# Proof: Total number of squares in a square?

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If you divide any square into power (2, 2N) equal squares then total number of squares formed is sigma(power(i,2)) where i iterates from 1 to power( 2, N).

E.g
1 square has total 1,
Divided into 4 has total 1^2 +2^2
Divided into 16 equal squares has total 1^2 + 2^2 + 3^2 + 4^2
Divided into 64 equal squares has 1^2 + 2^2 + 3^2 + ......... + 8^2

Can you prove if this is correct? I have solved it.

posted Jun 23, 2014

Total number of squares are defined as

``````1^2 + 2^2 + 3^2...n^2 or n*(n+1)(2n+1)/6
``````

Assumption square is divided equally using (n-1) horizontal and (n-1) vertical lines.

You can always prove this with induction with assuming the above statement is true for the n and adding one more vertical and horizontal line will get the statement is true for n+1..

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