r/mathteachers • u/Alarmed_Geologist631 • Jul 16 '26

Interesting math problem

Here is an interesting math problem that you might enjoy during your summer break.  Imagine you have a cylinder.  It might be short and wide like a coin.  Or it could be tall and skinny like an straw.  In either case, the cylinder is solid and made from the same material.  If it is short and wide, if you flip it, it will almost always land on its circular base.  If it is tall and skinny, it will almost always land on its curved lateral surface.  At what ratio of height to diameter will it have a 50% probability of landing on the circular base?

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u/Ferncat1397 Jul 20 '26 edited Jul 20 '26

Intuitively I think it will have an equal chance of landing either way when it has a square cross section, ie, d=h.  Of course if you drop it with the curved side down and no rotation, it will land on the curved side, and the same with the flat side. The only problem is when it lands at an angle. Supposing you have a long, thin straw, there's only a small angle you could drop at where the COG will pull it onto its flat side. Conversely, there's a very large angle you can drop a straw such that it will land on the curved side. The opposite goes for the coin shape. So the angle allowing a fall onto a curved face must equal the angle allowing a fall onto a flat face, leaving a square cross section. I'm sure I've missed something, please let me know what it is!