Given that $\theta$ is an integer and that $X_1$ and $X_2$ are independent random variables which are Uniformly distributed on the integers $1, 2, \ldots, \theta$, prove that $X_1 + X_2$ is not sufficient for $\theta$.
I'm pretty new on the topic of sufficiency and not sure how to argue reasonably. We could start from showing $P(X =x \mid T = t)$ (where $X$ is vector of $X$s and $T$ is sufficient statistic) does not depend on theta, but sure if this is the best way. Any help is appreciated.
Suppose $X_1+X_2=5.$ One might think that means $$ (X_1,X_2) =(1,4) \text{ or } (X_1,X_2)=(2,3) \text{ or } (X_1,X_2) = (3,2) \text{ or } (X_1,X_2) = (4,1). $$ And if $\theta\ge4,$ then that is correct. But what if $\theta=3$? Then you'd have $$ (X_1,X_2) = (2,3) \text{ or } (X_1,X_2) = (3,2). $$ Thus the conditional distribution of $(X_1,X_2)$ given $X_1+X_2$ depends on $\theta.$