The diagram below shows how you could split a unit square up into nine non-overlapping squares.
Prove that the only number of non-overlapping squares you cannot split a unit square into are 2, 3, or 5 smaller squares.
Consider the following geometrical sequence.
It can be seen that each new "term" in the sequence will have two more squares than the previous. So it is possible to split the unit square into 4, 6, 8, 10, ... non-overlapping squares. That is, all the even terms above and including four.
By splitting each of the shaded grey square into 2 by 2 smaller squares we lose one square and gain four, adding three squares to any term in this sequence. Therefore we can split a unit square into 7, 9, 11, 13, ... non-overlapping squares. That is, all the odd terms above and including seven.
It should be quickly evident that the unit square cannot be split into 2, 3, or 5 smaller squares. Hence we prove the result.