Consider $X_{1} \dots X_{n}$ sample from $U[\theta, \theta +1]$.
Let's consider $f_{\theta} = \operatorname{1}(\theta \le X_{(1)} \le X_{(n)} \le \theta +1)$. Now we may say that there is two sufficient estimates.
First question what estimate we should take. $X_{(1)}$ or $X_{(n)}$?
Now let's take the former one. We need to consider $\int_{\theta}^{\theta+1} f(x) n(1+\theta -x)^{n-1} $, easy to see that is completeness estimate.
But the same idea about $X_{(n)}$. But we know that there is exist only one UMVUE.
What should we do here ?