# Six Balls, Two Weighings

There are six balls of the same shape, but of three colors - two per color. For every pair of monochromatic balls, one is lighter. All light balls are of the same weight, and so are all the heavy balls. Use 2 weighings on a balance scale to determine which balls are light and which are heavy.

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Copyright © 1996-2018 Alexander Bogomolny

### Solution

There are six balls of the same shape, but of three colors - two per color. For every pair of monochromatic balls, one is lighter. All light balls are of the same weight, and so are all the heavy balls. Use 2 weighings on a balance scale to determine which balls are light and which are heavy.

Assume the colors are A, B, C and the balls labeled A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}. We have to replace the labels with more informative A_{L}, A_{H}, B_{L}, B_{H}, C_{L}, C_{H}, where "L" stands for "light" and "H" stands for "heavy".

Weigh A_{1} + B_{1} against B_{2} + C_{1}.

If A

_{1}+ B_{1}< B_{2}+ C_{1}thenB

_{1}< B_{2}because even if A_{1}< C_{1}, B_{1}> B_{2}would lead to A_{1}+ B_{1}= B_{2}+ C_{1}, at best. Thus B_{1}= B_{L}, B_{2}= B_{H}.If A

_{1}= A_{H}then C_{1}= C_{H}andif C

_{1}= C_{L}then A_{1}= A_{L}.

With this in mind, weigh A

_{1}+ C_{1}against B_{1}+ B_{2}.A

_{1}+ C_{1}= B_{1}+ B_{2}implies A_{1}= A_{L}and C_{1}= C_{H}.A

_{1}+ C_{1}< B_{1}+ B_{2}implies A_{1}= A_{L}and C_{1}= C_{L}.A

_{1}+ C_{1}> B_{1}+ B_{2}implies A_{1}= A_{H}and C_{1}= C_{H}.

If A

_{1}+ B_{1}= B_{2}+ C_{1}then weigh B_{1}against B_{2}.B

_{1}= B_{L}implies A_{1}= A_{H}and C_{1}= C_{L}.B

_{1}= B_{H}implies A_{1}= A_{L}and C_{1}= C_{H}.

The case where A

_{1}+ B_{1}> B_{2}+ C_{1}is similar to the first one.

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Copyright © 1996-2018 Alexander Bogomolny