S.O.S. Mathematics CyberBoard

I have the following question which needs help from Experts here.

In the final round of a computer game, two players, Professor X and Magneto played a game. Both of them just remember their own score. After that they had been told that the difference of their score was 100 marks. After knowing this

Magneto: i don't know Professor X's score.
Professor X: I also don't know Magneto's score.
Magneto: Now, I know Professor X's score, but if both of us score an extra one mark, I may not know the score.

If the minimum score of the game is 1 mark and the score is always an integer, find the total score of Professor X and Magneto.

Please guide me on how to solve the above question. And where can I find more of this type of questions to practice on?

What I have is:
Let m be Magneto's score, x be Professor X's score,
so m+100=x, or x+100=m,
then what is next??

The difference is 100 points.
If one of them has x points, where x>100, then the other person has x-100 or x+100 points.
If one of them has 100 points or fewer, then the other must have x+100 points.

So Magneto's first statement tells you that he has at least 101 points.
Professor X's statement tells you that he, too, has at least 101 points, but he now knows that Magneto has at least 101 points, but is still unable to work out if Magneto has more or fewer points, therefore Professor X must have at least ...
Magneto now deduces that Professor X must have more points than him. Magneto's score is as high as possible while permitting him to do that, so it must be ...

Moved from Proposed Problems to Miscellaneous

_________________

Hi aswoods,
Thank you very much for your guide.
But what i have been doing for the pass hoursss is to bang my head on a wall..i can see through that wall now...

I seriously cannot understand:
1) Why the statements means that they both knew each other has >101 points?
[quote="aswoods"]
If one of them has x points, where x>100, then the other person has x-100 or x+100 points.
If one of them has 100 points or fewer, then the other must have x+100 points.
[/quote]

From the above, I guess we have an equation of : M=X+100,
so the possible answers for that could be:
1+100=101
2+100=102
3+100=103
.
.
.

2) Which part of the statement from Magneto tells that he deduces that Professor X's score is higher than his score?

If one of them has fewer than 101 points, and each has at least one point, then because someone won by 100 points, the person with fewer than 101 points knows he lost, and since he knows the winner won by 100 points, he knows the other player has his score + 100. So both have at least 101 since they neither knew who won.

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

"Magneto: Now, I know Professor X's score, but if both of us score an extra one mark, I may not know the score."
What do you think about the above statement?

They both know their own scores.

If you know you got 50 points, and then somebody says the difference was 100, then the other guy must have got 150 points (because there aren't any zero or negative scores; -50 points isn't possible).

So if neither Magneto nor Professor X can (at first) figure out what score the other got, they must both have at least 101 points each.

And if Professor X hears that Magneto can't work it out, he can deduce that Magneto has 101 points or more. But Professor X still can't work it out, which means that Professor X must have 201 points or more (if he had 150, say, he would know that Magneto had 250, as 50 has been ruled out).

Magneto figures this out, and now (3rd statement) he apparently knows what Professor X's score must have been. Therefore, knowing that Professor X has at least 201 points is enough information. But if Magneto has 301 points or more, then it can't be enough information, because Professor X could have either 201 or 401 points. So Magneto must have fewer points.

Thank you aswoods for clarifying things out. That makes a lot more sense to me now.
By the way, is there a particular topic to search for this kind of questions? (Do I search for the keyword "logic"?) I want to learn more on these type of questions.

I've seen them called induction puzzles. In general, diagrams are used to solve them rather than algebra. There is a systematic approach in mathematical logic called public announcement theory (or dynamic epistemics, or common knowledge) but it is very hard to learn and apply.

This sort of problem is more along the lines of recreational mathematics and you may have trouble finding textbooks or anything like that. If you want a lot of practice, you could go to a newsagent and look for booklets of logic puzzles near the crossword section. Most of the puzzles are of a similar level of difficulty to that one, and there may be public announcement problems among them.