Is The Abbreviated Solution Proper?

Recently I presented a puzzle that relied on making assumptions to solve. It came on the tail end of a piece about the dangers of jumping into solving mode without looking at the underlying assumptions. The puzzle:

Two mathematicians are given slips of paper by a third colleague.

To the first mathematician, he says, “The number on your slip is the product of two numbers. They can be the same number or different, but they are whole integers.” To the second mathematician he says, “The slip I gave you has the sum of the same two numbers.”

The first mathematician says, “Well, I don’t know what the two numbers are.”

The second mathematician says, “Yeah, I don’t know either.”

Then the first one says, “Now I know what they are.”

The second one says, “And so do I.”

What are the numbers?

So we assume that both mathematicians are quick with numbers. That seems safe enough. Let’s also assume the puzzle has an honest answer. That assumption is trickier. It often works well enough in recreational puzzles, but is not a very good assumption in real life problems. In real life, there might not be an answer. Remember that some time ago I presented the puzzle that a sphere has a hole drilled through the center. The hole is 1 inch long measured after the material is removed. What is the volume of material left in the sphere after the drilling? This puzzle can be solved by brute force, but by assuming there is a solution, the answer can easily be worked out in your head.

The next assumption is to assume that the two numbers are reasonably small. The puzzle wouldn’t make much sense to have two 10-digit numbers. Again, that seems reasonable.

We assume both mathematicians are telling the truth.

Finally, we assume that the response of the second mathematician about his uncertainty gives the first one a clue, and the fact that he did get a clue enables the second one to solve the puzzle also.

Given these assumptions, then we can start to work our way through some combinations to see if anything works. A few moments of thought will show you that neither number can be zero. If either one were zero, the product would be zero, and that doesn’t give any clues to the second mathematician. The smallest combination that is possible is 1 and 1, but if the first mathematician would know the answer since there is only one way he/she could have 1 be the product of two numbers.

Okay, all zeros are ruled out, and 1 and 1 is ruled out. Suppose the product is 2. Again, there is only one way to get it, so we reject that hypothesis. Now try 3 and you get the same result. That brings us to 4, which could be 2 X 2 or 4 X 1. The sum would be either 4 or 5. But if the sum is 4, the product could be 3 or 4. It can’t be 3 because the first mathematician did know the answer. Therefore if the product is 4, the second mathematician would know his/her sum was composed of 2 + 2, but that possibility is ruled out because the first mathematician didn’t know the answer. What if the sum is 5? Then the product could be 4 or 6, and the second mathematician doesn’t know the answer. Since the first mathematician is looking at a 4, and the second one doesn’t know the answer, the 4 cannot be composed of 2 X 2 by the above reasoning. Therefore he/she knows that the numbers must be 1 and 4. When he/she says so, the second mathematician follows a line of reasoning that shows that 6 cannot be the product because then the first mathematician would not have deduced the answer.

So 1 and 4 work for the two numbers to give a product of 4 and a sum of 5. However, I have not proven this solution is unique. Is it? And have I done it correctly?

Of course we could have assumed that there is only one answer and that it is a low number. After eliminating 0, 1, 2, and 3 as possible products, we could have guessed 4 as being the lowest possible product and therefore it must be the correct one. Then the sum must be 5. Is that abbreviated solution proper?

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.