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# Hat guess the number

## Question

Two persons A and B are each wearing a hat with a number written on it. Neither A nor B can see the
number on their own hat but they can see the number on the other’s. A sees number 5 on B’s hat and
B sees number 4 on A’s hat. They are told that A has the product of two positive integers written on
her hat and that B has the sum of the same two numbers written on her hat.
First B is asked whether she knows for sure what the two numbers are. If her answer is ‘no’, A is
asked the same question and so on until someone answers ‘yes’. Assuming that both are perfect
logicians and answer truthfully, who says ‘yes’ first and when?

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B : No
A : No
B : Yes

Spoiler

B: see 4, there are 2 possibility : 1 x 4 or 2 x 2. so B do not know the 2 numbers
so B first say "No"

A: see 5, there are 2 possibility : 1 + 4 or 2 + 3.
From B answer, A still can't deduce the 2 numbers.
if B see 6 (2*3) B will say No (2 possibility 1*6,2*3)
if B see 4 (1*4) B will say No (2 possibility 1*4,2*2)
so A say "No"

if A see 4 from (2*2) there are 2 possibility : 1 + 3 or 2 + 2.
but A must say yes, because if the numbers are 1 and 3, B will answer "yes" imediately
so A must deduce the numbers are 2 and 2 in the first try.

So A must be see 5, and B deduce the numbers must be 1 and 4

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How and why

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On 2/18/2016 at 7:33 PM, sujith said:

How and why

B: see 4, there are 2 possibility : 1 x 4 or 2 x 2. so B do not know the 2 numbers
so B first say "No"

A: see 5, there are 2 possibility : 1 + 4 or 2 + 3.
From B answer, A still can't deduce the 2 numbers.
if B see 6 (2*3) B will say No (2 possibility 1*6,2*3)
if B see 4 (1*4) B will say No (2 possibility 1*4,2*2)
so A say "No"

if A see 4 from (2*2) there are 2 possibility : 1 + 3 or 2 + 2.
but A must say yes, because if the numbers are 1 and 3, B will answer "yes" imediately
so A must deduce the numbers are 2 and 2 in the first try.

So A must be see 5, and B deduce the numbers must be 1 and 4

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