= Solution
A <sufficient statistic> is a function of the sample whose conditional sample distribution, given that statistic, does not depend on the parameter. The <factorization criterion for sufficiency> states, for a dominated family, that $T$ is sufficient exactly when the joint density has form $g_\theta(T(x))h(x)$, where $h$ is independent of $\theta$.
Let $m=X_{(1)}=\min_iX_i$ and $M=X_{(n)}=\max_iX_i$. The joint <uniform density> is
$$
f_{a,b}(x_1,\ldots,x_n)=(b-a)^{-n}1_{\{a\leq m,\ M\leq b\}},
$$
so $\boxed{T=(m,M)}$ is sufficient for $(a,b)$ by the <factorization criterion for sufficiency>. The <uniform order statistics> have expectations
$$
\mathbb Em=a+\frac{b-a}{n+1},\qquad
\mathbb EM=a+\frac{n(b-a)}{n+1}.
$$
For example, the standardized minimum has survival probability $(1-t)^n$ on $[0,1]$, whose integral is $1/(n+1)$; the maximum follows by reflection. Solving these expectation equations gives the <unbiased estimator>
$$
\boxed{\widehat a=\frac{nm-M}{n-1},\qquad
\widehat b=\frac{nM-m}{n-1},\qquad
\mathbb E(\widehat a,\widehat b)=(a,b).}
$$
The assumption $n\geq2$ makes the denominator positive.
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