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.

Solution

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Copyright © 1996-2017 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 A1, A2, B1, B2, C1, C2. We have to replace the labels with more informative AL, AH, BL, BH, CL, CH, where "L" stands for "light" and "H" stands for "heavy".

Weigh A1 + B1 against B2 + C1.

  1. If A1 + B1 < B2 + C1 then

    • B1 < B2 because even if A1 < C1, B1 > B2 would lead to A1 + B1 = B2 + C1, at best. Thus B1 = BL, B2 = BH.

    • If A1 = AH then C1 = CH and

    • if C1 = CL then A1 = AL.

    With this in mind, weigh A1 + C1 against B1 + B2.

    • A1 + C1 = B1 + B2 implies A1 = AL and C1 = CH.

    • A1 + C1 < B1 + B2 implies A1 = AL and C1 = CL.

    • A1 + C1 > B1 + B2 implies A1 = AH and C1 = CH.

  2. If A1 + B1 = B2 + C1 then weigh B1 against B2.

    • B1 = BL implies A1 = AH and C1 = CL.

    • B1 = BH implies A1 = AL and C1 = CH.

  3. The case where A1 + B1 > B2 + C1 is similar to the first one.

Weighing Coins, Balls, What Not ...

  1. The Oddball Problem, B. Bundy
  2. Weighing 12 coins, Dyson and Lyness' solution
  3. Weighing 12 coins, W. McWorter
  4. Thought Less Mathematics, D. Newman
  5. Weighing with counterbalances
  6. Odd Coin Problems, J. Wert
  7. Six Balls, Two Weighings
  8. 12 Coins in Verse
  9. Six Misnamed Coins, Two Weighings
  10. A Fake Among Eight Coins
  11. A Stack of Fake Coins
  12. Five Coins - One Good, One Bad
  13. With One Weighing

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Copyright © 1996-2017 Alexander Bogomolny

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