I'm trying to prove that for all natural numbers $n \ge 6$, a square can be divided into $n$ smaller squares.
The smaller squares do not need to be of the same size.
So for induction, the base case is $P(6)$, which is that a square can be broken into $6$ squares (I can draw a picture to prove this).
Best Answer
Hint: You only need to do it for $6$, $7$, and $8$. For these, you need to produce explicit splittings.
But after that, anything differs by $3$ from an earlier case. and adding $3$ squares is easy, we just do the natural splitting of an existing square.
If one wants to do a formal induction, let $n \gt 8$. Suppose the result is true for all $i$ such that $6\le i \lt n$. We want to show it holds at $n$. By the induction assumption, it holds at $n-3$. Split one of the squares of the splitting into $n-3$ squares into $4$ squares. That gives us a splitting into $n$ squares.