Square Square everywhere!!
Find the total number of squares with all their vertices belonging to this 1 0 × 1 0 array of evenly spaced points.
Note: The sides of squares need not to be horizontal and vertical.
The answer is 825.
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1 solution
I'm not really sure about this...Does "need not to" means "cannot be" or "okay not to be"?
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“Do not have to be” would be best, I guess.
Your general formula is not correct : try on a square of nine points. With your formula the result is 4 instead of 5 in reality. In my opinion the result is 285 (sum of k squared from 1 to 9).
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With a 3 × 3 array of points, for any 1 ≤ N ≤ 2 , there are ( 3 − N ) 2 square arrays of ( N + 1 ) × ( N + 1 ) points, and each such array exactly contains N squares. My formula would give N = 1 ∑ 2 N ( 3 − N ) 2 = 4 + 2 = 6 squares which is correct (there are 4 unit squares, 1 square of side 2 , and 1 square of side 2 ). You are missing out on the squares that are not aligned horizontally/vertically
For each 1 ≤ N ≤ 9 , there are ( 1 0 − N ) 2 square arrays of ( N + 1 ) × ( N + 1 ) points here. For each such ( N + 1 ) × ( N + 1 ) array, there are N squares that can be formed that fit this array, while not fitting any smaller array. Thus there are N = 1 ∑ 9 N ( 1 0 − N ) 2 = 8 2 5 possible squares to be found.