To summarize, if A had 6,

Smuggling yourself aboard the rogue. submarine was the easy part.

Hacking into the nuclear missile launch. override— a little harder.

But now you’ve got a problem: . you don’t have the override code.

You know you need . the same two numbers

that the agents of chaos just used. to authorize the launch.

But one wrong answer will lock you out.

From your hiding spot, . you’ve been able to learn the following:

The big boss didn’t trust any minion . with the full information

to launch nuclear missiles on their own.

So he gave one launch code to Minion A, . the other to minion B,

and forbade them to share . the numbers with each other.

When the order came,

each entered their own number . and activated the countdown.

That was 50 minutes ago,

and there's only 10 minutes left. before the missiles launch.

Suddenly, the boss says, “Funny story— . your launch codes were actually related.

I chose a set of distinct positive. integers with at least two elements,

each less than 7, and told their sum. to you, A, and their product to you, B.”

After a moment of awkward silence,. A says to B,

“I don’t know whether you know my number.”

B thinks this over, then responds,

“I know your number, and now I know. you know my number too.”

That’s all you’ve got.

What numbers do you enter . to override the launch?

Pause now to figure it out for yourself.. Answer in 3

Answer in 2

Answer in 1

Ignorance-based puzzles like this are. notoriously difficult to work through.

The trick is to put yourself . in the heads of both characters

and narrow down the possibilities . based on what they know or don’t know.

So let's start with A's first statement.

It means that B could conceivably. have something with the potential

to reveal A’s number,. but isn’t guaranteed to.

That doesn’t sound very definitive, . but it can lead us to a major insight.

The only scenarios where B could know. A’s number

are when there’s exactly one valid way . to factor B’s number.

Try factoring a few . and you’ll find the pattern—

It could be prime— where the product. must be of 1 and itself—

or it could be the product of 1 . and the square of a prime, such as 4.

In both cases, there is exactly one sum.

For a number like 8, factoring it . into 2 and 4, or 1, 2, and 4,

creates too many options.

Because the boss’s numbers . must be less than 7,

A’s list of B’s possibilities. only has these 4 numbers.

Here’s where we can conclude a major clue.

To think B could have these numbers,. A’s number must be a sum of their factors—

so 3, 4, 5, or 6.

We can eliminate 3 and 4,. because if the sum was either,

the product could only be 2 or 3,

in which case A would know that B . already knows A’s number,

contradicting A’s statement.

5 and 6, however, are in play,

because they can become . sums in multiple ways.

The need to consider this is one of . the most difficult parts of this puzzle.

The crucial thing to remember . is that there’s no guarantee

that B’s number is on A’s list—

those are just the possibilities . from A’s perspective

that would allow B to deduce A’s number.

That ambiguity forces us to go through. unintuitive multi-step processes like:

consider a product, see what sums. can result from its factors,

then break those apart . and see what products can result.

We’ll soon have to do something similar . going from sums to products

and back to sums.

But now we know— . when A made his first statement,

he must have been holding. either 5 or 6.

B has access to the same information . we do,

so he knows this too.

Let’s review what’s in each. brain at this point:

everyone knows a lot about the sum,. but only B knows the product.

Now let’s look at the first part . of B’s statement.

What if A’s number was 5?

That could be from 1+4 or 2+3,

in which case B would have. either 4 or 6.

4 would tell B what A had, like he said,

because there’s only one option to make. the product: 4 times 1.

6, on the other hand, could be broken . down three ways, which sum like so.

7 isn’t on B’s list of possible sums,. but 5 and 6 both are.

Meaning that B wouldn’t know. whether A’s number was 5 or 6,

and we can eliminate this option . because it contradicts his statement.

So this is great— 5 and 4. could be the override code,

but how do we know it's the only one?

Let’s consider if A’s number was 6—

which would be 1+5, 2+4, or 1+2+3,

giving B 5, 8, or 6, respectively.

If B had 5, he’d know that A had 6.

And if he had 8, the possibilities for A . would be 2+4 and 1+2+4.

Only 6 is on the list of possible sums,. so B would again know that A had 6.

he still wouldn’t know whether. B had 5 or 8.

That contradicts the second half . of what B said,

and 5 and 4 must be the correct codes.

With seconds to spare you override . the missile launch,

shoot yourself out of the torpedo bay,

and send the sub . to the bottom of the ocean.