My probability is very rusty after a few years out of school it seems - I am sure I wouldn't have been stumped by this a while ago :(
I have two discrete, nonnegative random variables $X$ and $Y$. I know $P(X + Y = m)$ for all $m \geq 0$ and $P(X-Y = n)$ for all $n$.
Can I say anything about $P(X=x,Y=y)$? How about $P(X=x,Y=y|X+Y=m, X-Y=n)$?
Remark that $X + Y = m, X- Y= n$ implies $2X = m+n$ and $2Y = m-n$ so basically you have $$ \mathbb{P}(X = x, Y = y | X + Y = m, X - Y = n) = \mathbb{1}_{\{x = \frac{m+n}{2}\}} \mathbb{1}_{\{y = \frac{m-n}{2}\}}. $$ However the joint law of $(X + Y, X-Y)$ is definitively not the ``product'', in the sense that those two variables are of course not independent, so you can not conclude directly from the total probability formula I think. However you can probably say something for the single case $\mathbb{P}(X = x | X + Y = m)$, at least if $X$ and $Y$ are independent.