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        riddles >> hard >> Infinite Quarter Sequence - what about 3 piles?
        
(Message started by: profcool on Aug 5^th, 2010, 7:40pm)

Title: Infinite Quarter Sequence - what about 3 piles?
Post by profcool on Aug 5^th, 2010, 7:40pm

this time you have to split coins into 3 piles, not 2.

can there be neat solution? ::)

Title: Re: Infinite Quarter Sequence - what about 3 piles
Post by Grimbal on Aug 6^th, 2010, 2:27am

What was the problem?

Title: Re: Infinite Quarter Sequence - what about 3 piles
Post by towr on Aug 6^th, 2010, 3:59am

The original problem is that you have an infinite pile of quarters twenty of which are tails, the rest head. And you need to split it into two piles that both have the same number of tails. [edit]You can turn quarters over, but can't see or feel if they're heads or tails. (It would be a bit too easy otherwise.)[/edit]

I don't see a way to generalize the two-pile solution to three though.

Title: Re: Infinite Quarter Sequence - what about 3 piles
Post by Grimbal on Aug 6^th, 2010, 5:02am

If there was a solution, it would also work when switching a T and a H. But that can affect only 2 piles. The number cannot change in all 3 piles. So the number of T in a pile must be constant regardless of where the T are. This is clearly impossible.

Title: Re: Infinite Quarter Sequence - what about 3 piles
Post by towr on Aug 6^th, 2010, 6:13am

Ah, sorry. I wasn't being thorough in the explanation. You can turn over the quarters, but you have no way of knowing what state they're in (you're blindfolded, and have gloves on so you can't feel them).
So in that case it's not true the number of T's are constant.

Title: Re: Infinite Quarter Sequence - what about 3 piles
Post by Grimbal on Aug 6^th, 2010, 7:35am

What I mean is that if you switch 2 coins from the initial state and then you do the blindfolded transform, this changes the number of T's in the end in at most 2 piles. The remaining pile has the same number of T's with or without the switch. And therefore all 3 piles must have the same number of T's with or without the switch.

You can keep changing the initial state 2 coins at a time. After every switch you can see that if you do the blindfolded transform from there the number of T's must remain unchanged as compared to the same transform from the original initial state.

And therefore the number of T's in all piles must be constant regardless of the initial state.