Math Examples to get High-Schoolers Interested

Question

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!

Answer 1

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.

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: $$\sqrt[5]{1.05}=\sqrt[5]{1+0.05}\approx \sqrt[5]{1}+\frac{1}{5}\cdot0.05=1.01$$

Answer 2

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.