Balance Problem

Problem

math tutoring balance
You are given 10 balls, which are all identical. One ball is either heavier or lighter than the rest. Using a balance three times, can you determine which ball is different, and whether the weight is heavier or lighter?

This requires no advanced math or tricky way of using the balance. Think it’s too easy? Try the same problem with 12 balls.

Solution

Here is the solution for the 12 ball problem; the solution for the 10 ball problem is a simple extension.

First place four balls on each side of the balance.  2 possible situations occur: the scale tips (A)or the scale doesn’t tip(B).

Situation A

You know four balls which may be heavy (H1-4), for balls that may be light (L1-4), and four which are normal (N1-4).  On the left side of the scale, place H1-3 and L1, and on the right side, place N1-3 and H4 on the right side.

If the left side drops, then one of H1-3 is responsible.  Compare H1 and H2, if neither side drops, H3 is different.

If the right side drops, then either H4 or L1 is different.  Compare either with a normal ball.

If neither side drops, then one of L2-4 is different.  Compare H2 and H3.

Situation B

Neither side of the scale tips, therefore we have N1-8 and D1-4 (D may be either heavier or lighter).  On the left side place D1-3, on the right place N1-3.

If the left side goes up or down, then you have L1-3 or H1-3, test as above.

If neither side moves, compare D4 to any normal ball.

Depending on how the situation plays out, one of these methods will give you the answer in three tries.

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