The host chose 2 numbers x and y such that: - both of them are whole numbers - they are not smaller than 2, but not bigger than 100 - they are not the same number He gave the result of summing these numbers to the player no.1 and he gave the result of multiplying these numbes to the player no.2 2- i do not know what those numbers are 1- i do not know either 2- well... if you do not know, then I know what they are! 1- then I know them, too
What are these numbers?
A cleaner version than the OP follows:
1) Ann has been told the product and says: "I don't know the numbers."
2) Bob has been told the sum and says: "I knew you didn't know the numbers."
3) Ann says: "Now I know the numbers."
4) Bob says: "Now I also know the numbers."
Solving this puzzle requires sifting through all pairs of numbers and ruling them out all but one. F.i., 1) + 2) implies that the two numbers cannot be a prime number pair (otherwise Ann would know the numbers by their product).
The answer is that the two numbers are 4 and 13, with sum 17.
This puzzle is the theme of Chapter 7 ("Sum and Product") in: H. van Ditmarsch and B. Kooi, One Hundred Prisoners and a Light Bulb, 2015. See this reference for variants and a detailed argument.