Ghost Valley and Split Problem

1, Pang Juan can be sure that Sun Bin certainly doesn't know these two numbers, so there are several inferences.

(a) Pang Juan's hand number is between 5- 197.

(2) The sum of Pang Juan must not be divided by the sum of two prime numbers, otherwise there is no certainty. This can be divided into two points:

Pang Juan's hand is not even, only odd, because any even number greater than 4 can be divisible into the sum of two odd prime numbers, which is guaranteed by Goldbach's conjecture;

And the odd number in Pang Juan's hand is not 2+ prime number. For example, if Pang Juan has 28, it can be divided into 1 1+ 17. When Sun Bin gets the product of 18 1,

It can be guessed immediately that Guiguzi gave him two numbers, 1 1 and 17, which contradicted Pang Juan's assertion that Sun Bin didn't know these two numbers, so he ruled out all even numbers.

For example, the number in Pang Juan's hand is prime number +2, such as 2 1, but it happens to be 19+2, so the number in Bin Sun's hand is 38, and the decomposition method is only 2* 19.

So Sun Bin was able to confirm these two figures at the beginning.

(c) The sum of Pang Juan shall not be an odd number greater than 53. Because odd numbers greater than 53 can always be decomposed into the product of even numbers and 53 (which is a prime number),

This product can only uniquely infer the product of 53 and even number, otherwise it is greater than 99. The other 97 is a prime number,

Similarly, all odd numbers from 97+2 to 97+98 should be excluded. In the end, the odd numbers of 99+98 are left, because they are all the largest numbers.

Sun Bin could have been inferred, which contradicted the premise that Sun Bin didn't know, and was naturally excluded.

So odd numbers above 53 can be excluded. For example, if the number in Pang Juan's hand is 59, it may be 53+6.

When Sun Bin gets 3 18, there is only one decomposition method, which is 53*6, because 106*3 and 159*2 are both greater than the maximum number of 99.

So this contradicts Sun Bin's prior uncertainty. It can also be inferred that all odd numbers in 195=97+98 are excluded, because 97 is a prime number.

So when Pang Juan is an odd number above 53, there is no such certainty. Sun Bin certainly doesn't know these two figures.

(d) Only the number 10 satisfies the above conditions: 1 1, 17, 23, 27, 29, 35, 37, 47, 5 1, 53.

2. Sun Bin knew the product in his hand, saying that he didn't know it at first, but now he knows it. That is to say,

Bin Sun looked at the sum of all the combinations corresponding to the factorization factor after the product in his hand, and it can only be one of the above 10 numbers.

That is to say, the product of 10 sum is not the product of other sums, so it may be the product of Sun Bin.

There are many kinds of this product, and the key is Pang Juan's third sentence.

Pang Juan knows the sum in his hand. When Sun Bin said this, Pang Juan said that he also knew these two numbers.

The sum number in Pang Juan's hand has a characteristic, that is, except one possible product, all other possible products can't satisfy the above.

Otherwise, Pang Juan has no such confidence. That is to say, in the combination of 10 and 0, only one logarithm can satisfy the previous condition.

At this time, it is necessary to combine the second condition. How to use this condition? Take 17 as an example:

Suppose the decomposition is 3+ 14, then the product is 52, while 42=3* 14=2*2 1=6*7, and the corresponding sum is17,23, 13.

Where 17 and 23 are both candidate solutions, that is to say, if the number in Sun Bin's hand is 42, he can't know the correct decomposition.

So 17 cannot be decomposed into 3+ 14. Similarly, the following decomposed list satisfying the second condition can be constructed:

Possible decomposition of 1 1: (4,7), (3,8), (2,9),

Possible decomposition of 17: (4, 13),

Possible decomposition of 23: (10, 13), (7, 16), (4, 19),

Possible decomposition of 27: (13, 14), (1 1, 16), (10, 17), (9,/.

Possible decomposition of 29: (13, 16), (12, 17), (1,18), (/kloc)

Possible decomposition of 35: (17, 18), (16, 19), (14,21), (12.

Possible decomposition of 37: (17,20), (/kloc-0,6,2/kloc-0,27), (9,28), (8,29), (6,3/)

Possible decomposition of 4 1: (19,22), (18,23), (17,24), (16,25), (/kloc-0)

(9,32),(7,34),(4,37),(3,38),

Possible decomposition of 47: (23,24), (22,25), (20,27), (19,28), (18,29), (17,30), (650)

( 10,37),(7,40),(6,4 1),(4,43),

Possible decomposition of 53: (26, 27), (25, 28), (24, 29), (23, 30), (22, 3 1), (2 1, 32), (20, 33).

( 17,36),( 16,37),( 15,38),( 13,40),( 12,4 1),( 10,43),(8,45),(6,47),(5,48),

Only 17 has the only feasible decomposition, so Pang Juan can determine the number in hand.

So the answer to this question is 4 13.