Need a result of Euler that is simple enough for a child to understand
Solution 1
How about Euler's theorem on Eulerian paths in graphs, which originated from his solution to the Königsberg bridge problem?
Solution 2
$$V + F = E + 2$$
 V is the number of vertices
 F is the number of faces
 E is the number of edges
Solution 3
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.
Solution 4
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.
Solution 5
It will be great to show EuclidEuler's theorem about perfect numbers! It will be a good option to talk about Euclid too.
Related videos on Youtube
Fixee
Updated on August 08, 2022Comments

Fixee less than a minute
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 allaround 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 nonEuclidean geometry, etc.

Matt E over 11 yearsFermat's last theorem for third powers?

Cheerful Parsnip over 11 yearsHow about the VE+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.

FUZxxl over 11 yearsGood idea. How about euler tours?

PseudoNeo over 11 yearsHere'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.

quanta over 11 years@Matt, is that a joke?

quanta over 11 yearsYou need to understand modular arithmetic and units to get the Totient function. That takes some algebra to build.

t.b. over 11 years@quanta: I'm pretty sure that Matt meant the statement, not its proof.

Jonas Meyer over 11 yearsFrom Wikipedia (with references I haven't checked): "In 1770, Leonhard Euler gave a proof of [Fermat's last theorem for third powers], but his proof by infinite descent contained a major gap. However, since Euler himself had proven the lemma necessary to complete the proof in other work, he is generally credited with the first proof."

t.b. over 11 years@Jonas: There seems to be some discussion on who is to be credited. There's also an MOthread.

Matt E over 11 years@quanta: Yes, I meant the statement. I thought it might be accessible to a kid who could be explained the totient function, and of some interest because of its history (assuming this was also explained). But I agree that $V  E +F = 2$ is much better; the FLT result was just the first thing that came to mind that wasn't obviously too hard.

Phira over 11 years@Fixee Do you have a reference for the story about Gauss and his children?

Fixee over 11 years@user9325: Only this [citation needed] wikipedia entry: en.wikipedia.org/wiki/Carl_Friedrich_Gauss#Family (The relevant portion of this entry is repeated hundreds of times in web searches, but none has a citation.)

Phira over 11 yearsI am not convinced, I have also always heard that Gauss "eliminated all traces of the paths of his discovery", but then I read part of the disquisitiones in the original and found it certainly more motivated than an average modern treatise.

t.b. over 11 years@user9325: An excerpt from Oort's book review: "Facts about difficulties between Gauss and this son are well documented, e.g., the gambling debts, and the party Eugene threw for his fellow students, for which Gauss did not want to pay the expenses. (...) Gauss may not have been the most empathic of fathers, but he did what he could, within reach of his own social and moral limits. In 1830 he tried to find Eugene just before he left, and finally did meet him at Olbers’ place in Bremen, where he managed to hand him travel money and a trunk."

Steve Jessop about 8 yearsFWIW, that triangular sum isn't really a "result of Gauss". He found it independently as a child, but if that counts then it's also "a result of a" heck of a lot of other children, who admittedly might have taken longer to find it than Gauss's reported few seconds :). Pascal, for example, proved the factorial formula for binomial coefficients more than 100 years previously, and $\frac{(n+1)!}{(n1)!2!}$ is your familiar $\frac{n(n+1)}{2}$


Michael Lugo over 11 yearsThere's a nice bijection between the two, as well, which usually goes under the name of "Glaisher's bijection".

quanta over 11 yearsThis 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"

PseudoNeo over 11 yearsIs really modular arithmetic that far from magic?

Pete L. Clark about 11 yearsI have never met an $8$ year old child who knew any trigonometry whatsoever. So this does not seem like a very useful answer.

Pedro over 10 years@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/…

Jair Taylor over 9 yearsI 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 eightyearold, even if its proof is not.