This was asked on sci.math ages ago, and never got a satisfactory answer.
Given a number of sticks of integral length $ \ge n$ whose lengths add to $n(n+1)/2$. Can these always be broken (by cuts) into sticks of lengths $1,2,3, \ldots ,n$?
You are not allowed to glue sticks back together. Assume you have an accurate measuring device.
More formally, is the following conjecture true? (Taken from iwriteiam link below).
Cutting Sticks Conjecture: For all natural numbers $n$, and any given sequence $a_1, .., a_k$ of natural numbers greater or equal $n$ of which the sum equals $n(n+1)/2$, there exists a partitioning $(P_1, .., P_k)$ of $\{1, .., n\}$ such that sum of the numbers in $P_i$ equals $a_i$, for all $1 \leq i \leq k$.
Some links which discuss this problem: