Need a result of Euler that is simple enough for a child to understand Talking to my 8 yr old about "the greatest mathematician of all time", I said it was probably Gauss in my opinion, but that Gauss was not very kind to his kids (for example, forbidding them to go into mathematics because it would "ruin the family name").  So I recommended Euler as being a better choice (from what I've read, Euler was an all-around good guy).
My son already knows a result of Gauss: the trick that lets you sum the first $n$ integers. So he asked for a result from Euler, but the best I could do was the Euler Totient function.  Although my son now knows the totient function, he finds it pretty unmotivated and no where near as cool as the Gauss trick.
Can you suggest something from Euler that might appeal to an 8 yr old?  Number theory and calculus are ok, but no groups/rings/fields, no real analysis, no non-Euclidean geometry, etc.  
 A: How about Euler's solution to the Basel's problem which made him famous? Though your kid needs to know what a sine function is and what its roots are. Even though Euler's proof was not rigorous, it motivates a lot of other interesting questions to ponder over.
A: How about Euler's theorem on Eulerian paths in graphs, which originated from his solution to the Königsberg bridge problem?
A: $$V + F = E + 2$$


*

*V is the number of vertices

*F is the number of faces

*E is the number of edges

A: Euler's theorem on partitions: The number of ways to write $n$ as a sum of distinct positive integers is the same as the number of ways to write $n$ as a sum of odd positive integers. For example, for $n=6$ we have 6, 5+1, 4+2, 3+2+1 with distinct parts, and 1+1+1+1+1+1, 3+1+1+1, 3+3, 5+1 with odd parts; four ways of doing it in either case.
A: Euler discovered (by hand, of course) that $2^{32}+1=4294967297$ is divisible by 641, which disproved Fermat's guess that all numbers $2^{2^n}+1$ are prime.
A: It will be great to show Euclid-Euler's theorem about perfect numbers! It will be a good option to talk about Euclid too.
