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Legacy prototype (2002) - modern Canvas versions (2026)
The purpose of this document is to define, with precision, the notation, local structure, rules, and scoring conventions needed to state a reduced one-tile optimization problem in Trigame.
The ultimate goal is to determine the best possible single-tile play for the character Kit, a player whose preferences differ from those of a general strategist: Kit dislikes placing tiles, prefers placing tokens, may skip once per game, and is assumed to possess strong memory but weak large-scale planning.
A game may contain up to 12 players. Players are indexed by positive integers
and denoted in game records by forms such as 1., 2.,
3., and so on.
In two-player examples, players may also be referred to by color, such as red and green, particularly when discussing graphical realizations of the game.
A token is a player's placable game piece. All tokens are identical in form and are distinguished only by ownership. Ownership is represented visually by color.
The board is built from triangular tiles. Each tile is oriented either
point-up or point-down. The first tile placed is the center tile and is always point-up.
It is denoted by [C].
Each tile contains exactly seven distinguished token positions, called sockets:
0,The outer numbering convention is orientation-dependent but standardized:
1 is the top corner,1 is the bottom corner.The remaining outer sockets are then numbered in clockwise order. Thus the numbering rule is: socket 1 is always the point of the tile, and the rest proceed clockwise.
Each tile contains nine path segments:
These paths partition the tile into three internal regions, hereafter called subs (sub-tiles or sub-areas).
Tiles are named relative to the initial center tile [C].
Rows are indexed vertically:
C = center row,A1, A2, A3, ... = successive rows above center,B1, B2, B3, ... = successive rows below center.
Horizontal displacement within a row is then indicated by
L1, L2, L3, ... for left,
and R1, R2, R3, ... for right.
Examples:
[C] = center tile,[B1] = the tile directly below [C],[B1R1] = the tile immediately right of [B1],[CL1] = the tile immediately left of [C].
0 is the center.
Socket 1 is the point of the triangle.
The remaining sockets are numbered clockwise.
0 is the center.
Socket 1 is the point of the triangle.
The remaining sockets are numbered clockwise.
Each tile has exactly seven sockets: one central socket and six outer sockets.
The central socket is always numbered 0.
The outer sockets are numbered clockwise, beginning at the point of the triangle.
1 is the top corner.1 is the bottom corner.Thus the numbering is orientation-dependent, but the rule is always the same: start at the point and count clockwise.
A token placed on a socket is written by appending the socket number to the tile designation. Thus:
[C] denotes the tile only,[C]1 denotes a token on socket 1 of the center tile,[B1R1]0 denotes a token on the center socket of tile [B1R1].Examples:
[C]0 = token in the center of the center tile,[C]1 = token on the pointed corner of the center tile,[C]4 = token on socket 4 of the center tile, counted according to the tile's orientation,[CL1]3 = token on socket 3 of the tile immediately left of the center tile.An event is a single atomic action of one player. The possible events are:
A standard turn consists of two events.
If a player captures a full tile during the turn, one extra event is awarded immediately and added to that same turn.
Each line of a game record corresponds to one player's turn. The player number appears first, followed by the events of that turn, separated by colons.
Example:
1. [C] : [C]0
This means Player 1:
[C], then0 of that tile.To denote removal, prefix the event with a minus sign.
Example:
2. -[C]0 : [C]
This means Player 2 removed the token at [C]0, then placed tile [C].
A player may skip the remainder of a turn, including the whole turn, but:
If a player skips before any tile has been placed, then no game-record line is produced for that skipped turn. Thus if Player 1 skips the opening and Player 2 begins play, the record begins with Player 2.
When a player places a token on a socket, that token may induce additional tokens. The rule is:
A newly placed token creates tokens in every straight-line direction toward the next token already owned by the same player, provided that every path segment between them is already of that player's color.
Equivalently: along any straight direction, if a newly placed token and an already owned token of the same player bound a path whose constituent segments are already in that player's color, then every intermediate socket on that line is filled with that player's tokens.
Automatic token placement applies only along paths already of that player's color. It does not occur along neutral or opponent-colored paths.
When one tile is placed adjacent to another, the newly placed tile overlaps the common boundary edge. Consequently, that edge is thereafter treated as belonging to the new tile rather than the old one. For example, if a red tile originally has all nine red paths, then placing a green tile beside it causes the common edge to become green rather than red, thereby reducing the number of paths belonging to the red tile.
A tile may not be placed adjacent to another tile if any token occupies a socket on the common edge.
A sub is captured when a player occupies all four sockets that bound it. Because those four sockets determine all four of its boundary paths, capturing all four boundary sockets is equivalent to owning the entire sub.
A full tile is captured when a player has captured all three of its subs. Upon capturing a full tile, the player immediately receives one extra event on that same turn.
Scoring is as follows:
Thus the game record language itself does not encode scoring, but the problem under study does depend on it.
The full Trigame language supports arbitrarily many tiles and players, but the present problem is much smaller:
Determine the best possible play for Kit in every possible single-tile game against exactly one opponent.
In this reduced setting:
[C],Kit is not modeled as a universal strategist. Instead:
Thus Kit is best modeled not as a strong general optimizer, but as a specialist in a small solved subgame.
If Kit is forced to move first and to place the center tile, then his first token has seven apparent destinations, but these reduce, up to symmetry, to only three classes:
| Class | Representative | Description |
|---|---|---|
| Center | [C]0 |
Token at the center socket |
| Corner | [C]1 |
Any corner socket, reduced by rotation |
| Edge-midpoint | [C]2 |
Any edge-middle socket, reduced by rotation |
Therefore the opening problem reduces from seven first-token placements to three symmetry classes. The same reduction applies to the opponent's opening if the opponent moves first.
Problem.
Let the game be restricted to a single tile [C], with the rules and scoring above.
Determine, for Kit, an optimal policy for every possible single-tile opening state against one opponent.
In particular:
[C]?This problem belongs primarily to:
A suggested route is:
The intended spirit is not merely solve a toy problem, but:
Construct a complete and optimal single-tile memory table for Kit, so that Kit may be globally child-like and locally devastating.
Match 1: Player 1 2: Player 2 1. [C] : [C]2 2. [C]1 : [C]3 1. [C]4 : [C]5 2. [C]6 : [C]0
This represents a fully legal single-tile game in the Game Record language. The mathematical problem is not to read such a record, but to determine which such records can arise under optimal play for Kit.