Question:

You are given 12 identical looking balls. One of them is fake (could be heavier or lighter) than the rest of the 11 (all the others weight exactly the same). You a provided with a simple mechanical balance and you are restricted to only 3 uses. Find the fake ball.

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C

A

P

T

A

I

N

I

N

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E

R

V

I

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W

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Solution: For convenience sake, let’s name the balls 1-12. This question is quite tricky. If you want to warm up your skills, you can try this easier 8 ball problem first.Ok, let’s get on with it.First we weigh {1,2,3,4} on the left and {5,6,7,8} on the right. There are three scenarios which can arise from this.

If they balance, then we know 9, 10, 11 or 12 is fake. Weigh {8, 9} and {10, 11} (Note: 8 is surely not fake)If they balance, we know 12 is the fake one. Just weigh it with any other ball and figure out if it is lighter or heavier.

If {8, 9} is heavier, then either 9 is heavy or 10 is light or 11 is light. Weigh {10} and {11}.

If they balance, 9 is fake (heavier). If they don’t balance then whichever one is lighter is fake (lighter).

If {8, 9} is lighter, then either 9 is light or 10 is heavy or 11 is heavy. Weigh {10} and {11}.

If they balance, 9 is fake (lighter). If they don’t balance then whichever one is heavier is fake (heavier).

Phwww, that was confusing enough but we are not done yet.

If {1,2,3,4} is heavier, we know either one of {1,2,3,4} heavier or one of {5,6,7,8} is lighter but it is guarantees that {9,10,11,12} are not fake. This is where it gets really tricky, watch carefully. Weigh {1,2,5} and {3,6,9} (Note: 9 is surely not fake).

If they balance, then either 4 is heavy or 7 is light or 8 is light. Following the last step from the previous case, we weigh {7} and {8}. If they balance, 4 is fake(heavier). If they don’t balance then whichever one is lighter is fake (lighter).

If {1,2,5} is heavier, then either 1 is heavy or 2 is heavy or 6 is light. Weigh {1} and {2}. If they balance, 6 is fake (lighter). If they don’t balance then whichever one is heavier is fake (heavier).

If {3,6,9} is heavier, then either 3 is heavy or 5 is light. Weigh {5} and {9}. They won’t balance. If {5} is lighter, 5 is fake (lighter). If they balance, 3 is fake (heavier).

If {5,6,7,8} is heavier, it is the same situation as if {1,2,3,4} was heavier. Just perform the same steps using 5,6,7 and 8. Unless maybe you are too lazy to try and reprocess the steps, then you continue reading the solution. Weigh {5,6,1} and {7,2,9} (Note: 9 is surely not fake).

If they balance, then either 8 is heavy or 3 is light or 4 is light. Following the last step from the previous case, we weigh {3} and {4}. If they balance, 8 is fake(heavier). If they don’t balance then whichever one is lighter is fake (lighter).

If {5,6,1} is heavier, then either 5 is heavy or 6 is heavy or 2 is light. Weigh {5} and {6}. If they balance, 2 is fake (lighter). If they don’t balance then whichever one is heavier is fake (heavier).

If {7,2,9} is heavier, then either 7 is heavy or 1 is light. Weigh {1} and {9}. If they balance, 7 is fake (heavier). If they don’t balance then 1 is fake (lighter).

Voila!!!!

Head is spinning yet. Take out a paper and try it out just to get things clear. Now that you know how to solve this problem, you are ready to take on this world.