**Problem statement:**

Does there exist positive integers $a<b<c$ such that $$1 + 2 + \dots + (a-1) = (a+1) + \dots + (b-1) = (b+1) + \dots + c?$$ (Note that $a$ and $b$ are not in the sums.)

**Motivation:** this puzzle (some progress is made).

**Notable progress:** Gareth McCaughan proved using Pell's equations that if a solution exists, then $c>10^{600000}$. Techniques using elliptic curves were also developed.

**Remark:** This reminds of this easy-to-state Diophantine equation with large solutions (MO link).