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Answer by user141903 for Math talk for all ages

Good luck! I really like your first bullet idea. I would also add the amazing fact that Gödel proved that we cannot prove the consistency of mathematics - the 9-year-olds will get the weirdness of this...

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Answer by RaphaelB4 for Math talk for all ages

This summer a friend of mine gives a very nice talk about the game Nim. This is fun, you can start challenging and playing a few games with the youngests. But it also contains the deep and impressive...

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Answer by l3utterfly for Math talk for all ages

I always found the Collatz Conjecture to be both simple and fascinating (https://en.wikipedia.org/wiki/Collatz_conjecture)The operations are understandable to any 9 year old (who won a math...

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Answer by Ethan Dlugie for Math talk for all ages

I'm fond of the little problem posed in the beginning of this Quanta article: take $n$ generic points on a circle and draw the complete graph between those points. Into how many regions do the edges...

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Answer by user21820 for Math talk for all ages

You could try giving a talk in the spirit of Nets, Puzzles, and Postmen, which is a book targeted at the general audience without mathematical background, about graph theory and how it can be applied...

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Answer by usul for Math talk for all ages

The talk can be entirely congratulatory, or it can contain a bit of actual mathematics ... any alternative suggestions?How about some history of mathematics?It could be structured around a few notable...

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Answer by Pablo H for Math talk for all ages

I've been to quite a few of such talks (though most or all in the 12-to-18-year-old range). I feel and believe that a talk that just presents something nice (to a professional mathematician) is...

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Answer by Alexander Schmeding for Math talk for all ages

I have given a talk to slightly older students, but the subject might be appropriate also to 9 year old students.The talk was about bodies of constant width. Obviously circles have the properties that...

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Answer by Timothy Chow for Math talk for all ages

My inclination would be to convey that it's fun to be a professional mathematician.How many people in the world have a fun job that they love doing? Only a small percentage. I feel privileged to be in...

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Answer by Mike Shulman for Math talk for all ages

Another possibility would be to talk about a few unsolved problems in mathematics. It's easy for kids to think of math as a "finished" edifice, compared to fields like biology and physics where we hear...

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Answer by JoshuaZ for Math talk for all ages

Talk about some elementary number theory problems. Twin prime conjecture, infinitely many even perfect numbers, existence of odd perfect numbers. These are all ones which are simple enough.

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Answer by Andreas Blass for Math talk for all ages

I like to show how the same mathematics shows up in very different contexts. A topic that I've used with quite varied audiences (though never with as much variation in a single audience as you have) is...

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Answer by gidds for Math talk for all ages

Following on from Wlod AA's idea, there must be plenty more juicy morsels that can be understood (if not proved!) without much knowledge.  For example:The Hairy Ball Theorem (“You can't comb a hairy...

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Answer by Wlod AA for Math talk for all ages

I'd present a juicy morsel of mathematics, say -- the Euler characteristics theorem for $\ \mathbb S^2.\ $ I'd adopt a classic proof.Let $\ \mathbb S^2\ $ be divided into convex geodesic polyhedra, $\...

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Math talk for all ages

I've been asked to give a talk to the winners of a recent math competition. The talk can be entirely congratulatory, or it can contain a bit of actual mathematics. I'd prefer the latter. I'd also like...

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