Can you solve the rogue submarine riddle?

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### The Mission: Smuggling and Hacking

Smuggling yourself aboard the rogue submarine was the easy part. Hacking into the nuclear missile launch override was a little harder. However, now you have 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.

### The Scenario: Countdown to Launch

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

### The Problem: A Puzzle with Limited Information

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?

### Working Through the Ignorance-Based Puzzle

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.

### Understanding Character A’s Perspective

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. Because the boss’s numbers must be less than 7, A’s list of B’s possibilities only has these 4 numbers.

### Understanding Character B’s Perspective and Solution

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

Discussion Questions

  1. How did the protagonist manage to smuggle themselves onto the rogue submarine without being detected?
  2. How does the relationship between Minion A and Minion B, and the lack of full trust from the boss, affect the dynamics of the situation?
  3. If you were the boss, would you have also separated the launch codes? Why or why not?
  4. How did the boss’s decision to share the sum of the integers with Minion A and the product with Minion B complicate the situation?
  5. What makes ‘ignorance-based’ puzzles like this one particularly challenging to solve?
  6. How did the protagonist’s understanding of potential factors and sums of numbers help them in deciphering the launch codes?
  7. Why were the numbers 3 and 4 eliminated from the list of potential sums?
  8. How does the ambiguity of B’s number increase the complexity of this situation, and what strategy did the protagonist use to navigate this complexity?

Lesson Vocabulary

**1. Smuggling** – *The act of moving goods illegally into or out of a country.* – In the scenario, the protagonist was successful in **smuggling** themselves onto the rogue submarine undetected.

**2. Override** – *To take control or make decisions, regardless of someone else’s decision, control, or previous settings.* – The protagonist had to hack into the missile’s launch **override** to prevent a disaster.

**3. Minion** – *A follower or underling of a powerful person, usually used to do their dirty work.* – The big boss gave one launch code to **Minion** A and the other to Minion B.

**4. Forbade** – *Past tense of forbid, to command a person or group not to do something.* – The boss **forbade** Minion A and Minion B to share their numbers with each other.

**5. Distinct** – *Clearly different or separate in quality or kind.* – The boss chose a set of **distinct** positive integers for the launch codes.

**6. Ambiguity** – *The quality of being open to more than one interpretation; inexactness.* – The **ambiguity** of B’s number forced them to go through a multi-step process.

**7. Insight** – *The capacity to gain a deep understanding of a person or thing.* – Understanding that B could conceivably reveal A’s number gave the protagonist a major **insight**.

**8. Contradicting** – *To assert the opposite of a statement made by someone else.* – If A knew that B already knew A’s number, that would be **contradicting** A’s initial statement.

**9. Perspective** – *A particular attitude toward or way of regarding something; a point of view.* – They had to consider the situation from both A’s and B’s **perspective**.

**10. Unintuitive** – *Not easily understood or grasped by intuition.* – Solving the puzzle required going through several **unintuitive** multi-step processes.

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