ColorTaiko! is an Illinois Mathematics Lab (IML) project for Fall 2024 and Spring 2025 in the Department of Mathematics at the University of Illinois Urbana-Champaign (UIUC). It introduces a playful, interactive game that challenges players to explore color patterns and solve puzzles. Future levels will be added, so come back and try your skills as the game evolves!
The game is inspired by the research article "The topology and geometry of units and zero-divisors: origami" by Igor Mineyev. This work explores how certain colorings of bipartite graphs could potentially lead to counterexamples to the Kaplansky conjectures, long-standing open problems in algebra related to group algebras. In the full version of the game, solving increasingly difficult levels could potentially bring new insights or counterexamples related to these mathematical problems.
Particularly, ColorTaiko! could potentially shed light on Kaplansky's conjectures, which ask: given a field and a torsion-free group, does the group ring contain any non-trivial zero divisors or non-trivial units? The first counterexample to the unit conjecture was presented by Giles Gardam in 2021. This counterexample involved the fundamental group of the Hantzsche–Wendt manifold over a field of characteristic 2, which inspired the paper by Igor Mineyev and, subsequently, this game. The zero-divisor conjecture remains open to this day.
The goal for ColorTaiko! is to create a complete bipartite graph that satisfies certain conditions:
- Level 1 implements color merging when the condition is satisfied.
- Level 2 implements orientation of horizontal edges and detects when orientation fails.
- Level 3 (in progress) implements no-fold condition.
- Level 4NP (in progress) implements no pattern condition.
- Level 4.6 (in progress) implements girth-6 condition on both the top and bottom graphs.
- Level 5NP.6 (in progress) combines no pattern and girth-6 condition. Successful completion of this level will yield a counterexample to one of the conjectures.
You can play the game at the link below:
https://play.math.illinois.edu/ColorTaiko!/