A four by four grid of unit squares contains squares of various sizes (1 by 1 through 4 by 4), each of which are formed entirely from squares in the grid. In each of the 16 unit squares, write the number of squares that contain it. For instance, the middle numbers in the top row are 6s because they are each contained in one 1x1 square, two 2x2, two 3x3, and the one 4x4
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This only needs to be done for 3 of the squares, since the number of squares for each of the remaining 13 squares must be the same as for one of the 3, by symmetry. As has been already stated, the numbers for the middle squares in the top row are both 6. Thus the numbers for the middle squares on the LHS, RHS and bottom will all be 6 as well. For any corner square, the number of squares = 1 + 1 + 1 + 1 = 4 For any interior square, the number of squares = 1 + 4 + 4 + 1 = 10 Thus the number of squares for the entire 4 x 4 grid (writing them in a grid) is: 4 6 6 4 6 10 10 6 6 10 10 6 4 6 6 4 The sum of all 16 numbers in the grid is (4 x 4) + (8 x 6) + (4 x 10) = 16 + 48 + 40 = 104
To do this for a 10 x 10 square would take some time, but the idea is the same. This time the calculations only need to be done for 15 squares, and the remaining 85 are the same as one of the 15, by symmetry.
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