• Players take turns choosing cells. The chosen cell and its corresponding cell become your color.
• Your score is the Euler number of your colored region.
• At the end, the larger Euler number wins.
• After move 3, the second player can swap colors.
• You can view the same board as Sphere, Disk, or Cross-capModel to view it.

The projective plane is a surface that can be made by gluing edges of a square. For example, gluing the left and right edges of a square in the same direction and the top and bottom edges in the same direction gives a torus, a donut-shaped world.

A Klein bottle is obtained by reversing one of these gluings. It is a slightly twisted world where going off one edge brings you back on the opposite side with orientation reversed.

The projective plane uses a different gluing: left-right and top-bottom are both matched in opposite directions. In this game, instead of using a square diagram, we represent it by treating opposite places on the sphere as the same.

The technical name for such opposite points is antipodal points. In the game, however, Same cell on the opposite side or Opposite pairis enough to keep in mind.。

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

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 fine details of shape.

In Euler Getter, your score comes from how your stones are connected. In other words, you are competing for a topological quantity while playing on the board.

Roughly speaking, the Euler number measures how a shape is organized. The basic formula is vertices − edges + cells . For polyhedra and subdivided surfaces, this number helps distinguish types of shapes.

In Euler Getter, the faces in the formula are counted as colored cells. Intuitively, the Euler number is connected pieces − loops. Making isolated islands tends to increase it, while connecting your region into loops or holes can decrease it.

No mental calculation is needed. Watch the on-screen +1 and -1 animations and Observe whether separated moves increase it and how connecting regions changes it. You will gradually get a feel for it.

The ideas used in Euler Getter connect to the projective plane, torus, Klein bottle, topology, Euler number, and more. it connects to ideas in university mathematics.

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

Mathematics is not only calculation. It also broadens how we see the world. If this game sparks your interest, please also take a look at the book and public lecture materials.

Reference: Euler Getter Wiki / Book page: Introduction to University Mathematics through Games