/sci/ - Science & Math

>>8305840

Do you know algebra? If so, know that the nth square number can be represented by:

[math]n^2[/math]

and the (n-1)th square number is:

[math](n-1)^2[/math]

and the nth odd number (starting from 1)

[math]2n-1[/math]

Then, adding the (n-1)th square number and the nth odd number:

[math](n-1)^2 + 2n - 1 = n^2 - 2n + 1 + 2n - 1 = n^2 [/math]

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If you don't know algebra, see pic related, the green squares have area (n-1)^2, the entire grid has area n^2, and the entire grid is split into the green square, and the red boxes and the orange box.

There are 2 lots of the red box (for the 2 sides of the square), 2 * a number is always even, so 2 * (number of red boxes) is even, but then we have 1 orange box left, so 2 * (number of red boxes) + (orange box) = odd number.

So odd number + (green square area) = (entire square area).