*Originally posted by Hegemon*

**There are 12 marbles all weighing the same except one. You have an old fashioned counterbalanced scale. By placing the marbles on the scales you can determine which marble is different from the rest. You can only use the scale three times. How?**

There are two situations here. An easy one, and a hard one. I'll start with the easy situation. We will label the marbles 1 to 12. We will then weigh marbles 1 to 4 against marbles 5 to 8. The easy situation is that these two sets of marbles weigh the same. So, we can use marbles 1 to 4 as our control group. We know our test group is marbles 9 through 12. So, we weigh marbles 1 through 3 against marbles 9 through 11. If the weight matches, then marble 12 is different. If you need to know whether it is lighter or heavier, you can use the third weighing to weigh marbles 1 against marble 12. Situation 1 part B is that marbles 1 through 3 are heavier(or lighter, doesn't matter) than marbles 9 through 11. Then the different marble is lighter, and it is either 9, 10, or 11. Weigh marble 9 against 10, if they weigh the same, then marble 11 id lighter, if 9 is lighter than 10, then 9 is the different marble. If 9 is heavier, then marble 10 is lighter. That takes care of the easy situation. The hard situation is when marbles 1 through 4 are heavier (Or lighter) than marbles 5 through 8. We have marbles 9 through 12, which we know all weigh the same as the majority. So, we remove marbles 1, 5, and 6 from the scale, move marbles 2 and 3 to the lighter side, and place marbles 7, and 9 in the heavy side. Now, we weigh marbles 4, 7, and 9 against marbles 2, 3, and 8. If the two sides balance, then either 1 is heavy, or 5, or 6 is light. So, weigh 5 and 6, and the lighter of the two is different, unless they balance, in which case 1 is heavy. Again, that would be too easy. So, let's say that the side with marbles 4, 7 and 9 is heavier. Then either 4 is heavy, or 8 is light, so weigh 4 against 1, if they balance, 8 is light, otherwise, 4 better weigh more than 1, meaning 4 is heavy. If the side with 2, 3, and 8 is heavy, then either 2, or 3 is heavy, or 7 is light, so weigh 2 against 3 If they balance, then 7 is light, otherwise, the heavier of the two marbles is different. Whew!! That takes care of all the possibilities. 🙂

Worst case scenerio weighings:

1: 1, 2, 3, 4 against 5, 6, 7, 8.

Side 1 is heavier.

2: 4, 7, 9 against 2, 3, 8

side 2 is heavier

3: 2 against 3.