Word problem. Which Billiard Ball Is Rigged?

Question

You have 12 billiard balls. To the naked eye, they all look identical, and in your hand, they all feel identical. One of the balls, however, is slightly heavier or slightly lighter than the others, but you don’t know which ball or whether it is heavier or lighter.

You do, however, have a balance scale. You can place any equal number of balls on each side of the scale, and the scale will tilt if one side differs in weight. (Note: There is no use in weighing different numbers of balls against each other — the weight difference is so slight that if the scale has more balls on one side, that side will always be heavier.) However, you can only use the scale three times.



How can you determine which ball is different, and whether it is heavier or lighter?

Answer 1

Separate the balls into 3 groups of four. A(1,2,3 & 4), B(5, 6, 7 & 8) and C(9, 10, 11, 12).

Weigh groups A and B.

They balance. Bad ball is in C

Make new 3 groups A(1 & 9), B(2 & 10) & C(11 &12)

Weigh groups A and B.

They balance. Bad ball is in C again.

Make new 3 groups A(1), B(11) and C(12)

Weigh groups A and B Weigh groups A and B

They balance bad ball is 12, if not then bad ball is 11.

They do not balance. A is light & C group is good

Make new 3 groups A(7 2 & 3), B(8 5 & 6) & C(1 4 &9)

Weigh groups A and B

They balance. C group bad.

Make new 3 groups A(1), B(4) and C(9)

Balance, ball 9 is bad. If A is light ball A is bad, If B is light than ball 4 is bad.

Hopefully that is enough to figure out any other combinations.

Answer 2

There are 24 different combinations of which ball is heavier or lighter. Let's call them 1H, 1L, 2H, 2L ... denoting Ball-1 is heavier, Ball-1 is lighter, Ball-2 is heavier, Ball-2 is lighter... respectively. And there are 27 different outcomes for scaling. I will call them 000, 001, 002, 010... showing the outcome of each scaling. For instance 101 means the first(or left) arm got heavy in first and third uses of scale but arms were equal in the second use. 222 means second (right) arm got heavy in every use of the scale. The goal is to match ("code") each different combination of ball weights to scaling outcomes. Say I start with 1H and code it with 111, this means I will have Ball-1 on the first arm on every scaling. 1L is automatically coded with 222. And say I code 2H with 112, this means I need to place Ball-2 on the first arm in first two uses of the scale and on the second arm on the third use. You go on like this and code each ball with being heavier or lighter with scaling outcomes. The only thing you need to be careful is number of balls on each arm should be equal on every scaling. This didn't seem to be an issue and I arrived at the following scaling scheme (each line shows a use of the scale and balls shown with numbers on the left are to be placed on the left arm and the ones on the right to the right arm of the scale):

1,2,3,10,11 --- 4,7,8,9,12

1,2,4,5, --- 3,6,9,10

1,3,4,8 --- 2,5,6,7

Check if it works!