I had already worked this problem out for someone else using pills.

The setting of the problem, as I remember it, goes something like

this.

A wise man has committed a capital crime and is to be put to death.

The king decides to see how wise the man is. He gives to the man 12

pills which are identical in size, shape, color, smell, etc. However,

they are not all exactly the same. One of them has a different

weight. The wise man is presented with a balance and informed that all

the pills are deadly poison except for the one which has a different

weight. The wise man can make exactly three weighings and then must

take the pill of his choice.

Remember that the only safe pill has a different weight, but the

wise man doesn't know whether it is heavier or lighter.

Most people seem to think that the thing to do is weigh six pills

against six pills, but if you think about it, this would yield you no

information concerning the whereabouts of the only safe pill.

So that the following plan can be followed, let us number the pills

from 1 to 12. For the first weighing let us put on the left pan pills

1,2,3,4 and on the right pan pills 5,6,7,8.

There are two possibilities. Either they balance, or they don't. If

they balance, then the good pill is in the group 9,10,11,12. So for

our second weighing we would put 1,2 in the left pan and 9,10 on the

right. If these balance then the good pill is either 11 or 12.

Weigh pill 1 against 11. If they balance, the good pill is number 12.

If they do not balance, then 11 is the good pill.

If 1,2 vs 9,10 do not balance, then the good pill is either 9 or 10.

Again, weigh 1 against 9. If they balance, the good pill is number

10, otherwise it is number 9.

That was the easy part.

What if the first weighing 1,2,3,4 vs 5,6,7,8 does not balance? Then

any one of these pills could be the safe pill. Now, in order to

proceed, we must keep track of which side is heavy for each of the

following weighings.

Suppose that 5,6,7,8 is the heavy side. We now weigh 1,5,6 against

2,7,8. If they balance, then the good pill is either 3 or 4.

Weigh 4 against 9, a known bad pill. If they balance then the good

pill is 3, otherwise it is 4.

Now, if 1,5,6 vs 2,7,8 does not balance, and 2,7,8 is the heavy side,

then either 7 or 8 is a good, heavy pill, or 1 is a good, light pill.

For the third weighing, weigh 7 against 8. Whichever side is heavy is

the good pill. If they balance, then 1 is the good pill. Should the

weighing of 1,5, 6 vs 2,7,8 show 1,5,6 to be the heavy side, then

either 5 or 6 is a good heavy pill or 2 is a light good pill. Weigh 5

against 6. The heavier one is the good pill. If they balance, then 2

is a good light pill.

I think that one could write all of this out in a nice flow chart,

but I'm not sure that the flow chart would show up correctly over

e-mail.