## Tuesday, November 15, 2005

### whole numbers: m and n

Two whole numbers, m and n, have been chosen. Both are unequal to 1 and the sum of them is less than 100. The product, m × n, is given to mathematician X. The sum, m + n, is given to mathematician Y. Then both mathematicians have the following conversation:

X: "I have no idea what your sum is, Y."
Y: "That's no news to me, X. I already knew you didn't know that."
X: "Ahah! Now I know what your sum must be, Y!"
Y: "And now I also know what your product is, X!"

The Question: What are the numbers m and n?

1. For those of you who take the time to look, here's a gazoo for you.

2. m = toast
n = butter.

3. I think you need to have breakfast before you try these puzzles out. ;-)

4. I'm confused.. Does their conversation have to do with it??

5. It probably does, but i can't figure out what it means. He might not know if he has an even sum based on the fact that he can't factor the product into either two even numbers or two odd numbers.

6. Their conversation has everything to do with it. You have to follow the logic (this is not an easy one to do!)

7. I'm leaving some of the examples out, but this is the basic logic:

From the first remark (X: "I have no idea what your sum is, Y.") follows that a can be factorized in more than one way.

From the second remark (Y: "That's no news to me, X. I already knew you didn't know that.") follows that b cannot be written as the sum of two prime numbers.

From the third remark (X: "Ahah! Now I know what your sum must be, Y!") we conclude that the number a that X has got, is apparently found at only one value for b.

From the fourth remark (Y: "And now I also know what your product is, X!") we conclude that for the number b that Y has got, there is apparently only one value of a possible.

Conclusion: the numbers m and n are 4 and 13.

8. LOL that one was way over my head.

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