An Euler Gem: Distinct Partitions and Odd Partitions (VIDEO!)
There are four ways to break the number 6 down into a sum of distinct numbers: 6 = 5+1 = 4+2 = 3+2+1. There are four ways to break the number of 6 down into odd numbers: 5+1 = 3+3 = 3+1+1=1 = 1+1+1+1+1+1.
It is no coincidence that the count of ways are the same. In 1740 Euler proved it will always be so! His proof is ingenious and here it is!
[I've also added a challenge at the end to discover other bizarre results like this one. I bet you can do it!]