Chain of Circles
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 skeptical scientist
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Chain of Circles
Cosmologicon posted this puzzle on Confoundry, and it's one of my favorite puzzles on the site, so I thought I'd share it here. (My apologies if it's been posted already, but I didn't find it with my searches.) Surprisingly, only two people have solved it so far on that site.
Circle #0 has a radius of 1.
Circle #1 overlaps Circle #0. There is a chord of Circle #0 that is a diameter of Circle #1.
Circle #2 has a diameter that's a chord of Circle #1.
Circle #3 has a diameter that's a chord of Circle #2.
And so on up to Circle #15.
What's the largest possible value of the largest distance between a point on Circle #15 and the center of Circle #0?
Circle #0 has a radius of 1.
Circle #1 overlaps Circle #0. There is a chord of Circle #0 that is a diameter of Circle #1.
Circle #2 has a diameter that's a chord of Circle #1.
Circle #3 has a diameter that's a chord of Circle #2.
And so on up to Circle #15.
What's the largest possible value of the largest distance between a point on Circle #15 and the center of Circle #0?
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
Re: Chain of Circles
I think I have a solution:
Spoiler:
Last edited by Pelli on Wed May 25, 2011 11:37 pm UTC, edited 1 time in total.
 skeptical scientist
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Re: Chain of Circles
redrogue wrote:Spoiler:
Circles sharing a single point overlap, but the diameter of one will not be a chord of the other, as required by the problem.
Pelli has the right solution, and a nice exposition as well (although you should really use [sub] and [sup] tags!)
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
Re: Chain of Circles
I'll just chime in and say that I thought the same way as redrogue and at least in my case the confusion comes from that I generally don't work with diameters as line segments but as a measure of the maximum distance between two points on the circle. Sure, if the latter meaning had been intended then the problem would not be wellphrased and it would be trivial (assuming that in such a case chord should've been length of a chord), but still.
Re: Chain of Circles
Pelli wrote:I think I have a solution:Spoiler:
skeptical scientist wrote:Pelli has the right solution, and a nice exposition as well (although you should really use [sub] and [sup] tags!)
WOW!
Okay, I was the one who on counfoundry submitted four solutions, first one with a numeric answer, then when that was marked incorrect an explanation of my reasoning, and then after the hint was given, a different numeric answer, and then the explanation of the reasoning there.
(And for some reason even though that wasn't the answer, the solution was marked correct and then challenged which prevented me from attempting to solve it again for real this time ... I'm really annoyed at that since I would not have given up then)
I kept assuming the ratio of the diameters from each circle to the subsequent one in the series was constant, and although I removed that assumption for the ultimate circle, still kept it for the remaining circles, because I couldn't understand why it would be the case that if you were to maximize the ratio of the diameters for two circles, that wouldn't necessarily be optimal for a threecircle chain, and so on. Very nice puzzle and I'm still disappointed that I was just given the answer without continuing to have the chance to work it out myself.

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Re: Chain of Circles
Protip: don't click spoilers if you don't want to know the answer.Avin wrote:I'm still disappointed that I was just given the answer without continuing to have the chance to work it out myself.
Re: Chain of Circles
Moose Hole wrote:Protip: don't click spoilers if you don't want to know the answer.Avin wrote:I'm still disappointed that I was just given the answer without continuing to have the chance to work it out myself.
I'm not talking about this forum. I already knew the answer when someone incorrectly marked my answer correct on TheConfoundry.
 skeptical scientist
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Re: Chain of Circles
Avin wrote:I'm not talking about this forum. I already knew the answer when someone incorrectly marked my answer correct on TheConfoundry.
Yeah, that's unfortunate. I'm not sure if it was me who did that, but I know I've accidentally done it elsewhere, usually because I start adding a comment, and then I forget to change it from correct to incorrect after I add the comment. Maybe I'll make a suggestion to address this issue.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
 Cosmologicon
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Re: Chain of Circles
I was the one who mistakenly marked your answer correct. Really sorry about that, I could tell you were getting close. Was there a way I could have made it so you could answer again? I'm sorry, I couldn't figure it out.
I went and put in a feature request on the site to make it harder to accidentally mark answers correct, and also to be able to undo your own mistakes without approval from another judge. I don't know if that ever got implemented, I haven't been on the site in a while....
I went and put in a feature request on the site to make it harder to accidentally mark answers correct, and also to be able to undo your own mistakes without approval from another judge. I don't know if that ever got implemented, I haven't been on the site in a while....
Re: Chain of Circles
Apropos of nothing, Confoundry has got some nice things going on in it. I guess the description didn't grab me when it was first described here.
Re: Chain of Circles
I was initially very confused because I somehow interpreted it to mean that Circle #0's diameter was a chord of Circle #1, which (if I'm not mistaken) would mean that Circle #1 could be of any arbitrary size, thus making the answer infinite.
Anyway, it turns out the real answer involves basic geometry. I'm almost picturing it, but I'd love it if someone had a picture somewhere.
Anyway, it turns out the real answer involves basic geometry. I'm almost picturing it, but I'd love it if someone had a picture somewhere.
 skeptical scientist
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Re: Chain of Circles
Lenoxus wrote:Anyway, it turns out the real answer involves basic geometry. I'm almost picturing it, but I'd love it if someone had a picture somewhere.
Here you go:
Or, if you'd rather see all circles and the final point along a single line:
Mmmm, Geometer's Sketchpad is fun. It's kind of pricy, but you can play with it for 20 minutes at a time (no saving) without paying. I have no idea whether playing it will help solve the problem, but it's entirely possible.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
 phlip
 Restorer of Worlds
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Re: Chain of Circles
I went about it a bit differently:
Spoiler:
Code: Select all
enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};
void ┻━┻︵╰(ಠ_ಠ ⚠) {exit((int)⚠);}
Re: Chain of Circles
skeptical scientist wrote:Lenoxus wrote:I'm almost picturing it, but I'd love it if someone had a picture somewhere.
Here you go:
Excellent, thank you.
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