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.

• Fermat's last theorem for third powers? – Matt E May 3 '11 at 16:13
• How about the V-E+F=2 formula for any convex polyhedron? You can show your 8 year old the platonic solids and verify the formula holds in each case. – Cheerful Parsnip May 3 '11 at 16:15
• Good idea. How about euler tours? – FUZxxl May 3 '11 at 16:16
• Here's a worthless comment that doesn't answer the question: my advisor, who isn't particularly an Eulerian fan (probably because one often insists on the computing aspect of Euler's works rather than on his insights), said he has marveled at the reading of his "Letters Addressed to a German Princess", which are said to be accessible to a wide audience. – PseudoNeo May 3 '11 at 16:22
• @Matt, is that a joke? – quanta May 3 '11 at 16:28

How about Euler's theorem on Eulerian paths in graphs, which originated from his solution to the Königsberg bridge problem?

$$V + F = E + 2$$

• V is the number of vertices
• F is the number of faces
• E is the number of edges

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.

• There's a nice bijection between the two, as well, which usually goes under the name of "Glaisher's bijection". – Michael Lugo May 3 '11 at 17:28

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.

• This is a very beautiful and ingenious computation but it relies on deep properties of modular arithmetic so without that background it would be hard to explain without saying "then Euler did magic and the result happens" – quanta May 3 '11 at 19:00
• Is really modular arithmetic that far from magic? – PseudoNeo May 3 '11 at 21:06
• @quanta I once found a page that showed what was the motivation of choosing 641 as a factor. You can also find a proof in en.wikipedia.org/wiki/Fermat_number . Also maa.org/editorial/euler/… – Pedro Tamaroff Feb 5 '12 at 6:08

It will be great to show Euclid-Euler's theorem about perfect numbers! It will be a good option to talk about Euclid too.

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.

• I have never met an $8$ year old child who knew any trigonometry whatsoever. So this does not seem like a very useful answer. – Pete L. Clark Jun 25 '11 at 19:41
• I think the equation $1 + \frac{1}{2^2} + \frac{1}{3^2} + \ldots = \frac{\pi^2}{6}$ is simple enough to be understandable - and surprising - to an eight-year-old, even if its proof is not. – Jair Taylor Dec 24 '12 at 4:18