Crack the Code: Solving the Rogue Submarine Riddle

You've infiltrated a rogue submarine, and the fate of the world hangs in the balance. Your mission: override a nuclear missile launch. The challenge? You need the correct authorization codes, but one wrong move locks you out. This isn't just about luck; it's about logic, deduction, and understanding the intricate dance of information.

The Submarine Scenario

Imagine this: you're deep within enemy territory, staring at a console that controls global destruction. The launch sequence is initiated, and time is running out. The submarine commander, in a twisted game, divided the launch code between two minions, forbidding them from sharing their numbers directly. Each minion entered their respective code, unaware of the other's input, only knowing that the numbers are distinct positive integers less than 7.

The Crucial Clues

Just as the countdown accelerates, the commander reveals a critical piece of information: the launch codes are related. He gave the sum of the numbers to Minion A and the product to Minion B. A tense silence fills the control room. Minion A declares, "I don’t know whether you know my number." B, after a moment of contemplation, retorts, "I know your number, and now I know you know my number too."

With mere minutes remaining, can you decipher the codes and avert disaster?

Decoding the Logic

This puzzle isn't about brute force; it's about stepping into the minds of Minion A and Minion B. It's about understanding what they know, what they don't know, and how their statements reveal hidden truths.

A's Initial Statement: A Glimmer of Hope

A's statement, "I don’t know whether you know my number," is the first breadcrumb. It implies that there's a possibility B could deduce A's number, but it's not a certainty. This hinges on the factors of B's number. If B's number has only one valid factorization, B would immediately know A's number. This occurs when B's number is:

• A prime number (e.g., 2, 3, 5)
• The square of a prime number (e.g., 4 = 2 * 2)

In these cases, there's only one possible sum. Since the numbers are less than 7, A compiles a list of potential numbers B could have: 2, 3, 4, or 5.

Narrowing Down the Possibilities

To consider that B could have those numbers, A's number must be a sum of their factors (3, 4, 5, or 6). However, we can eliminate 3 and 4. Why? Because if A held either of those numbers, B's number would be either 2 or 3. In that case, A would know that B already knows A's number, contradicting A's initial statement. Therefore, A must be holding either 5 or 6.

B's Revelation: The Final Piece

Now, let's analyze B's statement: "I know your number, and now I know you know my number too." This is where the puzzle truly unravels.

• Scenario 1: A's number is 5

  • A could have arrived at 5 through 1 + 4 or 2 + 3.
  • This means B would have either 4 or 6.
  • If B had 4, there's only one way to make the product: 4 * 1. B would know A's number.
  • If B had 6, it could be 1 _ 6, 2 _ 3. The sums would be 7 and 5. 7 isn't on A's list, but 5 is. B wouldn't know if A had 5 or 6.
  • Since B does know A's number, we can eliminate the possibility of A having 6.

• Scenario 2: A's number is 6

  • A could have arrived at 6 through 1 + 5, 2 + 4, or 1 + 2 + 3.
  • This means B would have 5, 8, or 6, respectively.
  • If B had 5, he'd know A had 6.
  • If B 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.
  • If A had 6, he still wouldn’t know whether B had 5 or 8. That contradicts the second half of what B said.

The Solution

Therefore, the only possible solution is that A has 5 and B has 4. The launch codes are 5 and 4.

Conclusion

With seconds to spare, you input the codes, override the launch, and save the world. This puzzle demonstrates the power of logical deduction and the importance of understanding how information is shared and withheld. It's a reminder that even in the most complex situations, clear thinking can prevail.