My Number is $136$
Ana and Boris play a logic game. They both know that possibly identical positive integers, $a$ and $b$, are chosen by a third party. Their product is written on Ana’s forehead and their sum is written on Boris’s forehead. Each can see the other’s number but not his or her own. After each sees the number on the other’s forehead, one says to the other, “There is no way you can know your number.” The other responds, “I now know my number is $136$.”
According to the WSJ, here is the putative solution.
My Number is $136$ answer explained:
$a = 1$ and $b = 135$. Ana sees $136$ on Boris and knows whatever product Boris sees $(1 \times 135,\ 2 \times 134,\ 3 \times 133$,\ etc.), he can’t know his number. Boris sees $135$ on Ana and considers he may have $3 + 45 (= 47 + 1),\ 5 + 27 (= 31 + 1),\ 9 + 15 (= 23 + 1)$, or $1 + 135$. The first three possibilities are eliminated by adding to one more than a prime (which would disallow Ana from making her statement) so Boris deduces his number is $136$.
I THINK THERE IS AN ERROR IN THE LOGIC OF THE ANSWER.
Clearly we are to assume that both Ana and Boris are both math whizzes and that Ana, too, could deduce that if $135$ happens to be written on her forehead, then Boris can deduce that $136$ is written on his forehead. So she simply cannot make the claim that "whatever product Boris sees, he can't know his number."
It would seem that the problem could be much better posed.
Let $A$ be the number that Boris sees on Ana's forehead : as long as $A$ is not prime, it can be written as the product of two smaller numbers $x$ and $y$, which would give Boris a sum of $x+y$, but also as the product of $1$ and $A$, which would give Boris the (different) sum $1+A$. So the only way Boris could find the correct sum on his forehead would be if $A$ is prime, and then this number would be $A+1$. Since Ana can see that $136=135+1$ but $135$ is not a prime, she knows that Boris cannot know the number.