1. The chessboard is divided into 64 unit squares. Find the number of all squares on the chessboard that are formed from the unit squares.
Solution. The number of $1 \times 1$ squares on the chessboard is $8^{2}=64$. There are $7^{2}=49$ $2 \times 2$ squares, and $6^{2}=36$ $3 \times 3$ squares, $5^{2}=25$ $4 \times 4$ squares, $4^{2}=16$ $5 \times 5$ squares, $3^{2}=9$ $6 \times 6$ squares, $2^{2}=4$ $7 \times 7$ squares, and $1^{2}=1$ $8 \times 8$ square. Therefore, the total number of squares on the board is $64+49+36+25+16+9+4+1=204$.