A Leech tree is a tree together with a positive-integer edge-weighting satisfying the property that the set of pairwise distances between vertices is exactly the set of positive integers from 1 to 2n(n−1).
Leech trees were initially introduced in the 1970s by Leech, who was motivated by applications to efficient circuit design. (To see the connection to circuit design, think of the edges as resistors and the edge-weights as different resistances.) Only five Leech trees are currently known, and all were given by Leech in his initial paper.
This problem asks for a construction of a new Leech tree. The smallest n for which it is open whether there is a Leech tree on n vertices is n=18. A nice solution to this problem would be a construction showing how to build an infinite family of Leech trees, or giving some insight into how to construct Leech trees in general. Even a solution obtained through brute-force would still be of interest. The main risk for this problem is that no other Leech trees exist.