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Three Squares -

We quickly figured out that the squares won't work horizontally or vertically, so we place them diagonally. By touching the first two squares, and overlapping the third square in the middle, we had three big squares and four small squares in the middle for a total of seven squares.

We then realised that if we overlapped the first two squares, we would create an extra square in the middle for a total of eight.

Four Squares -

We continued with overlapping diagonally. We added a square at the bottom and managed to get a total of fifteen squares.