Not sure: finding counterfeit coin in three weighings

[Guest]

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...

 

[soroban]

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.

 

[Guest]

what if you don't know if the counterfeit bag is lighter or heavier though? Can you still do it in three weighings?

 

[Denis]

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?