Can 1×4 planks cover a 10×10 patio?

Question

You want to pave a square patio that is a $10 \times 10$ grid of unit cells using 1×4 planks. Each plank covers exactly four cells in a straight line, laid either horizontally or vertically, with no overlaps and no cells left uncovered. The patio has $100$ cells and each plank covers $4$ , so you would need exactly $25$ planks, the areas match perfectly.

Can it actually be done? Either give a tiling, or prove no arrangement of 1×4 planks can ever pave the 10×10 patio.

Accepted Answer

It cannot be done, and a clever coloring proves it. Color the columns in a repeating cycle of four. Number the columns 1 through 10 and give column c the color (c-1)\bmod 4, so the colors run 0,1,2,3,\ 0,1,2,3,\ 0,1. Count how many columns get each color: color 0 lands on columns 1,5,9 (three columns), and color 2 lands on columns 3,7 (only two columns). Each column has 10 cells, so: - Cells of color 0: 3 	imes 10 = 30. - Cells of color 2: 2 	imes 10 = 20. Now see what each plank covers. A horizontal plank spans four consecutive columns, and four consecutive colors in the cycle 0,1,2,3 are …

Can 1×4 planks cover a 10×10 patio?

Your answer

More Brainteasers questions