The middle activity stair in parallel chip-firing
Follow the latest progress by email.
In parallel chip-firing on a finite connected simple graph, all eligible vertices fire simultaneously. Does every initial configuration with 2|E|−|V|<|σ|<2|E| eventually have minimal period two and activity one half?
Why it matters and what counts as progress
Why it matters: The question asks whether the same middle activity stair occurs beyond previously solved graph families.
What counts as progress: A proof for every connected simple graph in the strict chip-count interval, or a configuration with a different eventual period or activity.
How the problem developed
- 20 Aug 2024 · Conjecture [1]
David Ji, Michael Li and Daniel Wang Proved period exclusions in the middle chip range and conjectured period two for every finite connected graph in that range; prior proofs covered selected graph families.
- 4 Aug 2026 · Reported claim [2]
Daniel Wang and Nathan Lannan Reported a proof that every finite connected simple graph with 2|E|−|V|<|σ|<2|E| has eventual period exactly two, resolving the stated middle-stair conjecture.
AI contribution
All parallel chip-firing games with 2|E|−|V| < |σ| < 2|E| have period 2
For every finite connected simple graph G=(V,E), a parallel chip-firing game with 2|E|−|V|<|σ|<2|E| has eventual minimal period two and activity one half.
Sources
Website timeline
Publication dates and editorial changes.
AI-found proof settles the middle stair of parallel chip-firing