Comments on: Where's a good logic puzzle discussion forum?

Question: Where's a good logic puzzle discussion forum?

sappidus — Sat, 11 Oct 2008 17:16:52 -0800

I recently ran across a wonderful logic puzzle, but no solution was provided. I eventually worked out a complete solution myself, but it was rather ugly, and I would like to see if people smarter than me can come up with something more elegant. Are there any good Internet puzzle discussion forums floating around?

I note that I already posted the puzzle to The Grey Labyrinth, a wonderful site, but I was looking for other such places.

And I wouldn't want to rob Ask MeFi of the puzzle itself, so I'll put it here. It's a minor modification of a problem from Julian Havil's lovely book, "Impossible?"

("Natural numbers" here means positive integers.)

Someone secretly picks two natural numbers (c and d, not necessarily distinct) and approaches two perfect logicians, A and B. A is told the sum of the two numbers, i.e. the number (c+d). B is told the sum of the squares of the numbers, i.e. the number (c^2+d^2). Both know the nature of the information conveyed but don't know the number given to the other person. The subsequent conversation (referring to the two natural numbers c and d) between them runs as follows:

B: I do not know the numbers.
A: I do not know the numbers.
B: I do not know the numbers.
A: I do not know the numbers.
B: I do not know the numbers.
A: I do not know the numbers.
B: I do not know the numbers, so...
A and B simultaneously: ...we'll never know them.

Why does it take so long for A and B to conclude this?

(The original problem, available here, is superior in that the answer seems to drop out of the sky from nowhere, but I like how my formulation begins in ignorance, ends in ignorance, has a whole lot of ignorance in between, and yet conceals a surprising amount of reasoning.)

By: 0xFCAF

0xFCAF — Sat, 11 Oct 2008 17:28:42 -0800

xkcd's forums aren't bad for this sort of thing. Someone will probably show you a solution written in Python.

By: andoatnp

andoatnp — Sat, 11 Oct 2008 17:36:14 -0800

Can you provide your solution?

By: sappidus

sappidus — Sat, 11 Oct 2008 17:37:47 -0800

The margins here are too narrow to contain it.

By: sappidus

sappidus — Sat, 11 Oct 2008 17:39:56 -0800

(I'm just kidding, but it is rather involved and, like I said, ugly. I'll let this post stew for a bit just in case anyone wants to think about the problem without my inelegant approach leading them astray.)

By: andoatnp

andoatnp — Sat, 11 Oct 2008 17:40:17 -0800

Can you post it somewhere and link to it or explain the gist of it?

By: b33j

b33j — Sat, 11 Oct 2008 17:52:40 -0800

I've found two websites that refer to this puzzle, and one says that the numbers are 8 and 9 (scroll down to Question 4), and the other that the numbers are 4 and 13. The second site linked seems to have an explanation that makes sense for why the logician could work it out.

By: TedW

TedW — Sat, 11 Oct 2008 17:53:58 -0800

I could swear there was a puzzler on car talk similar to this, but I have so far been unable to find it. Perhaps someone else remembers it with more detail.

By: b33j

b33j — Sat, 11 Oct 2008 17:54:24 -0800

And here is a discussion of the issue.

By: sappidus

sappidus — Sat, 11 Oct 2008 18:04:47 -0800

Fine links, b33j, but the 4 and 13 one is a different problem. Indeed, the chapter in Havil's book that I mentioned concerns itself with the solution to that problem and offers up the current one as an elaboration. The current problem is quite a bit different due to the fact that there are seemingly no bounds specified!

The answer to the Havil problem is indeed {8,9}, but proving that it is unique is the real issue here. My modification is intended to throw the spotlight away from finding A numerical solution (easily done with a quick computer simulation) and to instead force the would-be solver to consider why it is The solution. This can be conveniently done by just adding one more "I don't know".

I will try to come up with a brief outline of the basic idea behind my approach to a solution and post it here.

By: sappidus

sappidus — Sat, 11 Oct 2008 18:32:30 -0800

Aha, actually, drilling down through a page linked off one of beej's links leads to a reference that the almighty JSTOR has archived: Mathematics magazine problem. So thanks! The solution therein is pretty similar to mine but is indeed a little more elegant.

Anyway, here's a brief description of how I looked at it:

Let's denote by S_n the set of (unordered) pairs {c,d} it could still be after the n-th "I don't know". So, for example, {1,2} is not a member of S_1. Why? Because B, who in that case would know the number 5 (=1^2+2^2), knows that 5 can only be represented as the sum of two squares in one way -- {1,2} -- and therefore could not honestly say, "I don't know the numbers".

What I was looking for a proof of is that S_7 = S_8 = S_9 = ... In other words, after "I don't know" #7, no additional information can be gathered.

It is perhaps not obvious that information can be gathered earlier than that, i.e., that S_1 != S_2 != S_3, etc. But that there is information there is what the original problem depends on, as it is basically asking for the unique member of S_6\S_7 (where backslash denotes the set difference operator).

All the S_n are infinite sets, by the way. Yet there is no need to set explicit bounds in the statement of the problem. That makes it quite a fascinating problem to me. Even better is that there is a reason behind the number of "I don't know"'s in the original Havil formulation (6): any more, and it would not work! Quite surprising. (It does work for 3, 4, and 5, though.)

So, the core issue is figuring out enough of the structure of the S_n to be able to solve the problem. It can be shown that for certain upper bounds for the sum c+d, the structure guarantees that a brute-force search for solutions below that upper bound is sufficient, i.e., no new information is gained by searching higher. The upper bound I had derived was 40. The Mathematics magazine derives a lesser upper bound of 22 (in a single sentence that hides quite a lot of reasoning, admittedly).

By: sappidus

sappidus — Sat, 11 Oct 2008 18:36:18 -0800

(I am still interested in good puzzle discussion forums, by the way.)

By: FuManchu

FuManchu — Tue, 14 Oct 2008 14:11:29 -0800

[ wu :: riddles ] was pretty fantastic when I could get to it. Link isn't working for me anymore, though.

By: FuManchu

FuManchu — Tue, 14 Oct 2008 22:50:37 -0800

It's back up! Oh, and a corrected link: LINK