Prove that a square can be dissected into n squares for n>5
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It’s easy to find examples for n=6,7,8.
Suppose that a dissection of a square into k smaller squares exists. Choose one of these k squares and divide it into 4 equal smaller squares. In this process, the chosen square is replaced by 4 new squares, so the total number of squares increases from k to k-1+4=k+3 .
Thus, whenever a dissection into k squares exists, so does a dissection into k+3 squares.
Starting from 6, we can reach every number n that is congruent 0 (mod 3).
Starting from 7, we can reach every number n that is congruent 1 (mod 3).
Starting from 8, we can reach every number n that is congruent 2 (mod 3).
Hence, for every integer n > 5, there exists a dissection of a square into n smaller squares.
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