Very difficult math/logic puzzle.

Answer

Dear Martin,

This question is a lot easier than question 24834, though! So perfectly explainable.

Piet does not know the sum. This means that the product Piet knows is not the product of two prime numbers. Otherwise Pete would immediately know that they are the two prime factors, and their sum would then be Steve’s number. For example, if Pete got 35, he knows it can only be 5 and 7. Also, the product is not a cube of a prime number, because then the only possibility is that the pair of numbers are that prime number and its square. For example, if Pete got 27, he immediately knew it was 3 and 9.

Steve goes through all the possibilities for his sum. He says that he already knew that Piet does not know the sum. That means that all of Steve’s possibilities yield a pair of numbers where at least 1 is not prime and the pair is different from a prime number and its square. For example, if Steve got 13, the possibilities are: 6+7 (6 is not prime), 5+8 (8 is not prime), 4+9 (9 is not prime), 3+10 (10 is not prime) , 2+11 (two prime numbers, so 13 is not). So Steve got 13, he didn’t know from the beginning that Pete wouldn’t know, because it could be 2 and 11, and in this case Pete would have gotten 22 and would have known immediately that the numbers were 2 and 11. So the sum isn’t 13. But it’s not bigger either. We go through them all: the sum is at least 5 (I suppose if you say you take two numbers in your head you mean two different ones, otherwise you would have said 2 or 1; but also if the numbers can be equal, the reasoning holds: you then have to start with 4, and 4=2+2, twice the prime number 2). But 5=2+3 (two prime numbers), so the sum is not 5. Is the sum 6? 6= 2+4 (prime and its square). So it’s not 6. 7=5+2, 8=5+3, 9=7+2, 10=3+7, 12=5+7, so none of these are. So it must be 11. And indeed 11=2+9 (9 not prime), 3+8 (8 not prime), 4+7 (4 not prime), 5+6 (6 not prime).

So we already know the sum. But what is the product now? Possibilities: 18, 24, 28, 30. But Steve says he already knew the sum isn’t greater than 13. So, say, if he got 30, the numbers could be 2 and 15, and the sum would be 17, greater then 13. In that case he was not sure that the sum was at most 13, because it could be 17. If he got 28, it could be 14 and 2 (sum 16), if he got 24, then 12 and 2 (sum 14). So it must be 18 now!! And indeed, the possibilities here are 3×6 and 2×9, both sums 3+6 and 2+9 are not greater than 13! Since the sum is 11, the numbers are 2 and 9.

Have you been able to follow this how and why? Which two solutions were you actually hesitating between?

Regards,
Hendrik Van Maldeghem, Ghent University