Sum and Product Redux

Logic Level 4

Dan has written up 2 distinct positive integers, both of them are no greater than 10. Dan told Sam and Pam the sum and product of these 2 integers, respectively. Both Sam and Pam knew about the information in this paragraph.

The following conversation occurs.

Sam: I know that Pam doesn't know the 2 integers.
Pam: I know that Sam doesn't know the 2 integers.
Sam: I know that Pam doesn't know the 2 integers.
Pam: I know that Sam doesn't know the 2 integers.
.
.
.

This goes on and on for as long as possible until one of them shouted "Now I know the 2 integers!" and the other later shouted, "Me too!"

Assuming that Sam and Pam are perfect logician who never lied, which of the following is not a possible sum of the 2 integers?

Inspiration .

11 13 7 9

1 solution

Steven Perkins
May 22, 2017

If the sum were 11, then Sam would reason that the product could be 28 (4 x 7).

If it were, then Pam would know the 2 integers, since 4 and 7 are the only factors that satisfy the conditions.

That being the case, Sam would not have been able to state: I know that Pam doesn't know the 2 integers.

Yup, it's easy to eliminate 11 from the options.

You still have to prove that 7,9 and 13 cannot be the answer.

Pi Han Goh - 4 years ago

I now believe that 9 and 13 could be the answer (could not be Sam's sum). See Sid's response to my solution.

Steven Perkins - 4 years ago

Exactly no number except 7 can be Sams sum.

So When Sam says Pam doesn't know the 2 integers Pam will immediately know that Sam has the sum of 7.

Thus Pam will now know both the product and the sum of 7.

Thus he knows the 2 integers, However the pair of integers can be one of these

(1,6);(2,5);(3,4).

But only Pam will be able to tell the 2 integers.Not sure about how Sam gets the 2 integers after that...

A Former Brilliant Member - 4 years ago

(Correct me If I'm wrong).

If the sum were 9, then Sam would reason that the product could be 14 (2 x 7).

If it were, then Pam would know the 2 integers, since 2 and 7 are the only factors that satisfy the conditions.

That being the case, Sam would not have been able to state: I know that Pam doesn't know the 2 integers.

Same goes for sum of 13 that Sam would reason that product could be 36 (4 x 9), Other case for 13 is the product cant be 42 (6 x 7).

A Former Brilliant Member - 4 years ago

I believed you're missing out on a lot of other cases. Note that there are 45 possible pairs of (the 2 numbers).

Note that the solution provided by Steven is also missing out on a lot of steps (on proving that sum cannot be 11).

Pi Han Goh - 4 years ago

I think I'm getting something wrong, But here's what I got so far.

For Sam to be sure That Pam doesn't know the 2 integers, Sam will have to eliminate all the products that are "predictable" (i.e. knowing for sure what the 2 integers are by knowing the product).

Thus Pam can't know one of these products.

(2 = 1 x 2),(3 = 1 x 3),(4 = 1 x 4),(5 = 1 x 5),(7 = 1 x 7),(15 = 3 x 5),(14 = 2 x 7),(9 = 1 x 9) ,(16 = 2 x 8),(21 = 3 x 7),(28 = 4 x 7),(35 = 5 x 7),(36 = 4 x 9),(42 = 6 x 7),(45 = 5 x 9),(48 = 6 x 8),(50 = 5 x 10),(54 = 6 x 9),(56 = 7 x 8),(60 = 6 x 10),(63 = 7 x 9),(70 = 7 x 10),(72 = 8 x 9),(80 = 8 x 10),(90 = 9 x 10).

For Sam to be sure that Pam doesn't know the 2 integers.

The possible sums related to these products are 3,4,5,6,8,9,10,12,13,14,15,16,17,18,19. (i.e. all possible sums except 7).

Thus Sam must know the sum of 7.

But in that case I can't figure it out how either of them are going to figure out the 2 integers.

A Former Brilliant Member - 4 years ago

Hmm, I agree with you now that 9 and 13 also can NOT be the sum based on the cases you presented. Sam would not have been sure Pam didn't know as given in his very first statement.

11 can NOT be the sum based on the case (4 x 7) that I presented.

So I believe the question is flawed. There seem to be 3 correct answers.

Or I'm not understanding something important.

Steven Perkins - 4 years ago

Sum and Product Redux

1 solution

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