Imagine you’ve snuck onto a submarine and managed to hack into the missile launch system. But there’s a catch: you need a special override code to stop the launch, and you don’t have it. The only way to get it is by figuring out the two numbers used to authorize the launch. If you guess wrong, you’ll be locked out.
Here’s what you know: The leader of the submarine didn’t trust anyone with the full launch code. Instead, he gave one number to Subordinate A and another to Subordinate B. They weren’t allowed to share their numbers with each other. When the launch order came, each entered their number, and the countdown started. Now, there’s only 10 minutes left before the missiles launch.
The leader then mentions something interesting: “Your launch codes are related. I chose a set of distinct positive integers, each less than 7, and told their sum to A and their product to B.” After a pause, A says to B, “I don’t know if you know my number.” B thinks for a moment and replies, “I know your number, and now I know you know my number too.”
This puzzle is all about understanding what A and B know. Let’s break it down:
When A says, “I don’t know if you know my number,” it means that B could have a number that reveals A’s number, but it’s not certain. This gives us a clue: B’s number must be something that can be factored in only one way to reveal A’s number.
For example, if B’s number is a prime number, it can only be factored as 1 and itself. Or, if it’s a perfect square like 4, it can only be factored as 2 and 2. Numbers like 8, which can be factored in multiple ways (2 and 4, or 1, 2, and 4), are too ambiguous.
Since the numbers must be less than 7, A’s possible sums for B’s number are 3, 4, 5, or 6. We can eliminate 3 and 4 because if the sum was either, the product would be 2 or 3, and A would know B’s number, contradicting A’s statement. So, A’s number must be 5 or 6.
B says, “I know your number, and now I know you know my number too.” This means B has figured out A’s number based on the product he knows. If A’s number was 5, it could be from 1+4 or 2+3, giving B the product 4 or 6. If B had 4, he would know A’s number because 4 can only be 1 and 4. But if B had 6, it could be factored in multiple ways, so B wouldn’t be sure.
If A’s number was 6, it could be from 1+5, 2+4, or 1+2+3, giving B the product 5, 8, or 6. If B had 5, he would know A’s number is 6. If B had 8, the possibilities for A would be 2+4 and 1+2+4, and only 6 fits.
After analyzing all possibilities, the only numbers that fit both A’s and B’s statements are 5 and 4. With this knowledge, you enter the override code, stop the launch, and save the day!
This puzzle teaches us about logical reasoning and how to deduce information based on limited knowledge. It’s a great exercise in thinking critically and understanding different perspectives.
Imagine you’re Subordinate A or B. Pair up with a classmate and role-play the scenario. Discuss what each of you knows and try to deduce the numbers based on the information given. This will help you understand the logic behind each character’s statements.
Using the same principles from the submarine riddle, create your own logic puzzle involving numbers. Share it with the class and see if they can solve it. This activity will enhance your understanding of logical reasoning and puzzle creation.
Play a game where you list numbers and their possible factors. Work in groups to identify numbers that have unique factorizations versus those with multiple possibilities. This will reinforce your understanding of how B deduced A’s number.
Write a short story where you are a detective solving a mystery using math and logic. Incorporate elements from the submarine riddle, such as deducing information from limited clues. Share your story with the class to practice creative writing and logical thinking.
Participate in a workshop where you solve a series of logic puzzles. Work in teams to discuss strategies and solutions. This will help you develop critical thinking skills and learn how to approach complex problems systematically.
Sure! Here’s a sanitized version of the transcript, removing any sensitive or potentially harmful content while maintaining the essence of the puzzle:
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Smuggling yourself aboard the submarine was the easy part. Hacking into the 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 were just used to authorize the launch. But one wrong answer will lock you out.
From your hiding spot, you’ve learned the following: The leader didn’t trust any subordinate with the full information to launch missiles on their own. So, he gave one launch code to Subordinate A, the other to Subordinate B, and forbade them from sharing 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 launch.
Suddenly, the leader 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 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.
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.
Let’s start with A’s first statement. It means that B could conceivably have something that reveals A’s number, but isn’t guaranteed to. This 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.
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 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 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 unintentive 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.
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 mind 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, 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.
To summarize, if 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 launch, escape, and send the submarine to safety.
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This version maintains the puzzle’s integrity while ensuring it’s appropriate for all audiences.
Submarine – A term not typically used in mathematics, but can metaphorically describe a number or variable that is hidden or submerged within an equation or expression. – In solving the equation, the variable x was like a submarine, hidden beneath layers of numbers and operations.
Number – A mathematical object used to count, measure, and label, which can be an integer, fraction, or decimal. – The number 7 is an integer that can be used to represent a quantity or position in a sequence.
Product – The result of multiplying two or more numbers together. – The product of 4 and 5 is 20.
Sum – The result of adding two or more numbers together. – The sum of 8 and 12 is 20.
Integers – Whole numbers that can be positive, negative, or zero, but do not include fractions or decimals. – The set of integers includes numbers like -3, 0, and 7.
Factor – A number that divides another number without leaving a remainder. – The factors of 12 are 1, 2, 3, 4, 6, and 12.
Prime – A number greater than 1 that has no positive divisors other than 1 and itself. – The number 5 is a prime number because its only factors are 1 and 5.
Square – The result of multiplying a number by itself. – The square of 6 is 36.
Possibilities – The different outcomes or results that can occur in a mathematical situation or problem. – When flipping a coin, there are two possibilities: heads or tails.
Reasoning – The process of thinking about something in a logical way to form a conclusion or judgment. – Using reasoning, she determined that the solution to the equation was x = 3.
