A $23 \times 23$ square is divided into smaller squares of dimensions $1 \times 1,2 \times 2$, and $3 \times 3$. What is the minimum possible number of $1 \times 1$ squares?
Color the rows of the square black and white alternately, so the top and bottom rows are black. Then each $2 \times 2$ tile covers two cells of each color, and each $3 \times 3$ tile covers six of one color and three of the other. In particular, if only $2 \times 2$ and $3 \times 3$ tiles are used, the difference between the number of black and white cells covered is divisible by 3 . But the entire board has 23 more black than white cells (if the bottom row were removed, the colors would be equally represented). So there must

![](https://cdn.mathpix.com/cropped/2024_04_17_fc0cd41b371b3df54d59g-1.jpg?height=304&width=306&top_left_y=1859&top_left_x=1625)
be at least one $1 \times 1$ tile.

To construct the required tiling with only one $1 \times 1$ tile, first use $3 \times 3$ tiles to build a $9 \times 12$ rectangle and $2 \times 2$ tiles to build a $2 \times 12$ rectangle. Join these two rectangles to form an $11 \times 12$ rectangle. Then use four copies of this $11 \times 12$ rectangle, together with the $1 \times 1$ square, to build a $23 \times 23$ square as shown in the diagram.