Last week I presented a puzzle in which the solution required looking at what people don’t know to find the answer. In short, the puzzle requires finding two integers, both of which are in the range 1-9. Two contestants, “The Multiplication Person” and “The Addition Person” or “M” and “A” try to find two hidden integers. The moderator has selected two integers from 1 to 9 (not zero). The only clues are that “M” has been shown the product of the two integers and “A” has been shown their sum. “A” and “M” can only communicate by saying whether they have the answer or not. No other signals are permitted.

- “M” looks at his slip, says, “I don’t know the answer.”
- “A” looks at his slip, and replies, “I don’t know either.”

- “M” says, “I don’t know the answer.”
- “A” replies, “I don’t know either.”

- “M” says, “I don’t know the answer.”
- “A” replies, “I don’t know either.”

- “M” says, “I don’t know the answer.”
- “A” replies, “I don’t know either.”

“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?

I gave a hint last week by pointing out that before anyone spoke, 45 combinations are possible. If the first digit is a 1, there are nine possibilities for the second digit. If the first digit is 2, there are 8 possibilities for the second digit, and so on. The progression adds to 45.

When “M” first says he doesn’t know the answer, then “A” knows that the integers cannot be (9, 9) because there is only one way to get a product of 81, and if “M” had seen 81, he would have known the answer. By similar reasoning, “A” concludes other pairs such as (1, 1), (1, 2), (9, 8), (8, 8), etc. can be eliminated. I said last week that “A” can narrow the field to 17 possibilities this way. Is this correct?

By a similar process, “M” considers that the sum “A” has seen can only be one of 17 numbers (1+1 through 9+9 or 2-18). However if the sum was 2, then the integers must be 1, 1; so that is impossible because “A” responded that he did not know the answer. Sums of 3 and 18 can be eliminated in the same way. Other possibilities can be eliminated just by “A” not knowing the answer, but when “M” realizes that by saying he didn’t know the answer, “A” can use this new information to solve the problem if the sum had some certain values. What are they? Since “A” didn’t know the answer even given the additional information that reduced the possibilities, “M” can eliminate some of sums, but he still doesn’t know the answer, and he says so in his second statement. “A” assumes that “M” has done the math correctly therefore the answers that would have been forced by his first statement of not knowing are incorrect. They proceed this way each giving the other additional information by acknowledging they were unable to compute the answer until only the correct answer of (2, 8) was possible. That is, the product was 16 and the sum 10.

There might be a quicker way to derive this, but I think the brute force method of sequentially eliminating contenders is the most straightforward.

Part of the charm of this puzzle for me is that the solution depends on the assumption of competence by each player of the other. That assumption is enough to pass information leading to a solution when it looks hopeless.

Suppose “A” had gone first. What would the result be then?

*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]integer, puzzle, decision theory, math puzzle, math[/tags]