Told only how many coins in a stack show heads, how can a man, without looking, split the stack into two...

logic/heads.piles New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

Told only how many coins in a stack show heads, how can a man, without
looking, split the stack into two piles with equal numbers of heads?

I met a man in a pub bar who said that he could divide a stack of coins
into two stacks with the same number of heads in each.  And what is
more he could do this without looking to see whether the coins were
heads or tails.  So I produced a number of similar coins and put them
in a stack.  He asked me how many heads there were, so without turning
any coins over or letting him see them, I counted them for him and told
him the answer.  With that he took the pile of coins and without
looking at the faces made two piles from them.  On inspection they had
exactly the same number of heads in each pile.  I was totally
A-mazed!!!  How *DID* he do this?  Eh!

-- John Yeadon

Show solution

However many heads there are, he counts that many coins into a pile,
and then turns the pile over.

When he first splits, there are x heads in the pile with h coins, and
therefore h-x heads in the other pile.  When he turns over the pile,
the h-x tails in that pile become heads, and the piles match.  He
himself does not know x, and therefore does not know how many heads in
each pile.

Try it at home.

-- Jonathan Dushoff <dushoff@gate.sinica.edu.tw>

Hence, you should have presented him a stack with an odd number of
coins; all heads.

-- Abigail <abigail@fnx.com>

The trickster can easily detect this, because after Jonathan's
procedure he has one pile.  He knows this pile to be all tails, so he
must merely add one additional step: divide this pile into two piles.
By construction each pile contains zero heads.

-- David A Karr <karr@shore.net>