I'll attempt to explain image 2 in a way that almost anyone could understand.
Thank you to Sam Bender for the expanding parentheses. constructive feedback welcomed
The picture is an illustration of the following equation $ (1+2+3+4+5+6+7+8)^2 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 +6^3 +7^3 + 8^3 $
Here's how the image corresponds to that equation:
The area of the whole thing is the first quantity:
\[
(1+2+3+4+5+6+7+8)^2.
\]
The sum of cubes part is slightly trickier, but it's easy to see what's going on for the odd numbers. There's only one space marked with a one. That represents the 1^3 term.
Now observe that the spaces marked with 3's, 2 black squares and 1 red square consist of three 3x3 squares 3 * 3^2 = 3^3 . Similarly, the spaces indicated by the 5's make up five 5x5 squares 5^3 . This pattern will hold for any odd. (rigorous proof to be inserted here)
How about the evens? That is, those terms 2^3, 4^3, 6^3 etc? These work similar to the odds except that one of the kxk squares gets split into two parts.
For example, for the twos: you've got the gray 2x2 square, and two halves (blue) of a 2x2 square. In total you get two 2x2 squares (2^3). Your questions would be greatly welcomed!
