= Solution
Both sides are alternating in $i,j,k$ and in $l,m,n$. If either triple repeats an index, both sides vanish. Otherwise each triple is a permutation of $(1,2,3)$; the <determinant> of the corresponding permutation <matrix> is exactly the product of the two permutation signs. Hence
$$
\boxed{
\epsilon_{ijk}\epsilon_{lmn}
=\begin{vmatrix}
\delta_{il}&\delta_{im}&\delta_{in}\\
\delta_{jl}&\delta_{jm}&\delta_{jn}\\
\delta_{kl}&\delta_{km}&\delta_{kn}
\end{vmatrix}}.
$$
There are $3!=6$ nonzero components of $\epsilon_{ijk}$, each equal to $\pm1$, so
$$
\boxed{\epsilon_{ijk}\epsilon_{ijk}=6}.
$$
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