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So, I want to start a mathematics club at my high school. To get people interested in attending some of the beginning meetings, I want to set up posters with either

An interesting/beautiful visual example of mathematics

or

A nice puzzle or riddle (that students could come to the club to attempt to solve or have the solution revealed for them)

I'm sure that lots of people on this site have great examples that would fit nicely. I've seen some questions like this on the exchange but I'm really looking for something that is

Attention grabbing on a poster and understandable to anyone.

I look forward to seeing the answers!

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    $\begingroup$ An example used on my campus drawn on the sidewalk in chalk: "Can you tile a chessboard using dominoes if the top-left and bottom-right corners of the board were removed?" $\endgroup$ – JMoravitz Feb 12 '18 at 2:39
  • $\begingroup$ A variant of @JMoravitz would be the following: Can you tile a chessboard with an arbitrary case removed using only triominos (L-shapes)? $\endgroup$ – C. Falcon Feb 12 '18 at 2:42
  • $\begingroup$ Something I like about that example is how unlike "math" it feels like to the untrained student. If you do want one that is more "mathy" one I remember leaving on a chalkboard once was "Find the sum of the digits of the sum of the digits of the sum of the digits of $4444^{4444}$" $\endgroup$ – JMoravitz Feb 12 '18 at 2:46
  • $\begingroup$ I like the oil slick problem. An oil slick on the water spreads and grows. Prove that if yo compare the oil slick one day with the slick 24 hours later at least one drop must be in the same place it was in the beginning. $\endgroup$ – fleablood Feb 12 '18 at 2:48
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    $\begingroup$ @ThomKiwi: I would look at mathoverflow.net/questions/8846/proofs-without-words $\endgroup$ – Moo Feb 12 '18 at 5:47
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I think simple but beautiful ideas (not too difficult) are needed for hs students. A possible example: $$1^3+2^3=(1+2)^2=9$$ $$1^3+2^3+3^3=(1+2+3)^2=36$$ $$1^3+2^3+3^3+4^3=(1+2+3+4)^2=100$$

$$.\,.\,.\,.\,.\,.\,.\,.\,.\,.\,.\,.\,.\,.\,.\,.\,.$$

Some graphics will also be nice including such things as the witch of Agnesi, tesseract, Morley's theorem, rose curves as graphs of sine and cosine in polar coordinates. enter image description here

A possible riddle may be like this: An ice cream-stand offers $7$ varieties of ice-cream cones. If someone is two buy exactly $3$ ice-cream cones, how many options will he/she have?

Another good thing is to introduce symmetry and change of variables with a problem like this system

$$\left\{\begin{align*} &x^2+xy+y^2=4\\ &x+xy+y=2 \end{align*}\right.$$

Also something like this may add intrigue: $$\sqrt[5]{1.05}=\sqrt[5]{1+0.05}\approx \sqrt[5]{1}+\frac{1}{5}\cdot0.05=1.01$$

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You could easily make a poster or two about the sum of square integers.

Or some intuition on the area of a circle.

Or check out this previous post.

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