Can you solve the riddle of Pandora’s box? - Alex Rosenthal

Pandora was having a decisively bad day.

She didn’t set out to open the box;

if anything, she’d resisted with every ounce of her will.

But her curiosity got the better of her,

and a cavalcade of evils of every shape and size spilled out.

By the time she'd slammed the front door of her home shut,

only 10 minor imps— of wrath, greed, gluttony, and sloth varieties—

and three greater demons remained.

Pandora tried desperately to get them under control,

but they resisted and pounded at the windows.

“This is hopeless!” she cried.

The first greater demon turned to Pandora and said,

“Hope? Hope is still here. I just made her look like an imp.

Because that’s the kind of bad guy I am!”

He whispered in the ears of the two other demons and continued.

“To make things interesting,

I just told Abaddon which variety of imp Hope is,

and Beelzebub how many eyes she has. Bye!”

Abaddon taunted Beelzebub by saying,

“I know you don’t know which one she is.”

Beelzebub responded, “Well, I didn’t before, but now I know.”

To which Abaddon replied in a huff, “Then I also know which one she is.”

And both left in a hurry, without so much as a farewell.

Pandora knows her only chance to fix this mess is to capture Hope

before the imps all escape her home.

But which one is she?

Pause here to figure it out yourself. Answer in 3

Answer in 2

Answer in 1

This problem originally took the world by storm in 2015

when Dr. Joseph Yeo Boon Wooi posed it as Cheryl’s birthday problem,

and for good reason.

The logic requires simultaneously modeling three different perspectives—

ours, Abaddon’s and Beelzebub’s, whom we’ll call A and B from now on.

This puzzle is much more approachable if we start by doing what Pandora could not:

cramming all of these imps into a box...

in this case, of the table variety.

One key thing to note is that unlike many logic problems,

the order of statements matters here,

because the two demons’ knowledge changes over the course of the dialogue.

So let's start with the first statement

and milk it for all it's worth before moving on.

In it, A says to B, “I know you don’t know which one she is.”

At this point, A knows the imp type,

but since there are multiple imps of each type, A doesn’t know who Hope is.

The eyes, however, work differently;

if Hope had 3 or 5 eyes, B would know who she is.

This step is where a lot of people succumb to a common pitfall.

Because the important takeaway here isn’t to just eliminate

the three- and five-eyed imps.

It’s to eliminate the entire rows of the table that they’re in.

Why?

The trick is to think about B’s perspective from A’s perspective.

Through A’s eyes, the entire table boils down to one of these four options,

because he knows which type the imp is.

If he knew Hope was a greed imp, she could have five eyes.

There’s only one imp with five eyes, so B could know who Hope is.

But A knows that B doesn’t know,

so greed isn’t possible, nor, by the same reasoning, is wrath.

Now we’re down to just five possible imps, and we can move on to the next statement,

which we have to consider through B’s eyes.

Critically, B is an expert logician, so anything we’ve figured out—

like that wrath and greed are off the table—

he’s also figured out.

That, plus knowing how many eyes Hope has,

allows him to figure out who she is.

So she must be in a column that has only one imp left,

meaning we can eliminate both one-eyed imps.

Finally, let’s examine the last statement and switch back to A’s perspective.

For A to now know who Hope is,

there must be only one option left in that row.

Therefore, Hope must be the six-eyed sloth imp.

Before anyone else can escape,

Pandora leaps into action and wrestles Hope to the ground.

The world may be a mess,

but at least she’s got a glimmer of... well, you know.