The key is to remember that there is only one bad marble; if any weighing balances, the bad marble is not on the balance. When a weighing does not balance, the bad marble is on the balance, and furthermore, we know some information about whether it is heavy or light -- for instance, if the bad marble is one of the ones on the heavy side, it is heavy.
For the first weighing, weigh four marbles (call them A, B, C, and D) against four other marbles (call these E, F, G, H). There are three possible results. Either they balance, in which case the bad marble is one of the other four, or the side with A, B, C, and D is heavier, in which case one of those four marbles is heavy *or* one of E, F, G, H is light, or the E, F, G, H side is heavier, in which case one of those four is heavy or one of A, B, C, D is light.
In the case that A, B, C, D balances with E, F, G, H:
Weigh three remaining marbles (I, J, K) against three known good marbles (A, B, and C, for instance).
If these balance, they are all good also, so the one unweighed marble (L) is the bad one -- weigh it against a good marble to determine if it is heavy or light.
Otherwise, one of I, J, K is bad and now we know if it is heavy or light. Weigh I against J; if they balance, K is bad (and the second weighing tells you if K is heavy or light), otherwise one of I and J is heavier than the other, and the second weighing tells you if the heavy or light marble is the bad one.
If A, B, C, and D are heavier than E, F, G, and H:
We know I, J, K, and L are good now. Weigh A, B, and E against C, D, and F.
If these balance, they are all good, so weigh G against H to determine which is the light one.
If A, B, and E were heavier, then we know that the bad marble is one of A, B, and F (A or B is heavy, or F is light); now weigh A vs. B to determine which is heavier -- if they balance, F is light.
If C, D, and F were heavier, we now know that either C or D is heavy, or E is light; weigh C vs. D to find the heavy one, or if they balance, E is light.
If E, F, G, H are heavier than A, B, C, D:
Use the procedure for when A, B, C, and D are heavy, but swap the words heavy and light throughout.Submitted 3/24/2000 9:52:01 PM