TITLE: Giving a 1-hour talk to highschool math club: any topic suggestion?
QUESTION [12 upvotes]: I've been invited (by my kid) to give a one hour talk to her highschool math club.  Last year (right before the pandemic hit) I did two such talks on probability, and they loved it.  I'm looking for topic suggestions.
Logistics: There will be about $10-15$ kids total, from grades $9-12$.  These kids are mostly "A" students, and some of them have skipped a math grade, but they're nowhere near Olympiad standard.  The talk will be on March $1$st, $2021$, and will be on zoom.  I will be making slides this weekend.
I don't want this to become an "educational" lecture, so I'm looking for fun examples to motivate certain math areas.  E.g. last year my talks on probability culminated with Polya Urn and Buffon's Noodle.  Also, I'd like everything (well, $95\%$) to be fully explainable within the one hour, i.e. I don't want to just spew facts and then tell them "just trust me".
My own math skill tends toward discrete: comp-sci, combinatorics, etc.  However I'm open to any idea - esp. if it has worked for you before! - as long as I can learn it during one weekend.  :)
Some ideas:

modulo arithmetic esp. finite fields: Perfect shuffle is fun.  Would cryptography (e.g. RSA) be too difficult?  What about codes e.g. Hamming?  Any other fun examples of modulo arithmetic?

non-Euclidean geometry: especially how a POINT in elliptic geometry is actually $2$ "ordinary points".  Can the alternates to the parallel postulate (and some consequence e.g. angle sum in a triangle) be explained sufficiently at highschool level in an hour?

finite geometry: Personally I find Finite projective planes very beautiful.  Do they (or finite affine planes) have any application?  (Besides the kiddie game Spot It?)

algorithm: This is actually my work area, but I don't know what I can cover in an hour.  Maybe the $O(N \log N)$ lower bound for sorting?  Some of these kids don't even have coding experience... :(

graph theory: Edge coloring of a complete graph is one of my favorite pictorial proofs.  Eulerian tour.  Euler characteristic (of a planar graph) is related.  What else?  Shortest path would require discussion of algorithm.  Hall's marriage theorem is surprising and neat but I don't think I can prove it from scratch in an hour.

combinatorics: Start with ${N \choose m}$ and then stars-and-bars for sure.  What are your favorite (elementary) examples?  Can Burnside's lemma be proved from scratch (at highschool level, without group theory) in an hour?


Comments on above and/or any other suggestion most welcome!
(Apologies if I'm tagging too widely...)

REPLY [5 votes]: Again, thanks all of you for answering and commenting.  I thought some of you might be interested in what actually happened.
I decided to go with a theme of "finite mathematics" and talk mostly about modulo arithmetic, using it to illustrate $3$ different topics:
(A) I started by performing a crowd-pleasing "magic trick" related to the perfect shuffle.  I actually cannot do a perfect shuffle at all, so instead I did a sort of reverse: I dealt a deck of cards into 4 piles, re-assembled them into a deck, and repeated the process $3$ more times.  As expected, the kids were very surprised to see that the original deck order was restored!  Which I then explained via $4^4 = 256 = 1 \pmod{51}$ as in the wikipedia article.  (BTW if some of you are not aware of this trick, it's actually quite neat and also easy to do in practice.) This took about $15$ minutes.
(B) Then I talked about error correction codes, starting with the background: why you need it, message, encoding, codeword, error, decoding, etc.  For motivation, I found out that the ASCII for lower-case "l" and lower-case "d" differ by just $1$ bit, so I made up a story how, with a $1$-bit error, "Trump lied" would become "Trump died". :D
Anyway, I discussed the Hamming $(7,4)$ code in detail, with actual examples of encoding and decoding, doing parity checks (which I called mod $2$ addition) by hand, and introducing the concept of Hamming distance.  I did not use linear algebra, as most kids haven't learned that.
Then I discussed the crucial fact that, when $N$ is prime, mod $N$ arithmetic supports division (I did not use the term finite field), which in turn means you can solve systems of equations.  This segued into polynomials and Reed Solomon codes, which I covered briefly using the approach of this excellent blogpost by @JyrkiLahtonen - thanks!
The whole error correction code segment took about $30$ minutes.
(C) Lastly, I talked about the kiddie game Spot-It - I have asked my kid to poll her friends before hand and many are familiar with it.  While the game itself is uninteresting, the math behind the card design is fascinating: it's based on the finite projective plane of order $7$.  I spent many slides using modulo $5$ arithmetic to construct the finite affine plane of order $5$, then showing how it satisfies some of Euclid's axioms and therefore deserves the fancy name of "finite plane".  Then, I showed how it can almost be turned into the cards of Spot-It, discussing point-line duality briefly.  Finally I added the points and line at infinity to turn it into a finite projective plane for the full solution.  This segment took about $25$ minutes.
The whole thing took about $70-75$ minutes and was quite well received, both during the event, and when my kid told me the others' reactions afterwards.  She said the older kids mostly got it, while the younger kids had some trouble but still got the flavor.
I spent way, waaaaay too much time over the weekend making the slides, as I was having such a great time!  I just couldn't bear to leave out any of the $3$ topics since they're all fun (and even "educational" in the case of error correction codes) and also show a great variety between them.  I ended up with about $60$ slides!  Had to talk really fast to go through them in $70$ minutes.  :)