# Counterfeit Coin Revisited

by Kevin J. Lin

Okay, as some noticed, we're really recycling an older puzzle in a harder version,
but the solution was already available on the site. But hopefully now that the 12
coin version is a puzzle in its own right, we won't get as much email asking us to
solve it. There have been several wordings of the solution, but the clearest written
was written for our old FAQ (now obsolete) by "James", a Washington University
student:

First, weigh 4 coins againt 4
coins.
If the balance tips, we have 4
coins we know are correct, 4
coins which might be too heavy
but cannot be too light, and 4
coins which might be too light
but cannot be too heavy.

Second, (assuming we tipped the
balance on the first weighing)
place 2 possibly light coins
and 1 possibly heavy coin on
each side. If the balance tips,
we have 9 known correct coins,
2 possibly light coins, and 1
possibly heavy coin.

Third, weigh the two possibly
light coins againt each other.
If the balance tips, the
lighter coin is counterfeit and
too light. Otherwise the
possibly heavy coin is
counterfeit and is too heavy.

If the balance in the second
weighing does not tip, we have
10 known correct coins and 2
possibly heavy coins. Weigh
those two coins against each
other, the heavier coin is
counterfeit and too heavy.

If the balance in the first
weighing does not tip, we have
8 known correct coins and 4
possibly heavy or light coins.
For the second weighing, weigh
3 of the possibly bad coins
againt 3 known correct coins.
If the balance does not tip, we
have 11 known correct coins and
one possibly light or heavy
coin.

Third, weigh the counterfeit
coin against a correct coin to
determine if the counterfeit is
too heavy or light.

If the balance in this second weighing tips, we have 9 known correct coins and
either 3 possibly heavy coins or 3 possibly light coins. For the third weighing, weigh
one of the possibly counterfeit coins against another, and from that we know which
coin is counterfeit. (From the last weighing we already knew whether it would be too
heavy or light.)