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

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

Weighing Coins, Balls, What Not ...

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