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February 2nd, 2013, 09:00 PM   #1
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Number of Squares proof

A square with side length n is divided into n^2 equal squares. Prove by induction that the total number of squares (of any side length) in this array is equal to
(n(n+1)(2n+1))/6


Help is appreciated
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February 2nd, 2013, 10:32 PM   #2
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Re: Number of Squares proof





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February 3rd, 2013, 05:06 AM   #3
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Re: Number of Squares proof

Quote:
Originally Posted by Jakarta
(n(n+1)(2n+1))/6
That's the standard sum of squares formula.
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February 3rd, 2013, 09:30 AM   #4
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Re: Number of squares proof

Quote:
Originally Posted by Jakarta
A square with side length n is divided into n^2 equal squares. Prove by induction that the total number of squares (of any side length) in this array is equal to
(n(n+1)(2n+1))/6
The inductive step is as follows.

Consider an grid of squares obtained by adding to an grid an extra column on the right and an extra row at the top. This creates:

additional squares
additional squares
additional squares
additional square

Total number of additional squares is






The number of squares of all sizes in the original grid is by the inductive hypothesis.

Hence total number of squares of all sizes in the grid is which greg1313 has shown is equal to .

Verify that the formula is true for and you’re done.
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