  This work is about a partition problem which is an instance of the distance
magic graph labeling problem. Given positive integers $n,k$ and $p_1\le p_2\le
\cdots\le p_k$ such that $p_1+\cdots+p_k=n$ and $k$ divides $\sum_{i=1}^ni$, we
study the problem of characterizing the cases where it is possible to find a
partition of the set $\{1,2,\ldots,n\}$ into $k$ subsets of respective sizes
$p_1,\dots,p_k$, such that the element sum in each subset is equal. Using a
computerized search we found examples showing that the necessary condition,
$\sum_{i=1}^{p_1+\cdots+p_j} (n-i+1)\ge j{\binom{n+1}{2}}/k$ for all
$j=1,\ldots,k$, is not generally sufficient, refuting a past conjecture.
Moreover, we show that there are infinitely many such counter-examples. The
question whether there is a simple characterization is left open and for all we
know the corresponding decision problem might be NP-complete.
