Pushup ladder number theory

I have two (very muscular) friends who like to do pushups. In particular, they do a pushup ladder consisting of:

1 pushup (rest)
2 pushups (rest)
... (etc.)
9 pushups (rest)
10 pushups (rest)
9 pushups (rest)
... (etc.)
2 pushups (rest)
1 pushup

Naturally, I wondered how many they were doing. I added it up and it turns out they did 100 pushups.

I was not as hard-core as my friends, so I only went up to 5 and back down. (1, 2, 3, 4, 5, 4, 3, 2, 1.) How many, I wondered, was I doing? I added it up and it turns out I was doing 25 pushups.

Huh. A pattern. Could it be true that if you go from 1, up to n, back down to 1, you are always doing n² pushups?

Like for instance, if you do 1 pushup, 2 pushups, ... all the way up to 16 pushups and then back down to 1, will you really have done exactly 256 pushups?

Yes!

Proof #1 (algebra). (The less-illuminating proof.)

There is a formula for adding up the numbers from 1 to n: (n+1)*n/2. So we simply add up the numbers from 1 to n (for the pushups on the way up), and then add up the numbers from 1 to n-1 (for the pushups on the way down).

on the way up: (n+1)*n/2 = n²/2 + n/2

on the way down: n*(n-1)/2 = n²/2 - n/2

If you add these two expressions together, the "+ n/2" and the "- n/2" cancel out, the "n²/2" and "n²/2" add up to n², and you just get n².

An acceptable proof, because it tells us that the number of pushups is n², but it is somehow unsatisfying. Okay, it's true, but why? Why?

Proof #2 (geometry). (The wonderful proof!)



The picture above illustrates the proof for n=6. Here's how it works: The vertical stacks of red blocks represent the pushups you do on the way up. First you do one (top right red block), then two, etc. up to six (the stack along the left side). On the way down, you do five (the blue blocks along the bottom), then four, etc. down to one (near the top right).

It is easy to see how you could draw a square picture like this for any side length n, i.e. maximum number of pushups n, and the total number of pushups would always be n², because the picture is a square.

Proof by picture! Very satisifying! Now I understand!