You have 10 coins; maybe they all weigh the same, maybe they don't.

logic/weighing/ten.coins.two.weights New

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

Problem

You have 10 coins; maybe they all weigh the same, maybe they don't.  If
they don't, they occur in only two different weights.  Given a pan
balance and three weighings, determine whether or not they all weigh
the same.

Show solution

First, note that if we are weighing two equal amounts of coins, and the
scale doesn't balance, we can stop (since that proves there are
different weighted coins).  So the assumption is that, at each
weighing, the scale balances.

Let's call the coins ABCDEFGHIJ.
Let's call the two possible weights x and y (where x might equal y).

1) Weigh ABC against DEF.  Either they all have the same weight, or
   they have a distribution of (WLOG) x+2y.

2) Weigh ABCD against GHIJ.  ABCD can have 2 distributions: x+3y or
   2x+2y.  Likewise, GHIJ will have the same distribution.

3) Weigh ABCEF against DGHIJ.

Here we have two cases:

a) If ABCD had distribution x+3y, then we know that: D = y
   ABCEF = 2x+3y = DGHIJ = y+(GHIJ) = y+(x+3y) = x+4y
   => x=y

b) If ABCD had distribution 2x+2y, then we know that D = x
   ABCEF = x+4y = DGHIJ = x+(GHIJ) = x+(2x+2y) = 3x+2y
   => x=y

In either case, either x=y, or the scales will not have balanced.

[ Editor: the contributor is not named in the collected copy. ]