1-20 of two numbers and told a, the product told B,a said do not know how much, B also said do not know, at this time a said I know, B said I also know, please guess this two number and how much

1~20 Choose 2 numbers and tell b the two numbers and the product of the 2 numbers.

Then ask a know how many of these two numbers are. A said do not know, ask B to know the two number is what? B also said not to know.

Then a suddenly reasoned out how many of these two numbers were, and told B he knew the answer.
B then also know how many of these two numbers are.


May I ask how they reasoned, and what are the 2 numbers?


Get the problem believe a lot of people are ignorant force, I also (just strange), last night in the roadside and other friends, see this problem, in the brain to think about, and then on the Internet to check the answer, a lot of people in seeking ideas, but get only the answer, not thinking. Based on my feeling that the problem is not simple, so dissect it.


In solving this problem, we need to have the ability to have deductive reasoning.


Interpretation:
1~20 Choose 2 numbers, these two numbers are definitely special numbers. If it's not special, then there's no problem.
Reasoning:
If it's not special, then there's no problem.


Interpretation:
A composition of this and the number must have multiple combinations.
Reasoning:
Otherwise a can not say at the beginning not to know, so these 2 numbers can not be (19,20), that is, this and the scope is "4~38" between.


Interpretation:
B said at the outset that I do not know this number, the two-digit product is certainly not a prime number.
Reasoning:
If it is prime, then you can directly infer that these two numbers are 1 and he itself, so these two numbers cannot be (1, prime numbers). So excluding all combinations of 1 and prime numbers, prime numbers in 1~20 are: 2,3,5,7,11,13,17,19. So the product of these two numbers cannot be 2,3,5,7,11,13,17,19.
In addition, we can know from 20 to choose 2 numbers there are 190 possible, here we just ruled out 8+2 species possible. We also have 180 possibilities, if we are A and B "absolutely" will be silent.


Is that really the case?
Are we really A and b?
No.....


Why.
Because a in B said "do not know" when the reasoning out of the two numbers are how many, from the 180 kinds of possible reasoning out of ...


Why.
Because we are not the same as a! IQ is different. No, we're smart.


What's the difference between us and AB?
A knowing the two and 180 combinations, B knows the product of these two numbers and 180 combinations, we only know there are 180 combinations.


So what a knows about this two-digit number is definitely very special.


Interpretation:
These 2-digit and, can decompose at least 2 combinations (for example, 5 can be decomposed into (1,4) (2,3)), and the combination of the decomposition can certainly belong to these 180 combinations, and can filter out the unique group data according to the existing known conditions.
Reasoning:
If the only set of data cannot be inferred from the existing conditions, then a cannot be immediately known.


Which condition is so special?
Interpretation:
It must be the latest in the answer from B.
Reasoning:
If it is not the latest reasoning, a will not answer after B to know the answer (because the IQ of a 200+), so this special condition is: The two number of the product is certainly not prime.


What combinations of arrays can be filtered out only by the "product of these two numbers is definitely not prime" condition.
Interpretation:
The product of almost all combinations is prime, and only the product of a single group is not a prime number.
Reasoning:
The product of 2 numbers is prime, the factor of prime number is 1 and he itself, that is, if the sum of these two numbers is fixed, the other addend is also the fixed value 1, the other addend must also be fixed value (for example, tell you, and is 20, one addend is 1, the other addend must be 19), Conclusion: This two-digit and can only be decomposed into 2 groups, one of which is (1, prime number).


We know that this number and must be in the "4~38", while this and certainly is the 1+ prime, then the 1+ prime number must not be more than 38, less than 38 is more than 4 of prime numbers by not a lot, so we can fully infer this and how much it might be. Then to these and decomposition, requirements can only be decomposed out of 2 groups is the result we want.
Prime number less than 38 greater than 4:5,7,11,13,17,19,23,29,31,37;
Then this and probably 6,8,12,14,18,20,24,30,32,38.2 addend cannot be broken after decomposition cannot have 1, while 2 addend are not the same, and both addend must be between 1~20.
And we've found that the larger the number, the more likely the decomposition will be, so we start with a 6 decomposition.


6 possible combinations (1,5) (2,4) (3,3). Exclude the last group and the first group, so 6 inferred that the two numbers are 2 and 4, their product 8 is not prime, which seems to meet our requirements.
If you don't rest assured, go down.
8 can be decomposed into (1,7) (2,6), (3,5) ....
12 can be decomposed into (1,11) ... Here we still give up the calculation, only 6 is our true love.

So we reasoned out that these two numbers are (2,4)

The complexity of the whole problem is very high and requires constant scrutiny, documentation, and a solution that really needs to be high on the brain.