Five Coins - One Good, One Bad

There are four coins of which one is counterfeit. There is a fifth coin known to be good. Find the counterfeit coin with two weighings on a balance scale with two pans.

Solution

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Copyright © 1996-2018 Alexander Bogomolny

There are four coins of which one is counterfeit. There is a fifth coin known to be good. Find the counterfeit coin with two weighings on a balance scale with two pans.

Denote the four coins of which one is the counterfeit A, B, C, D, and the fifth - good - coin G.

Solution 1

Weigh, say, {A, B} against {C, G}.

In case of balance, weigh, say, D against G. D is counterfeit; the weighing will show whether it's heavier or lighter than a normal coin.

Otherwise, weigh, say, {A, C} against {D, G}.

In case of balance, B is counterfeit and is lighter or heavier depending the result of the previous weighing.

If the balance position the same as before, A is counterfeit and is lighter or heavier depending the result of the previous weighing.

If the balance position is reversed, C is counterfeit and is heavier or lighter depending the result of the previous weighing.

Solution 2

Same first weighing; same conclusion in case of balance. In the absence of balance, either one of {A, B} is heavier and C is lighter, or one of {A, B} is lighter and C is heavier. This is resolved by weighing A against B.

References

  1. B. Averbach, O. Chein, Problem Solving Through Recreational Mathematics, Dover, 2000, #9.7

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 © 1996-2018 Alexander Bogomolny

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