Game of Y

Canonical Sets of a Symmetry Group in Y

Here is a brief paper I wrote about how Game of Y symmetries could be captured in Group Theory. It defines a canonical set.

Vertices of a lopped-off triangular pyramid

What I created here is a 93-point 3D graph of each point's distance from the 3 sides.
The 3 corners of the base are the Y-board. The lopped-off face are the points in the center: You can play Game of Y on this graph, but see more easily how far you are from an edge.

Current Research (AGI Green)

Using Game of Y as a testbed, we've been researching:

C++ engine using hand-coded AI
Python engine using Monte Carlo search
Pytorch engine using trained neural net with weight-sharing for symmetries
Mitosis in neural nets
Autogen-based multi-LLM-agent chat for game server

Magazine Review of Game of Y

I wrote a review of the Game of Y for "World Game Review" magazine, published in Vol. 13. While I'm looking for it, here is a citation of my review in the book "Hex Strategy", 2000 A.K. Peters, Ltd. by Cameron Browne.

Jan 2, 2023

Imagine a 3-dimensional hollow triangular pyramid (4-sided die).

Its faces are not completely solid, but rather defined by a set of vertices graphed on integers. There is a line segment of 9 vertices that defines each edge.

The bottom face is empty.

Instead of tapering off to a point, the top of the pyramid has been lopped off, so the top is actually a triangle face. It has 5 vertices on each side.

To help you visualize:

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Top of our shape, looking down

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Each of the 3 side faces

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Bottom of our shape

How many vertices is this?

The bottom is empty, so it contains only vertices that are part of the side faces.

Each of the 3 side faces has 9+8+7+6=5=35 vertices, for a total of 3*35=105

However, side faces share side edges, so to not count those twice we have to subtract: 105 - 3*5 = 90 unique vertices.

The top face has edges which share vertices with the sides. Only the 3 vertices in the center are new.

That gives a total of 90 + 3 = 93 unique vertices.

That should sound familiar. What's it a 3D graph of?