The Objective

Move the glowing nodes until zero lines intersect. Every puzzle generated in this game is mathematically proven to have at least one completely non-crossing planar layout!

Euler's Polyhedral Formula

For any connected planar graph drawn in a plane without intersecting edges, the number of vertices (V), edges (E), and faces (F) satisfies:

This fundamental invariant was discovered by Leonhard Euler in 1752 and is the cornerstone of algebraic topology and computational geometry.

Kuratowski's Theorem

Can every graph be untangled? No! In 1930, mathematician Kazimierz Kuratowski proved that a graph can be drawn in the plane without crossing if and only if it does not contain a subgraph homeomorphic to K₅ (the complete graph on 5 vertices) or K_3,3 (the complete bipartite utility graph).