Mathematical_Recreations-Kraitchik-2e

324 THE FALSE COIN.
57. The false coin. The general enunciation of this
problem is the following: We have N coins one of which
is lighter or heavier than the other, all identical. We are
asked to identify the false coin and find its defect by n
balancings using an ordinary scale.
We shall use the following notations: The coins are
A, B, C ... and we write A>O to say that the coin A is
heavier than the other, A < 0 is a lighter coin, A = 0 is a
good coin.
We shall consider the following 3 problems.
1. N = 4, and in addition we have 3 good coins. By
n = 2 balancings we can identify the false coin and find its
defect.
Let A, B, C, D be the 4 coins and a, b, c the 3 good
coins. We balance ABC and abc and then A and B or
A and D. The following events may occur:
ABC>abc ,A>B then A>O
A<B B>O
A=B C>O
ABC<abc A>B B<O
A<B A<O
A=B C<O
ABC = abc A>D D<O
A<D D>O
There are 8 possible cases, each coin may be the false
one and lighter or heavier than the other.
2. There are 9 coins one of which is heavier than the
other. We can identify it by 2 balancing.
\'Ve distribute the 9 coins in 3 groups and we balance 2
groups. If A, B, C are the 3 groups, we balance 2 groups,
A andB.
If A> B, the group A contains the false coin
A <B " "B " "" "
A=B " "C " "