a tiling by $1\times4$ tiles exists if and only if $4$ divides $n$.

Lesson. When a tile has length $k$, colour the board with $k$ colours along diagonals; each tile then covers all the colours once.