## Problem 737

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hacatu
Posts: 1
Joined: Tue Dec 08, 2020 9:06 pm

### Problem 737

How is whether or not coins are stackable determined? I figured that, for the assembly of the first n-1 coins to be stackable on the nth coin, the center of mass of the first n-1 coins should lie within the nth coin. Knowing the nth center of mass thus gives a quick way to compute the nth angle. It seems natural to store the center of mass in polar coordinates. I found a more or less closed form solution for the radius, as well as an iterative solution, and a formula for the angle of the center of mass and rotation angle in terms of the radius.

But my concern is twofold. My formulae seem a little bit off: the total angles for the numbers of coins given are about 3% off, but the error decreases for more coins. I know there is probably a precision problem as well. Unfortunately I haven't been able to find any clean form for the sum of angles. Have others used a laurent series or some other approximation?

I won't put my exact formula for the radius, but the n=2 radius can be immediately calculated assuming the center of mass interpretation is correct.
Last edited by hacatu on Wed Dec 09, 2020 8:13 pm, edited 1 time in total.
MuthuVeerappanR
Posts: 493
Joined: Sun Mar 22, 2015 2:30 pm
Location: India
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### Re: Problem 737

It's very early to post values hacatu. Please keep working on the problem and remove the values. Possibly, someone will reply and you can get the chat in a private forum.

It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment.
pjt33
Posts: 70
Joined: Mon Oct 06, 2008 6:14 pm

### Re: Problem 737

hacatu wrote: Tue Dec 08, 2020 9:34 pm My formulae seem a little bit off: the total angles for the numbers of coins given are about 3% off, but the error decreases for more coins.
I don't think you should be concerned about that, as long as it's 3% over rather than 3% under. Bear in mind that those numbers given are for the first point at which a discrete number of coins passes the nth full rotation. There's no expectation that they should exactly overlap.