Required Mathematical Intelligence

Chapter 66

Mr. Q is playing games with Mr. S and Mr. P. Mr. Q wrote a number on two small pieces of paper.Both numbers are positive integers and the difference is 1.
He stuck one piece of paper on Mr. S's forehead and the other on Mr. P's forehead.Therefore, two people can only see the number on each other's forehead.

Mr. Q kept asking: Can any of you guess the number on your head? Mr. S said: "I can't guess." Mr. P said: "I can't guess either." Mr. S said: "I still can't guess." Mr. P said: "I can't guess either." S Mr. still can't guess, and Mr. P can't guess either. Both Mr. S and Mr. P have failed to guess three times.However, at the fourth time, Mr. S shouted: "I know!" Mr. P also shouted: "I know too!"

Question: What numbers are on the heads of Mr. S and Mr. P?

[Answer: "I can't guess." There is an important message in this sentence.

If Mr. P has 1 on his head, of course Mr. 5 knows that he has 2.5 on his head. The first time Mr. 1 said "I can't guess", it is tantamount to telling Mr. P that the number on your head is not [-].
At this time, if Mr. S has a 2 on his head, Mr. P knows that he should have a 3 on his head. However, when Mr. P says "I can't guess", it is equivalent to saying: Mr. S, you don't have a 2 on your head.
The second time Mr. S said he couldn’t guess it, it was equivalent to saying: Mr. P’s head is not 3, if so, I must have 4 on my head, and I can guess it.

Mr. P said he couldn’t guess it, which meant that Mr. S’s head was not 4. Mr. S said he couldn’t guess it, which meant that Mr. P’s head was not 5. Mr. P said he couldn’t guess it, which meant that Mr. S’s head was not 6.
Why did Mr. S guess it now?It turned out that Mr. P had a 7 on his head. Mr. S thought: Since there is not a 6 on my head, he has a 7 on his head, and of course an 8 on my head!
Then Mr. P also understood: he can guess that it is 6 from the fact that his head is not 8, of course it is because I have 7 on my head!

In fact, even if the two people have 100 and 101 written on their heads, as long as they repeatedly exchange information and say "I can't guess", they can always guess in the end.

There is another confusing thing about this kind of question: at the beginning, when Mr. P saw the 8 on the opponent's head, he knew for sure that his head would not be 1, 2, 3, 4, 5, 6; Mr. S will also know that 1, 2, 3, 4, 5 will not be on his head. In this way, the first few sentences of the two "can't guess", and the exchange of information is definitely useless.But it is wrong to say that it is useless, because if one sentence is missing, you will end up guessing wrong. ]
(End of this chapter)