One Dozen

One Dozen

A dozen ball is given which have equal shape and size. Out of 11 balls one has slight difference in weight?

If a weighing machine is provided how can we trace the odd one using only three chances?

True, if you don't know if the odd ball is heavy or light it is a bit more complicated. This is a well-known, interesting puzzle. Here's a nice concise solution.

Divide the balls into three groups: A, B, and C. Number the balls in each group, so we can refer to the balls as A1, B3, C2, etc.

Step 1 Weigh group A against group B. Assume that they balance. The suspect ball is in group C.

..Step 1.1 Weigh A1, A2, C1 against C2, C3, A3. If they balance, the suspect ball is C4.

....Step 1.1.1 Weigh C4 against any other ball to find out whether it is heavy or light.

..Step 1.2 If A1, A2, C1 go down, either C1 is heavy or one of C2 or C3 is light. Weigh C2 against C3. If they balance the suspect ball is C1; if they don't balance, then the one that goes up is the the suspect ball, and is light.

..Step 1.3 If A1, A2, C1 go up, see step 1.2, reversing "heavy" and "light," "up" and "down."

Step 2 If, in step 1, group A goes down, then weigh A1, A2, B1, against A3, B2, C1

..Step 2.1 If they balance, then either A4 is heavy, or B3 or B4 is light. Weigh B3 against B4 find out which is true. See step 1.2.

..Step 2.2 If, in step 2, A1, A2, B1 go down, then either A1 or A2 is heavy, or B2 is light. Weigh A1 against A2 to find out which is true.

..Step 2.3 If, in step 2, A3, B2, C1 go down, then either A3 is heavy or B1 is light. Weigh A3 against B1 to find out which.

Step 3: If, in step 1, group B goes down, then see step 2, reversing "heavy" and "light," "up" and "down."