   # Proof: Total number of squares in a square?

+1 vote
181 views

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.. answer Jun 23, 2014

Similar Puzzles

Difference between squares of two numbers is 8. Twice the square of first number by square of second number is 19. What are the numbers?

64 numbers (not necessarily distinct) are placed on the squares of a chessboard such that the sum of the numbers in every 2x2 square is 7.

What is the sum of the four numbers in the corners of the board?

+1 vote

The given figure shows a circle, centred at O, enclosed in a square. Find the total area of shaded parts?  