Here is a puzzle I'm taking from a magazine I get. I thought I intuitively knew the answer, and I did, but I can't prove it and the solution given in this months issue does a magic hand wave over the part I can't prove.
There is a setup with a coin slot that leads to a pair of balance scales. The arms of the scales are slanted so the coin may fall into a bucket on the left or a bucket on the right. The probability of the coin rolling into a bucket is proportional to the number of coins already in the bucket. (That is, if the left bucket has 13 and the right bucket has 4, the probability of the 18th coin falling into the left bucket is 13/17).
We seed each bucket with 1 penny, so the first coin doesn't determine the result of all the other coins. We take 998 other pennies, and put them 1 by 1 into the coin slot, allowing them to fall into a bucket.
What is the expected value of the money in the lighter bucket when this process is finished?
Coins and Scales ... but not what you think.
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Coins and Scales ... but not what you think.
"It doesn't matter who you vote for, the government always gets in."  Elizabeth May, Canadian Green Party Leader
Re: Coins and Scales ... but not what you think.
The puzzle is not welldefined. In the case where you wind up with 500 pennies and 500 pennies, do we arbitrarily decide that one is the lighter side, do we say there is no lighter side, or do we declare both sides to be the lighter side?
Some of us exist to find out what can and can't be done.
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Re: Coins and Scales ... but not what you think.
btilly wrote:The puzzle is not welldefined. In the case where you wind up with 500 pennies and 500 pennies, do we arbitrarily decide that one is the lighter side, do we say there is no lighter side, or do we declare both sides to be the lighter side?
In the case of equal weight buckets, just arbitrarily pick one.
"It doesn't matter who you vote for, the government always gets in."  Elizabeth May, Canadian Green Party Leader
Re: Coins and Scales ... but not what you think.
Spoiler:
You've been here a while and you still don't spolierise?????????????????????????
Last edited by AvalonXQ on Fri Sep 05, 2008 8:54 pm UTC, edited 2 times in total.
Re: Coins and Scales ... but not what you think.
I'm getting a value (with method) of
To AvalonXQ:
Maybe I'm missing something, but isn't the above assuming each coin is equally likely to hit either bucket? If I'm understanding what I'm reading correctly, you seem to be saying, after seeding (R,L,L.....L) and (L,L,...L,R) are equally likely, which they aren't.
Spoiler:
To AvalonXQ:
AvalonXQ wrote:Spoiler:
Maybe I'm missing something, but isn't the above assuming each coin is equally likely to hit either bucket? If I'm understanding what I'm reading correctly, you seem to be saying, after seeding (R,L,L.....L) and (L,L,...L,R) are equally likely, which they aren't.

 Posts: 199
 Joined: Tue Aug 19, 2008 11:23 pm UTC
 Location: Salem, MA
Re: Coins and Scales ... but not what you think.
mbrownmx wrote:I'm getting a value (with method) ofSpoiler:
To AvalonXQ:AvalonXQ wrote:Spoiler:
Maybe I'm missing something, but isn't the above assuming each coin is equally likely to hit either bucket? If I'm understanding what I'm reading correctly, you seem to be saying, after seeding (R,L,L.....L) and (L,L,...L,R) are equally likely, which they aren't.
This agrees with my intuition and the answer they give. I hadn't even considered looking at an inductionstyle argument, and now it seems obvious.
"It doesn't matter who you vote for, the government always gets in."  Elizabeth May, Canadian Green Party Leader
Re: Coins and Scales ... but not what you think.
mbrownmx wrote:
Maybe I'm missing something, but isn't the above assuming each coin is equally likely to hit either bucket? If I'm understanding what I'm reading correctly, you seem to be saying, after seeding (R,L,L.....L) and (L,L,...L,R) are equally likely, which they aren't.
Actually, I'm saying they are.
Spoiler:
Also, fixed my math above, replacing an (n1) with an n.
Re: Coins and Scales ... but not what you think.
AvalonXQ wrote:mbrownmx wrote:
Maybe I'm missing something, but isn't the above assuming each coin is equally likely to hit either bucket? If I'm understanding what I'm reading correctly, you seem to be saying, after seeding (R,L,L.....L) and (L,L,...L,R) are equally likely, which they aren't.
Actually, I'm saying they are.Spoiler:
Also, fixed my math above, replacing an (n1) with an n.
Ah, ok. Right, that makes sense.
I think the math typo was what confused me, since that didn't agree with what I was getting.
Re: Coins and Scales ... but not what you think.
Good puzzle!
+1 would solve again.
Spoiler:
+1 would solve again.

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Re: Coins and Scales ... but not what you think.
Well I get an answer and a nice general formula, but can't prove either of them (could prove the answer by calculating everything out, but I don't want to).
Spoiler:
 Godskalken
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