- 06 Mar '11 00:10I give two smart guys each a piece of paper with a number on. I tell them that their numbers are consecutive integers and between 1 and N (inclusive). They can only see their own number.

Their conversation is as follows;

1st guy: "I dont know your number"

2nd guy: "I dont know your number"

1st guy: "I dont know your number"

2nd guy: "I know your number"

What was N? - 06 Mar '11 00:56

Clarification: Have you told them what N is?*Originally posted by wolfgang59***I give two smart guys each a piece of paper with a number on. I tell them that their numbers are consecutive integers and between 1 and N (inclusive). They can only see their own number.**

Their conversation is as follows;

1st guy: "I dont know your number"

2nd guy: "I dont know your number"

1st guy: "I dont know your number"

2nd guy: "I know your number"

What was N? - 06 Mar '11 05:54 / 4 editsFrom the 1st guy's first comment, we know n != 1.

From the 2nd guy's first comment, we know n!=2.

From the 1st guy's second comment, we know n!=3.

From the 2nd guy's second comment, we know n=4.

I assumed neither guy made any logical or factual errors in their reasoning or their comments. - 06 Mar '11 11:25

1st guy: "I don't know your number"*Originally posted by wolfgang59***I give two smart guys each a piece of paper with a number on. I tell them that their numbers are consecutive integers and between 1 and N (inclusive). They can only see their own number.**

Their conversation is as follows;

1st guy: "I dont know your number"

2nd guy: "I dont know your number"

1st guy: "I dont know your number"

2nd guy: "I know your number"

What was N?

2nd guy knows 1st guy's number is not 1 or N

2nd guy: "I don't know your number"

1st guy knows 2nd guy's number is not 1 or 2, or N or N-1

1st Guy: "I don't know your number"

2nd Guy know 1st guy's number is not 1 or 2 or 3, or N or N-2 or N-3.

2nd Guy: "I know your number"

2nd guy must have either 3 (so he knows first guys number must be 4), or 4 (so he knows first Guy's number must be 5), or N-3 (so he knows first Guys number must be N-4), or N-4 (so he knows first guys number must be N-5)

We know N must be at least 7, but it can be any number larger than that.

I think we need some other bit of information to get it such as:

"If I now told you the value of N you would know exactly what the 2nd guys number was". - 06 Mar '11 12:21

Yep, that was exactly my reasoning as well.*Originally posted by iamatiger***2nd guy must have either 3 (so he knows first guys number must be 4), or 4 (so he knows first Guy's number must be 5), or N-3 (so he knows first Guys number must be N-4), or N-4 (so he knows first guys number must be N-5)**

We know N must be at least 7, but it can be any number larger than that. - 06 Mar '11 19:37

I think tht's the right logic but is there a mistake at this step?:*Originally posted by mtthw***Yep, that was exactly my reasoning as well.**

Quote:

1st Guy: "I don't know your number"

2nd Guy know 1st guy's number is not 1 or 2 or 3, or N or N-2 or N-3.

Unquote

shouldn't it be:

"2nd Guy know 1st guy's number is not 1 or 2 or 3, or N or N-1 or N-2."?

I think this leads to 1st guy's number being 4 and N being 7. - 06 Mar '11 19:57

Yes, I agree iamatiger made a typo there. But I do not agree this leads to the answer N = 7, although it is consistent with N = 7. The problem still seems under-determined.*Originally posted by JS357***I think tht's the right logic but is there a mistake at this step?:**

Quote:

1st Guy: "I don't know your number"

2nd Guy know 1st guy's number is not 1 or 2 or 3, or N or N-2 or N-3.

Unquote

shouldn't it be:

"2nd Guy know 1st guy's number is not 1 or 2 or 3, or N or N-1 or N-2."?

I think this leads to 1st guy's number being 4 and N being 7.

The content of your correction to iamatiger's typo is certainly consistent with the 1st guy's number being 4 and N being 7. But, isn't is it also consistent with the 1st guy's number being 4 (or 5) and N being basically any integer larger than 7? Iamatiger's reasoning still seems good, despite his little typo. - 06 Mar '11 20:06

That was trhe answer I was after but my initial problem did lead to multiple answers. I should have phrased it thus;*Originally posted by JS357***I think tht's the right logic but is there a mistake at this step?:**

Quote:

1st Guy: "I don't know your number"

2nd Guy know 1st guy's number is not 1 or 2 or 3, or N or N-2 or N-3.

Unquote

shouldn't it be:

"2nd Guy know 1st guy's number is not 1 or 2 or 3, or N or N-1 or N-2."?

I think this leads to 1st guy's number being 4 and N being 7.

1st guy: "I dont know your number"

2nd guy: "I dont know your number"

1st guy: "I dont know your number"**3rd guy to 1st guy:**"I know your number"

What was N? - 06 Mar '11 20:24

Well, I've bomber out on other P&Ps.*Originally posted by LemonJello***Yes, I agree iamatiger made a typo there. But I do not agree this leads to the answer N = 7, although it is consistent with N = 7. The problem still seems under-determined.**

The content of your correction to iamatiger's typo is certainly consistent with the 1st guy's number being 4 and N being 7. But, isn't is it also consistent with the 1st guy's ...[text shortened]... ny integer larger than 7? Iamatiger's reasoning still seems good, despite his little typo.

But take a look at this:

Quote (with corrected line):

2nd Guy know 1st guy's number is not 1 or 2 or 3, or N or N-1 or N-2.

2nd Guy: "I know your number"

The only way guy 2 can KNOW the number guy has, is if he has converged on the same number from both directions. The one number that is converged on from both directions is the number that is both one greater than 3, and is also one less than N-2, but this is the case if and only if it is 4 and N = 7. It is also consistent with guy 1 still not knowing whether guy 2 has a 3 or a 5, even though, all along, having the 4, he knew guy 2 had one or the other. - 06 Mar '11 21:31 / 5 edits
*Originally posted by JS357***Well, I've bomber out on other P&Ps.**

But take a look at this:

Quote (with corrected line):

2nd Guy know 1st guy's number is not 1 or 2 or 3, or N or N-1 or N-2.

2nd Guy: "I know your number"

The only way guy 2 can KNOW the number guy has, is if he has converged on the same number from both directions. The one number that is converged on from bot ...[text shortened]... 2 has a 3 or a 5, even though, all along, having the 4, he knew guy 2 had one or the other.**The only way guy 2 can KNOW the number guy has, is if he has converged on the same number from both directions.**

I disagree. The way the problem was initially offered, N could be 7 but it could also be any integer over 7.

Maybe look at it this way. Let's suppose you are Guy2. You know your number is 3 (suppose). And you also know that your number and Guy1's number are consecutive integers. So, if you also then find out that Guy1's number is not 2 (which is of course entailed by his number's not being 1 or 2 or 3), then you know his number is 4. This is all consistent with your also knowing that his number is not 7 or 6 or 5, but it is also perfectly consistent with your also knowing that his number is not (say) 7, 777, 777 or 7,777,776 or 7,777,775. So, N could be 7 or it could be 7,777,777. (Or take your pick of any other integer above 7.)

EDIT: By the way, as wolfgang has clarified, the convergence from both sides would be necessary for his modified problem. Wolfgang's modified problem works (does constrain the problem to N = 7) as long as it is assumed that the observer guy3 starting out knew N and also knew that the other two players have consecutive integers between 1 and N, inclusive. But the key difference is that this third observer starting out does not know guy2's number whereas guy2 of course does; hence, one problem (with guy2's annoucing he knows guy1's number) does not constrain N for us, whereas the other problem (with guy3 announcing he knows guy1's number) does. - 06 Mar '11 21:56

OK Gotcha. Still learning...*Originally posted by LemonJello***[b]The only way guy 2 can KNOW the number guy has, is if he has converged on the same number from both directions.**

I disagree. The way the problem was initially offered, N could be 7 but it could also be any integer over 7.

Maybe look at it this way. Let's suppose you are Guy2. You know your number is 3 (suppose). And you also know that your ...[text shortened]... N for us, whereas the other problem (with guy3 announcing he knows guy1's number) does.[/b]