Can you solve the counterfeit coin riddle? - Jennifer Lu
You’re the realm’s greatest mathematician,
but ever since you criticized
the emperor's tax laws,
you’ve been locked in the dungeon
with only a marker to count the days.
But one day you're suddenly brought
before the emperor,
who looks even angrier than usual.
One of his 12 governors has been convicted
of paying his taxes
with a counterfeit coin,
which has already made its way
into the Treasury.
As the kingdom's greatest mathematician,
you’ve been granted a chance to earn
your freedom by identifying the fake.
Before you are the 12 identical looking
coins and a balance scale.
You know that the false coin will be very
slightly lighter or heavier than the rest.
But the emperor’s not a patient man.
You may only use the scale three times
before you’ll be thrown
back into the dungeon.
You look around for
anything else you can use,
but there's nothing in the room,
just the coins,
the scale, and your trusty marker.
How do you identify the counterfeit?
Pause now to figure it out yourself!
Answer in 3
Answer in 2
Answer in 1
Obviously, you can't weigh each coin
against all of the others,
so you’ll have to weigh several coins
at the same time
by splitting the stack
into multiple piles,
then narrowing down where
the false coin is.
Start by dividing the 12 coins
into three equal piles of four.
Placing two of these on the scale
gives us two possible outcomes.
If the two sides balance,
all eight coins on the scale are real
and the fake must be among
the remaining four.
So how do you keep track of these results?
That’s where the marker comes in.
Mark the eight authentic
coins with a zero.
Now take three of them and weigh them
against three unmarked coins.
If they balance, the remaining unmarked
coin must be the fake.
If they don’t, draw a plus
on the three unmarked coins
if they’re heavier,
or a minus if they’re lighter.
Now take two of the newly marked coins
and weigh them against each other.
If they balance, the third coin is fake.
Otherwise, look at their marks.
If they are plus coins,
the heavier one is the impostor.
If they are marked with minus,
it's the lighter one.
But what if the first two piles
you way don't balance?
Mark the coins on the heavier side
with a plus
and those on the lighter side
with a minus.
You can also mark the remaining four
coins with zeros,
since you know the fake one
is already somewhere on the scale.
Now you'll need to think strategically
so you can remove all remaining ambiguity
in just two more ways.
To do this, you’ll need
to reassemble the piles.
One method is to replace three
of the plus coins
with three of the minus coins
and replace those
with three of the zero coins.
From here, you have three possibilities.
If the previously heavier side
of the scale is still heavier,
that means either the remaining
plus coin on that side
is actually the heavier one,
or the remaining minus coin on the
lighter side is actually the lighter one.
Choose either one of them and weigh it
against one of the regular coins
to see which is true.
If the previously heavier side
became lighter,
that means one of the three minus coins
you moved is actually the lighter one.
Weigh two of them against each other.
If they balance, the third is counterfeit.
If not, the lighter one is.
Similarly, if the two sides balanced
after your substitution,
then one of the three plus coins
you removed must be the heavier one.
Weigh two of them against each other.
If they balance, the third one is fake.
If not, then it's the heavier one.
The emperor nods approvingly
at your finding,
and the counterfeiting lord
takes your place in the dungeon.
