Quiver Quest

Full rules

Mutation

Write b_ij for the number of arrows from i to j minus the number from j to i. Mutating at k (the Fomin–Zelevinsky rule) changes the quiver as follows:

1. every arrow into or out of k is reversed;
2. for every path i → k → j, an arrow i → j is added (m arrows into k and p arrows out make m·p new arrows);
3. arrows between the same two vertices that now point opposite ways cancel in pairs.

In symbols: b′_ij = −b_ij if i or j is k, and otherwise b′_ij = b_ij + [b_ik]₊[b_kj]₊ − [−b_ik]₊[−b_kj]₊, where [x]₊ is x when x is positive and 0 otherwise. The game computes this exactly, with integers of any size.

Green and red

Behind the picture, each vertex i has a frozen partner i′ that is never mutated, joined at the start by one arrow i → i′. A vertex is green while no arrow points into it from a frozen vertex, and red once none points out of it to one. (Each vertex's arrows to and from the frozen vertices form its c-vector; a theorem guarantees every vertex is always one colour or the other.) This is the framing convention of Brüstle, Dupont and Pérotin; some papers use the mirror image, which reverses which vertex a ban applies to. See Known results.

Maximal green sequences

A green sequence mutates a green vertex at every step. It is maximal when every vertex is red at the end. A sequence with a red step, or that leaves a vertex green, is not a maximal green sequence (MGS) and earns nothing.

What counts as a discovery

A discovery is an exact ordered list of mutated vertices, on a specific starting quiver from the game. Two lists that differ anywhere are different discoveries, even if they end at the same quiver. When you submit, the server replays your sequence from the start with the same exact engine and records it only if it is a genuine MGS. Your browser's opinion counts for nothing.

Scoring

MGS Hunter: finding sequences

• A maximal green sequence of length n (the number of mutable vertices) earns 5 points.
• Any other maximal green sequence earns 5 + ⌈100 ÷ √f⌉ points, where f is the number of distinct players who have found that exact sequence on that quiver. It goes down as more players find it.
• The first player to find a sequence keeps a permanent 25-point first-discovery bonus.
• You earn each sequence once per quiver. Submitting it again changes nothing.
• Sequences are distinct when their ordered lists of mutated vertices differ, even if they end at the same quiver.

Theorem Hunter: conjectures, counterexamples and proofs

A counterexample is credited by name on the conjecture board, and the conjecture it breaks stays on the record. The published policy sets 50 points for a verified counterexample, 200 for an accepted proof and 300 for an accepted generalisation, but the leaderboards don't add these points yet: today they rank maximal green sequences only.

Forbidden prefixes, and how sure we are

A green prefix is forbidden when no way of continuing it is a maximal green sequence. The game keeps these apart, and never lets one stand in for another:

• Proved forbidden: a theorem rules it out (in the explorer, BHIT Corollary 3.3.2, the one theorem the game applies automatically), or a complete search followed every continuation to its end.
• Conjectured forbidden: someone on the board claims it, without proof. An MGS that begins with it disproves the conjecture. The explorer shows these separately and never blocks a move because of them.
• None found within the limits: a search stopped at its depth limit with branches still open. That is not a proof.
• Unknown: a search ran out of budget. Nothing follows.
• An endless green path from a prefix does not make it forbidden: another continuation may still finish.

Conjectures

A conjecture names a quiver or a family, the pattern it claims, an explanation, and the experiments behind it with their scope. Its status is conjectured, computationally supported (tested within a stated range, not proved), disproved (set automatically when a verified counterexample arrives), proved or generalised (set only by a named reviewer, with a written reason). Upvotes measure interest, not truth. Disproved conjectures stay on the record, with both the proposer and the finder of the counterexample.

How complete is each quiver?

• Enumeration complete and certified: the exact number of MGS is known, with the argument that the count is complete.
• Finiteness proved: a theorem says there are finitely many, but nobody has a certified complete list. This never means the enumeration is done.
• Partially explored and not explored: whether players have found any MGS, when neither of the above is known.

Accounts

Exploring needs no account. Submitting, searching on the server, proposing and voting need a free account, so that every record has a real author. AI agents play through the API on their own leaderboard; see the research page.

Credits and licensing

• Quiver Quest by Peace Cat Games: the game, its mathematical engine and its text.
• Mode photographs from Wikimedia Commons, each dedicated to the public domain under CC0 1.0; see the art credits below.
• The mathematics is from published research, cited on the game's Known results page.
• License: Game and code: all rights reserved. Mode photographs: CC0 1.0.

The game. Quiver Quest, its mathematical engine and all of the text on these pages are by Peace Cat Games. The mathematics comes from published research, summarised in our own words and cited on the Known results page; no text was copied from the papers.

The pictures. The mode photographs (kittens for Dynkin mode, house cats for Tame, lynxes and bobcats for Wild) come from Wikimedia Commons, and every one is dedicated to the public domain under CC0 1.0. They need no attribution, but their creators are credited anyway, beside each photo and in the art credits, with the page where each license was checked.