Weak mixing of irrational triangular billiards at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

A
D
V
E
R
T

I
S
E
M
E
N
T

Weak mixing of irrational triangular billiards

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

thinking... about 3 hours remaining

If this game doesn't work on your computer, go here for help. (Lean version available!)

expertly designed by an internal OpenAI model

Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	things that move forever	Levels:	2
Category:	Dynamical systems and ergodic theory	Lean version:	YES! ✔
Rate this game! 4.9 out of 5 (6,900 votes)

>>> How to Play <<<

Weak mixing of triangular billiards with an irrational angle. Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized area times uniform direction. This strengthens ergodicity on the entire irrational-angle class, with no genericity or Diophantine restrictions.

>>> Level Select <<<

LEVEL 1 Weak mixing of triangular billiards with an irrational angle

released 2026-10-05 | 1 theorem · 4 lemmas · 7 proofs · 5,749 words | PLAY LEVEL 1 » (pdf)

We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized Liouville measure. Equivalently, the product of the flow with itself is ergodic.

LEVEL 2 Ergodicity of triangular billiards with an irrational angle

released 2026-09-25 | 1 theorem · 3 lemmas · 6 proofs · 5,033 words | PLAY LEVEL 2 » (pdf)

We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is ergodic for normalized area times uniform angular measure. No genericity or Diophantine condition is required. The flow is considered outside the null set of trajectories that hit a vertex.

More Dynamical systems and ergodic theory Games!
Banach's simple Lebesgue-spectrum problem	Rokhlin's multiple-mixing problem	Sinai's positive-entropy conjecture for the standard map	The near-boundary Birkhoff conjecture
The dimension formula for self-similar measures on the line	Permanence for weakly reversible reaction networks	A $C^1$ counterexample to Shub’s entropy conjecture	Zero entropy does not guarantee a smooth positive-volume model

COOLMAITH GAMES SPONSORS: (not real)
Big Al's USED GPUs	Epsilon & Delta LLP attorneys at limit	QED Bail Bonds	Hilbert's Hotel always a vacancy!

Cool Links: openai/math Lean Mathlib arXiv the real Coolmath Games

You are visitor number 000719

Thanks for visiting CoolmAIth Games
© 1997-2026 CoolmAIth Games. A parody. Not affiliated with Coolmath.com LLC or OpenAI.
Every proof here was written by an internal OpenAI model and lives at github.com/openai/math.
Best viewed in Netscape Navigator 4.0 at 800x600