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I was watching this video about a "Rogue submarine riddle" and wondering about the first part of the solution for the problem. The idea is that you have two people $A$ and $B$ that are given codes and the "boss" who gave the codes then spits out the following paragraph

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.”

In the solution the explainer states that

The only scenarios where $B$ could know $A$'s number is when there is exactly one way to factor $B$'s number.

He then states that the "pattern" here is that $B$'s number must be a prime of the square of a prime.

I don't quite understand why is this the case? $B$'s number can be anything of the form $nk$, where $1 \le n,k\le 6$ so how can we reduce this to the case where the number must be either a prime or a square of a prime?

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  • $\begingroup$ One complicating issue is that $1$ is allowed as a factor. Thus, for instance, $8$ does not settle the issue for $B$. $8$ could be the result of $2\times 4$ or it could be the result of $1\times 2\times 4$. (it can't be $1\times 8$ because $8$ is too big, and it can't be $2\times 2\times 2$ because the factors must be distinct). No way for $B$ to tell, without more information. $\endgroup$
    – lulu
    Jul 2 at 23:58

1 Answer 1

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A's sum number is at least $1+ 2=3$ and B's product number is at least $1\times 2=2$. Similarly A's sum number is no more than $1+2+3+4+5+6=21$ and A's sum number is no more than $1 \times2 \times3 \times4 \times5 \times6=720$. But B having a product number of $720$ would not tell B what A's sum number was, as it could be $21$ or it could be $2+3+4+5+6=20$. Developing this idea:

  • If B's product number were prime i.e. $p$ then $B$ would know that is was the product $1 \times p$ and so would know A's sum number was $1+p$

  • If B's product number were the square of a prime i.e. $p^2$ then $B$ would know that is was the product $1 \times p^2$ (it could not be $p \times$ something as that something would include multiplying by a second $p$) and so would know A's sum number was $1+p^2$

  • If B's product number were some other number, say $ab$ with $2\le a \le b$ then $B$ would not know whether it was $a\times b$ or $1 \times a \times b$ (and in some cases there would be further possibilities) and so would not know whether A's sum number was $a+b$ or $1+a+b$ (or possibly something else)

So if A (knowing the sum number) cannot tell whether or not B can work out the sum number, then the sum number must allow a product number which is prime or the square of a prime (in which case B can work out A's sum number) and must allow a product number which is neither (in which case B cannot work out A's sum number). A's initial thinking is:

  • A sum number of $3$ would not do as the only product number would be $1\times 2=2$ allowing B to work out the sum number is $1 + 2$
  • A sum number of $4$ would not do as the only product number would be $1\times 3=3$ allowing B to work out the sum number is $1 + 3$
  • A sum number of $5$ would do as the product number could be $1\times 4=4$ allowing B to work out the sum number is $1 + 4=4$, or the product number could be $2 \times 3=1\times 2 \times 3=6$ not allowing B to work out whether the sum number is $2 + 3=5$ or $1+2 + 3=6$
  • A sum number of $6$ would do as the product number could be $1\times 5=5$ allowing B to work out the sum number is $1 + 5=6$, or the product number could be $1\times 2 \times 3=6$ not allowing B to work out whether the sum number is $2 + 3=5$ or $1+2 + 3=6$, or the product number could be $2 \times 4=8$ not allowing B to work out whether the sum number is $2 + 4=6$ or $1+2 + 4=7$
  • A sum number of $7$ would not do as the product number could be $1\times 6=6$ or $1\times2\times4 =8$ or $2 \times 5=10$ or $3 \times 4=10$, none of which are prime or the square of a prime (put another way, $7$ is not $1$ more than a prime or $1$ more than the square of a prime), so A knows that B cannot work out the sum number from the product number.
  • Any larger sum number that is actually possible with elements less than $7$ runs into the same issue

So from A's initial statement we know that the sum number must be $5$ or $6$. Importantly for the rest of the puzzle, not only do we know that B know this, but also that B also knows that A knows B knows.

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