# 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.

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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

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

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Copyright © 1996-2018 Alexander Bogomolny