How many squares in a four-by-four grid of dots?

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Sixteen dots form a 4 by 4 square array. How many squares can be drawn with all four corners on dots? Count the ordinary upright squares of every size, and also the tilted ones whose sides are not parallel to the grid.

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20 squares: 14 upright and 6 tilted. Upright squares. In a 4 by 4 array of dots there are 3 by 3 = 9 squares of side 1, 2 by 2 = 4 of side 2, and 1 of side 3: 14. Tilted squares. A tilted square with dot corners has its corners on the sides of an upright bounding square, one corner on each side, at the same distance along each side. A bounding square of side 2 (there are 4 of them) contains exactly one tilted square: the "diamond" joining the midpoints of its sides. A bounding square of side 3 (there is 1) contains two tilted squares: corners at distance 1 along each side going clockwise, or …

How many squares in a four-by-four grid of dots?

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