4. If a $23 \times 23$ square is composed of 1000 squares of sizes $1 \times 1, 2 \times 2, 3 \times 3$, how many $1 \times 1$ squares are needed at minimum?
4. At least 1 $1 \times 1$ square is needed. First, it needs to be shown that a $23 \times 23$ square can be composed of the following: $2 \times 2, 3 \times 3$ squares, and exactly 1 $1 \times 1$ square. Then, it also needs to be proven that a $1 \times 1$ square is indispensable.