Deduction puzzle

[Fredo Corleone]

Let's say that Gwern chooses two natural numbers above 1.
Bora knows the product and Leo know their sum, and the following conversation takes place:
Bora: I don't know the sum.
Leo: I knew that, the sum is less than 14.
Bora: I knew that. However I know the numbers.
Leo: And so do I.
What numbers did Gwern choose?

[Majd]

10 and 3?

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[Majd]

I maybe misinterpreting part of the question though, 

I’m assuming Bora saying “I knew that” meant Bora knew that Leo knew that Bora didn’t know the sum, as opposed to meaning that Bora knew that the sum is less than 14, which was just revealed by Leo

So I could be on the wrong path here

[Majd]

2nd attempt:

5 and 8

[Majd]

lol, that’s wrong too, 

ok, gotta go to sleep now, cool puzzle 

[Fredo Corleone]

This is indeed a cool puzzle and a very hard one IMO.

It's all about deduction

[Pedro Cooper]

It is impossible to determine the exact numbers that Gwern chose based on the information provided. The conversation between Bora, Leo, and Gwern does not contain enough information to determine the numbers.

To view this discussion on the web visit https://groups.google.com/d/msgid/brain-training/401abda9-d893-4c78-9edc-b4bbe17c6bb1n%40googlegroups.com.

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Pedro Cooper

University of Michigan, Class of 2017

Sociology

College of Literature, Sciences and the Arts

[Fredo Corleone]

Shut up Chat GPC

[Fredo Corleone]

By the way it's possible. But you need a brain to do that, a powerful fleshy human brain!

[Alex C.]

Leo knows the sum and Bora claims he doesn't know it. So they are opposites?

Bora's first claim and second claim seem contradicting. Is he lying about knowing the numbers?

From Leo's claim we get 55 pairs of numbers. I don't know.

[Leonardo]

There are some pairs of numbers that can be deleted, as they would have been obvious to Bora without further information. Think of every unique product of two numbers that sum less than 14. For example, 21 is a product that can result from 3 and 7, so it can't be 21 or Bora would know right away. That is how the problem operates, you go back and forth thinking about what of the two would know in every point in time.

The exercise is solvable this way, but it is extremely difficult. It took me more than an hour to arrive to the last possible numbers and I basically took a chance with the last two ones, but I got it right. 

[Fredo Corleone]

Beep beep beep... BUMP!

[M]

By the way it's possible. But you need a brain to do that, a powerful fleshy human brain!

Citation needed.

This python code produces a single answer.

def f(X, Y):
    # two numbers above 1
    assert 2 <= X
    assert 2 <= Y

    P = X * Y
    S = X + Y

    # I don't know the sum
    assert len({x + P // x for x in range(2, P // 2 + 1) if P % x == 0}) > 1

    # I knew that
    products = {x * (S - x) for x in range(2, S - 1)}
    assert all([len({x + p // x for x in range(2, p // 2 + 1) if p % x == 0}) != 1 for p in products])

    # the sum is less than 14
    assert S < 14

    # I knew that
    assert max({x + P // x for x in range(2, P // 2 + 1) if P % x == 0}) < 14

    print(X, Y)

for x in range(14):
    for y in range(x, 14):
        try: f(x, y)

        except: pass 

[Fredo Corleone]

Well done

[Tofe]

2 and 2?

[Sriharsha Darbha]

It has to be 6 and 7.

As Bran knew the product to be 42, the numbers could’ve been 14&3 or 6&7. When he came to know the sum to be <14, it’s became obvious that it was 6&7 to Bran.

Because Bran could reduce the numbers without a doubt after hearing <14, he could rule out the option of 10&3

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