“I Don’t Know What The Two Integers Are.”

After several weeks of serious things involving decision theory, I tried to find a light-hearted puzzle for this week. Ideally it would be one involving statistics and a paradox. But I had no clue. So I did what any right-thinking person would do and googled a bit and found several versions of an old puzzle that emphasizes the value of knowing what is not there. I don’t know the original source. If anyone does, please let me know and I will give credit.

The puzzle is simple to state and solve, but like any puzzle I present here, it has some nice twists. This is my version. Assume a conference on puzzles. Some of the attendees are challenged to a contest. The moderator, Will Shortz, selects two contestants. Call them “The Multiplication Person” and “The Addition Person” or “M” and “A” for short. The moderator announces he has selected two integers from 1 to 9 (not zero). The contestants must discover the unknown integers. The only clues are that “M” has been shown the product of the two integers and “A” has been shown their sum.

The rules are that “A” and “M” can only communicate to each other by saying whether they have the answer or not. No other signals are permitted.

• “M” looks at his slip, shrugs his shoulders and says, “I don’t know what the two integers are.”
• “A” looks at his slip, thinks a moment and replies, “I don’t know either.”

Time passes.

• “M” looks at his slip, shrugs his shoulders and says, “I don’t know what the two integers are.”
• “A” looks at his slip, thinks a moment and replies, “I don’t know either.”

More time passes.

• “M” looks at his slip, shrugs his shoulders and says, “I don’t know what the two integers are.”
• “A” looks at his slip, thinks a moment and replies, “I don’t know either.”

The moderator is getting impatient.

• “M” looks at his slip, shrugs his shoulders and says, “I don’t know what the two integers are.”
• “A” looks at his slip, thinks a moment and replies, “I don’t know either.”

Finally “M” says, “I know the integers.” He writes the correct values on a blackboard and wins the applause of the crowd.

What is the solution and how did “M” find it?

The answer will appear next week (unless you cheat and look it up on Google), but this can help you get started. Note that before the communications start, Both “A” and “M” know there are 45 possible combinations of integers possible (why?), but only one is the correct answer. “M” knows that “A” can have 17 possible sums (why?). When “M” starts off by saying he doesn’t know the answer, “A” immediately eliminates 28 of the possible combinations (how?). And so it goes. [Note: if "A" is gutsy, he could guess the values immediately after "M's" first announcement because his knowledge of the sum (which we don't have) allows him to eliminate even more combinations, but not all. "A" obviously has a better than 1 out of 17 chance of winning with a guess.]

If you solve this puzzle, you can apply the same thought process to the game of Clue. Instead of depending only on what you see, learn from what others do not see.

In response to the interest my original tutorial generated, I have completely rewritten and expanded it. Check out the tutorial availability through Lockergnome. The new version is over 100 pages long with chapters that alternate between discussion of the theoretical aspects and puzzles just for the fun of it. Puzzle lovers will be glad to know that I included an answers section that includes discussions as to why the answer is correct and how it was obtained. Most of the material has appeared in these columns, but some is new. Most of the discussions are expanded compared to what they were in the original column format.

[tags]decision theory, integer, puzzle, brain teaser[/tags]