Suppose you wish to place 7 equally-sized cylinders of length l and diameter d in such a way that they all touch each other. What is the largest value of l/d such that this is impossible?

Originally posted by royalchicken Suppose you wish to place 7 equally-sized cylinders of length l and diameter d in such a way that they all touch each other. What is the largest value of l/d such that this is impossible?

Do you mean that they all touch at least one other cylinder? or that they each touch every other cylinder?

Originally posted by royalchicken Actually, ingenuity and trigonometry suffice. Are you guys going to give it a go, or wait around for Acolyte ðŸ˜‰?

Not me, the only interest I have in mathematics consists in its reduction to first-order logic with identity and set-theory. I'm a foundational type of guy.ðŸ˜€

Originally posted by royalchicken No takers? Acolyte?

Erm... I don't have the dexterity to construct a physical model with biros, and I can't really picture how the 7 cylinders are arranged, much less prove that they have to be arranged in a particular way. ðŸ˜•

A wee hintie: 7 cylinders is the largest number for which this can be done (easy to prove), and there is only one configuration (consider that as fact; it is). To visualize it, use unsharpened pencils, or cigarettes, or just draw them (that's what I did-probably the easiest). Figure out 3, then 4, etc. Then look at limiting positions.

Originally posted by royalchicken A wee hintie: 7 cylinders is the largest number for which this can be done (easy to prove), and there is only one configuration (consider that as fact; it is). To visualize it, use unsharpened pencils, or cigarettes, or just draw them (that's what I did-probably the easiest). Figure out 3, then 4, etc. Then look at limiting positions.