Unfortunately, you've all been captured by the tyrannical rulers

and brought to the ancient colosseum for their deadly entertainment.

Before you're thrown into the dungeon,

you see many numbered hallways leading outside.

But each exit is blocked by an electric barrier

with a combination keypad.

You learn that one of you will be allowed to try to escape by passing a challenge

while everyone else will be fed to the mutant salamanders the next morning.

With her perfect logical reasoning, Zara is the obvious choice.

You hand her a concealed audio transmitter so that the rest of you can listen along.

As Zara is led away,

you hear her footsteps echo through one of the hallways,

then stop.

A voice announces that she must enter a code

consisting of three positive whole numbers in ascending order,

so the second number is greater than or equal to the first,

and the third is greater than or equal to the second.

She may ask for up to three clues,

but if she makes a wrong guess,

or says anything else,

she'll be thrown back into the dungeon.

For the first clue, the voice says the product of the three numbers is 36.

When Zara asks for the second clue,

it tells her the sum of the numbers

is the same as the number of the hallway she entered.

There's a long silence.

You're sure Zara remembers the hallway number,

but there's no way for you to know it,

and she can't say it outloud.

If Zara could enter the passcode at this point, she would,

but instead, she asks for the third clue,

and the voice announces that the largest number appears only once

in the combination.

Moments later, the buzz of the electric barrier stops for a few seconds,

and you realize that Zara has escaped.

Unfortunately, her transmitter is no longer in range,

so that's all the information you get.

Can you find the solution?

Pause on the next screen to work out the solution.

3

2

1

You're worried about the fact that you don't know Zara's hallway number,

but you decide to start from the beginning anyways.

From the first clue, you work out all of the eight possible combinations

that come out to a product of 36.

One of these must be right, but which one?

Now comes the hard part.

Even though you don't know which number you're looking for,

you decide to work out the sum of each combination's three numbers.

That's when it hits you.

All but two of the sums are unique,

and if the hallway number had matched any of these,

Zara would have known the correct combination right then and there

without asking for the third clue.

Since she did ask for the clue,

the hallway number must have matched the only sum

that appears more than once in the list:

thirteen.

But which of the two combinations that add up to thirteen is correct:

1,6,6,

or 2,2,9?

That's where the third clue comes in.

Since it tells us that the largest number must be unique,

2,2,9 must be the code.

When night falls, you and the others escape through hallway thirteen

and rejoin Zara outside.

You've freed yourselves through math and logic.

Now it's time to free the rest of the world.