Show how a 13 by 13 square can be decomposed into 11 non-overlapping squares whose side

lengths are integers (with sides parallel to the sides of the big square), if the largest subsquare
is 7 by 7.

To decompose a square into smaller squares, we need to find a pattern or arrangement that satisfies the given conditions. Let's break down the problem step by step:

1. Start with the largest square with side length 13. We are given that the largest subsquare is 7 by 7. This means that we need to find a way to decompose the remaining area (6 by 6) into smaller squares.

2. Since we are looking for 11 non-overlapping squares, we can distribute them in many ways within the 6 by 6 area. One possible arrangement is to divide the remaining area into three 2 by 2 squares and one 1 by 1 square. This adds up to 9 squares so far.

3. Now that we have decomposed the 6 by 6 area, we need to place the remaining two squares. We can place them in any available space that does not overlap with the existing squares. One option is to place one square in the bottom right corner of the 7 by 7 subsquare and the other square diagonally opposite to it in the top left corner. This completes the arrangement.

Here is the breakdown of the squares and their side lengths:

1. 7 by 7
2. 2 by 2
3. 2 by 2
4. 2 by 2
5. 1 by 1
6. 1 by 1
7. 1 by 1
8. 1 by 1
9. 1 by 1
10. 1 by 1
11. 1 by 1

Please note that this is just one possible arrangement that satisfies the given conditions. There might be other valid solutions as well.