The classical problem of eliminating a false coin out of 12 given can be solved using three steps of weighing by a simple levelbalance, even without prior knowledge of the false coin being heavier or lighter than the standard coins.
Solve the same problem for 13 coins (i.e. 12 normal, one false , 3 steps to establish which is the different one and whether it is lighter or heavier), provided you may use a balancescale with nonequal arms.
Set the scales to 2a=b, assuming all coins were good.
Weigh ABCDEF against GHI.
If they balance, mark all those good coins x; then
aweigh JKLx against xx.
bIf they balance, M is bad and the 3rd weighing
cM against any pair of good coins will determine whether it is heavy or light.
dif they don't balance, we will know whether the bad coin is light or heavy, and also that it is J,K, or L. Then
eWeighing J against Kx will either balance,
fin which case the bad coin is L, and we will know whether L is light or heavy; or not,
gin which case it is whichever is lighter/heavier (see d) of those 2.
If ABCDEF against GHI does not balance: we mark all 4 remaining (good) coins x, leaving either abcdef<GHI or ABCDEF>ghi using small letters for the light side.
(1) Since we have already marked light/heavy, if we can tell which coin is bad, we will at once know whether it is lighter/heavier.
(2) We have a method for determining which of 3 coins is bad in one weighing, if we know light/heavy  see d to f above.
Assume WLOG that abcdef<GHI, since this makes no difference to the method .
Take out one group of 3, say abc.
Weigh defGHI against xxx. If light, we use the last weighing to test d against f; if heavy we use the last weighing to test G against H; if they are the same then we use the last weighing to test a against b.

Posted by broll
on 20110218 04:48:36 