Can you solve the counterfeit coin riddle? - Jennifer Lu

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You’re the realm’s greatest mathematician, but ever since you criticized the Emperor’s tax laws, you’ve been locked in the dungeon. Luckily for you, one of the Emperor’s governors has been convicted of paying his taxes with a counterfeit coin, which has made its way into the treasury. Can you earn your freedom by finding the fake? Jennifer Lu shows how.

Lesson by Jennifer Lu, animation by Artrake Studio.

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 twelve 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 twelve 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 here if you want
to figure it out for 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 twelve 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 imposter.

If they are marked with minus,
it's the lighter one.

But what if the first two piles you weigh
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 weighings.

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.