# Tile a square as small as possible using two different sizes of square tiles

There are two types of square tiles. One type has a side length of 1 cm and the other has a side length of 2 cm. What is the smallest square that can be made with equal numbers of each type of tile?

This puzzle is from a UK Junior Mathematical Olympiad.

an integer.

If we use $$k$$ tiles of type 1 (side length 1) and $$k$$ tiles of type 2 (side length 2), the total area is $$5k$$, which means the square made this way must be of side length $$5x$$.

Now consider

the square of side length 5, which is the smallest of the kind. In order for that to be the answer, we need to put five non-overlapping 2x2 squares into the 5x5 region.

But

it is impossible by the following coloring argument: every 2x2 square must cover one of the Xs, but there are only four of them.

.....
.X.X.
.....
.X.X.
.....

Therefore,

the answer is 10. A square of side length 10 allows the following tiling of 20 2x2 tiles (marked by lowercase letters) and 20 1x1 tiles (marked by dots):

aabbccddee
aabbccddee
ffgghhiijj
ffgghhiijj
kkllmmnnoo
kkllmmnnoo
..........