The area of a square is the length times the length, Correct? The area of the large square is large length times large length. The area of the small square (black tile only) is small length times small length.

The area of the white tiles must be the difference of the two areas found above.

We need to find two numbers a and b such that a x a - b x b = 65. Here, a is the length of the large square and b is the length of the small square (black tile). You can look up your multiplication table for the list of values.

1 x 1 = 1
2 x 2 = 4
3 x 3 = 9
4 x 4 = 16
5 x 5 = 25
6 x 6 = 36
7 x 7 = 49
8 x 8 = 64
9 x 9 = 81
10 x 10 = 100

And so on.

Of course, the difference in areas is 65 so the large square cannot be 8 x 8.

We try 9 x 9 first. We then see that

9 x 9 - 1 x 1 = 81 - 1 = 80
...
9 x 9 - 4 x 4 = 65

Bingo! There we are! We realise that if the large square is 9 x 9 and the small square is 4 x 4, then their difference is 65, what we want! Beyond that, the two squares must be larger than before so that we can find another pair whose difference is 65.

Therefore, the small square (the black tile) must have a length 4 cm.