數學題(15分)

Frist number the balls from 1 to 12. For the first weighing let us put on the left pan balls
1,2,3,4 and on the right pan balls 5,6,7,8.

There are two possibilities. Either they balance, or they don't. If they balance, then the counterfeit ball 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 counterfeit ball is either 11 or 12.

Weigh pill 1 against 11. If they balance, the counterfeit ball is number 12. If they do not balance, then 11 is the good pill.

If 1,2 against 9,10 do not balance, then the counterfeit ball is either 9 or 10. Again, weigh 1 against 9. If they balance, the counterfeit ball is number 10, otherwise it is number 9.


If the first weighing 1,2,3,4 vs 5,6,7,8 does not balance, then any one of these balls could be the counterfeit ball. 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 counterfeit ball is either 3 or 4. Weigh 4 against 9. If they balance then the counterfeit ball 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 heavy ball, or 1 is a light ball.

For the third weighing, weigh 7 against 8. Whichever side is heavy is the counterfeit ball . If they balance, then 1 is the counterfeit ball . If 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 heavy ball or 2 is a light ball. Weigh 5 against 6. The heavier one is the counterfeit ball . If they balance, then 2 is a counterfeit ball .

Back to the second weighing, if 1, 2, 3, 4 is the heavy side. We now weigh 5,1,2 against 6, 3, 4. If they balance, then the counterfeit ball is either 7 or 8. Weigh 8 against 9. If they balance then the counterfeit ball is 7, otherwise it is 7.

Now, if 5,1,2 against 6, 3, 4 does not balance, and 6, 3, 4 is the heavy side, then either 3 or 4 is a heavy ball, or 5 is a light ball.

For the third weighing, weigh 3 against 4. Whichever side is heavy is the counterfeit ball . If they balance, then 5 is the counterfeit ball .
If the weighing of 5,1,2 against 6, 3, 4 show 5,1,2 to be the heavy side, then either 1 or 2 is a heavy ball or 6 is a light ball. Weigh 1 against 2. The heavier one is the counterfeit ball . If they balance, then 6 is a counterfeit ball .