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let's target kids at most of primary school, what are the surprising things that would let them wow ?

Mobius strip would be one. first, it has only one side; then, as Dr Tadashi Tokieda shows, cutting through the centre line will end up as one circle, not two.

Magic Square will be another one, it's amazing that all rows, columns and diagonals have the same sum.

Pls suggest other ideas that sounds surprising and fun to kids, assuming no more knowledge than primary school level?

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I had a lot of luck when running a primary school math club (ages 9-10) with the game of Nim. It's not much harder than tic-tac-toe to play, but there is a lot more math underneath that sounds "hard" but doesn't actually have any prerequisites.

It's not as quick of a payoff as your ideas in the question, but the outline goes like this:

  • Show them how to play the game
  • Learn some reeally basic strategy like "don't leave just one pile"
  • Notice that matching piles basically might as well not be there at all (so "3 + 3 = 0")
  • Notice that piles of 1, 2, and 3 together might as well not be there at all (so "1 + 2 + 3 = 0")
  • Develop the whole addition table for Nim with piles up to size 7 by just playing the game
  • If mathematical maturity allows, go into binary so that the addition table's structure can get some sense to it, but this isn't really necessary; just building an addition table that says 3 + 5 = 6 is pretty awesome and fun for a group of kids, especially when it isn't just made up, it has an application (the game)

If they get really into it, you can teach them Kayles next and really there is no end to the things you can do; see Winning Ways and their Mathematical Plays by Conway, Berlekamp, Guy.

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  • $\begingroup$ How many piles would you suggest play with , in Nim? $\endgroup$ – athos Dec 6 '17 at 3:35
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    $\begingroup$ Start with 4+5+6. It feels "natural" and the winning moves are non-obvious (either take 3 things from pile 5 or take 1 thing from pile 4 or take 5 things from pile 6). $\endgroup$ – Chris Cunningham Dec 6 '17 at 15:34
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    $\begingroup$ Also let them decide how to set up the game; they can make just one pile if they want. The game is not very good, but that's still really valuable experience. $\endgroup$ – Chris Cunningham Dec 6 '17 at 15:34
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Another math circle activity that is fine for kids (warning: I have not tried this specifically on primary school, so the patience required might be too high):

Modular arithmetic modulo small numbers is required, but kids are more than happy to accept it. Write on a piece of paper that (only on this piece of paper) 3 = 0. Then talk about it; if 3 = 0, then for sure also 4 = 1. And 11 = 8 = 5 = 2. When they have to do 2 + 2, they get 4, but you say "wait a minute, on this paper, what is a simpler way to write 4?" and you get 2 + 2 = 1 mod 3.

Then you build Pascal's Triangle modulo 3, or modulo 4, or whatever modulus they want to try. The algorithm for building the next row is easy, and the patterns you get out of this are pretty awesome. For some visuals in advance, check out https://www.maa.org/press/periodicals/loci/joma/patterns-in-pascals-triangle-with-a-twist-first-twist-what-is-it . You can get these patterns by just not writing the 0's in Pascal's triangle, or by writing each number in a different color.

For maximum prettiness of the pictures, you will want to use hexagonal grid paper with very light lines so that kids can perfectly space the numbers. At the time of this post, https://incompetech.com/graphpaper/hexagonal/ still lets you generate free hexagonal graph paper and choose the weight of the lines. If too much time has passed, hopefully some other kind internet soul has some for you.

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Guessing the result of $V-L+F$ of a solid figure without hesitation leaves them pretty surprised, usually...

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