Not sure: finding counterfeit coin in three weighings

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The king has five bags of gold. He gave each of his most trusted people one bag. Now one contains counterfeit gold. The king has a scale to weight the gold. We knows for sure the counterfeit bag weighs less. The king wants to know which bag weighs a different amount. The king wants to use the scale only three times to figure this out.

Is this feasible?

I apologize if the above is confusing... :(
 
Hello, booniesgirl_29!

This is a classic problem.
(And I assume that the "scale" is a balance scale.)

The king has five bags of gold. \(\displaystyle \;\) He gave each of his most trusted people one bag.
Now one contains counterfeit gold. \(\displaystyle \;\)The king has a balance scale.
We knows for sure the counterfeit bag weighs less.
The king wants to know which bag weighs a different amount.
The king wants to use the scale only three times to figure this out.

Is this feasible? . . . Yes, but only two weighings are needed!
Label the bags: A, B, C, D, E.

First weighing: \(\displaystyle \;\) Balance A and B against C and D.

There are three possible outcomes.

\(\displaystyle [1]]\;AB\,=\,CD.\;\) Then \(\displaystyle E\) is the light bag.

\(\displaystyle [2]\;AB\,>\,CD.\;\) Then either \(\displaystyle C\) or \(\displaystyle D\) is light.
\(\displaystyle \;\;\;\)Second weighing: \(\displaystyle \;\)Balance \(\displaystyle C\) against \(\displaystyle D\) to determine the light bag.

\(\displaystyle [3]\;AB\,<\,CD.\;\) Then either \(\displaystyle A\) or \(\displaystyle B\)is light.
\(\displaystyle \;\;\;\)Second weighing: \(\displaystyle \;\)Balance \(\displaystyle A\) against \(\displaystyle B\) to determine the light bag.
 
what if you don't know if the counterfeit bag is lighter or heavier though? Can you still do it in three weighings?
 
booniesgirl_29 said:
what if you don't know if the counterfeit bag is lighter or heavier though? Can you still do it in three weighings?
YES.
(Using Soroban's way)
after 2 weighings you'll be left with a bag either lighter or heavier;
as 3rd weighing, simply weigh it against any other bag : capish?
 
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