This is a challenging, and quite popular, question of pure logic you could easily encounter as a hedge fund job interview question. We suggest you don’t skip right to the answer, but take some time to digest the problem if it stumps you. Also, remembering the principles of the solution may be easier than remembering the details. Good luck!
Question:
You are given a set of scales and 12 marbles. The scales are of the old balance variety. That is, a small dish hangs from each end of a rod that is balanced in the middle. The device enables you to conclude either that the contents of the dishes weigh the same or that the dish that falls lower has heavier contents than the other.
The 12 marbles appear to be identical. In fact, 11 of them are identical, and one is of a different weight. Your task is to identify the unusual marble and discard it. You are allowed to use the scales three times if you wish, but no more.
Note that the unusual marble may be heavier than the others, or it may be lighter. You are asked to both identify it and determine whether it is heavy or light.
Answer:
This question has been very popular indeed. Sometimes it is golf balls, sometimes marbles, sometimes coins. Most people find it very challenging.
The first step is to split the 12 marbles into three groups of four. Each group of four has two subgroups, a singleton and a triplet: {{1}a,{3}a}, {{1}b,{3}b}, and {{1}c,{3}c}.
Compare {{1}a,{3}a} to {{1}b,{3}b}. If they balance, then the odd ball is in group C. In this case, compare {3}c to {3}b. If {3}c is heavier (or lighter), then comparing any two marbles from within {3}c immediately locates the odd one; if {3}c balances {3}b, then compare {1}c to {1}b to see whether {1}c is heavier or lighter.
If the initial comparison is unbalanced, say {{1}a,{3}a} is heavier than {{1}b,{3}b}, then rotate groups {3}a, {3}b, and {3}c and compare {{1}a,{3}b} to {{1}b,{3}c} (while holding out {{1}c,{3}a}). If they balance, then a heavy marble is in {3}a and comparing any two marbles from within {3}a immediately locates the odd one. Suppose they do not balance. If {{1}a,{3}b} is heavy, then either {1}a is heavy, or {1}b is light. Compare {1}a to {1}c to finish. If {{1}a,{3}b} is light, then {3}b is light and comparing any two marbles within {3}b immediately locates the light one.
In each case, only three weighings are needed.
Special thanks for this interview question to Timothy Crack, author of Heard on the Street: Quantitative Questions from Wall Street Job Interviews.
{ 3 comments }
I think you can do it starting with two groups of six.
Round 1: put 6 marbles on each side of scale. the lighter set of 6 contains the light marble.
Round 2: Split the 6 marbles into two groups of three. Weigh the two groups. The lighter group has the light marble.
Round 3: Choose any two of the three marbles that from the lighter group in round two. If they weigh same the light marble is the one you didn’t weigh. If one is lighter that is the light marble.
The odd marble maybe heavier or lighter than the others.
REALY APRECIATE YOUR INTERVIEW QUESTION,KEEP OT UP.
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