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You have 9 snooker balls. All appear identical. One of them however is a slightly different weight to the others but feels the same to your hand. This ball is either lighter OR heavier. You don't know. To determine the odd ball you have a balance scales. You must find the ball in three weighings or less. How can you do it? Answers must explain FULLY how this is done.
(Answer on next page.)
Okay let's say we have three groups of three balls, A, B and C.
There are a number of cases.
A and B same weight
First, take A and B and weigh them. If A and B are the same, the odd ball is in C. We call the C group C1, C2 and C3.
Then take two balls from C and weigh them (e.g. C1 and C2). If C1 is the same as C2, then C3 is the heavier or lighter ball. If they are different, then take C2 and C3.
If C1 was heavier than C2, then if C2 is lighter than C3, C2 is the lighter ball [C2=odd]. If C1 is lighter than C2, then if C2 is heavier than C3, C2 is the heavier ball [C2=odd]. If C2 is the same weight as C3, then C1 is the heavier or lighter ball [C1=odd].
A and B not the same weight
Now if A and B were not the same weight, well then we know that the odd ball is in either A or B.
So we take say group B, and weigh it against C. If it's the same weight as C, the odd ball is in A. If not, the odd ball is in B (since B cannot be different from both A and C and not have the odd ball). Let's call the group that ball is in (be it A or B) X. You should now know whether the odd ball is lighter or heavier:
- If A > B and B < C, ball is lighter.
- If A > B and B = C, ball is heavier.
- If A < B and B > C, ball is heavier.
- If A < B and B = C, ball is lighter.
Take two balls X1 and X2 from group X (X1, X2 and X3). Going on the condition above, if X1 is not the same as X2 you should know which ball is heavier or lighter (and therefore the odd ball). Otherwise, if X1 is the same as X2, X3 is the odd ball.