• Players take turns choosing cells. The chosen cell and its paired cell become your color.
• Your score is the Euler characteristic of your territory.
• At the end, the player with the larger Euler characteristic wins.
• After 3 moves, the second player may swap colors.
• You can view the same board as a Sphere model, Disk model, or Cross-cap model.

The projective plane is a surface that can be described by gluing edges of a square. For example, gluing opposite edges of a square in the same direction gives a torus, the surface of a donut.

A Klein bottle is obtained by reversing one of those gluings. It is a twisted world where going out through one side brings you back on the other side with orientation reversed.

The projective plane uses another gluing pattern, identifying both pairs of opposite edges with reversed direction. This game represents the projective plane by treating opposite points on a sphere as the same place.

The technical term for opposite points on a sphere is antipodal points. In the game, it is enough to think of them as the same cell on the opposite side, or an opposite pair.

Topology studies properties of shapes that do not change when they are stretched or bent. It cares less about distances and angles, and more about connectedness, holes, and loops.

For example, a coffee cup and a donut can both be viewed as shapes with one hole. Topology focuses on this broad pattern of connectedness rather than small geometric details.

In Euler Getter, the connectedness of your stones becomes the score. So while playing on the board, you are competing over a topological quantity.

Euler characteristic is a number that roughly measures how a shape is connected. The basic formula is vertices − edges + faces. For polyhedra and subdivided surfaces, this number helps distinguish types of shapes.

In Euler Getter, connected regions can raise the score, while loops and holes can lower it. Roughly, making separate islands can increase it, while connecting regions into loops or holes can change it.

You do not need to calculate it mentally. Watch the +1 and -1 effects on the screen, and observe what happens when you place cells apart or connect them. Gradually, you will learn to read the shape.

Euler Getter is related to ideas such as the projective plane, the torus, the Klein bottle, topology, and Euler characteristic. These ideas connect to university-level mathematics.

A gentle introduction to these ideas through games and puzzles is Takehiko Yasuda, Introduction to University Mathematics Through Games. Through the Tower of Hanoi, Sprouts, Nim, topology games, the projective plane, and Euler Getter, the book introduces the fun of mathematics beyond calculation.

Mathematics is not only about calculation; it can also expand how we see the world. If this game sparks your interest, please also look at the book and related lecture materials.

References: Euler Getter Wiki / Book information page