Geometric Principle Games
Posted: 13 January 2026, 14:58
Some abstract games, especially connection games, are based on geometric/topological principles. A principle can be, in a sense, equivalent to a theorem.
Famously, the game of Hex cannot end in a draw. If a Hex board is filled with stones of two colors, in any proportion to each other, a winning path will form in exactly one of the colors. Connection games with this characteristic are called fundamental.
The Hex theorem in itself is not a game. Players aren't simultaneously cramming handfuls of stones onto the board. Players take turns adding stones of their own color, one stone per turn. Hex needs to employ the pie rule to mitigate first move advantage. (Player 2 has the option of switching colors, claiming the first placed stone as his own, and not placing a stone, as his first turn.) [By "his" I mean to include all genders. I don't use "they" in my rule sheets or game discussions because an ambiguous, possibly plural pronoun can be confusing.] Without pie, Player 1 could claim the center cell and have both a positional advantage and the numerical advantage of having an extra half stone on the board on average. Pie forces Player 1 to place his first stone on a somewhat equitable cell, somewhere between the center and an edge.
While Hex partitions the board perimeter into four contiguous, alternating color segments, my own fundamental connection games, Conect and Atoll, partition the perimeter into two colored segments and eight colored segments, respectively.
Conect
https://www.marksteeregames.com/Conect_rules.pdf
Atoll
https://www.marksteeregames.com/Atoll_rules.pdf
In Conect, you try to connect your one segment to itself with a path that wraps around (or occupies) the center cell.
The less perimeter segments a connection game has, the more susceptible it is to first move advantage. Simple pie alone is not sufficient to balance Conect. So it's played on the projection of a hexagonally tessellated cone, which depowers the central area. Hence the name.
Group isn't a connection game, but it's based on a geometric/topological principle. The main difference between the game and its founding hypothesis is the inclusion of an anti-blob rule, to deny Player 1 a runaway advantage. Pie alone might only switch the advantage to Player 2. There might not be an equitable first placement cell.
Group
https://www.marksteeregames.com/Group_rules.pdf
Maximum Minimum Group Hypothesis
https://www.marksteeregames.com/Maximum ... thesis.pdf
Matt Emerson identified an inconsistency in the MMGH, which has been corrected.
Thanks for the feedback, everyone. I had originally used the word "conjecture." Then I upgraded it to "theorem." Now I compromised with "hypothesis."
Famously, the game of Hex cannot end in a draw. If a Hex board is filled with stones of two colors, in any proportion to each other, a winning path will form in exactly one of the colors. Connection games with this characteristic are called fundamental.
The Hex theorem in itself is not a game. Players aren't simultaneously cramming handfuls of stones onto the board. Players take turns adding stones of their own color, one stone per turn. Hex needs to employ the pie rule to mitigate first move advantage. (Player 2 has the option of switching colors, claiming the first placed stone as his own, and not placing a stone, as his first turn.) [By "his" I mean to include all genders. I don't use "they" in my rule sheets or game discussions because an ambiguous, possibly plural pronoun can be confusing.] Without pie, Player 1 could claim the center cell and have both a positional advantage and the numerical advantage of having an extra half stone on the board on average. Pie forces Player 1 to place his first stone on a somewhat equitable cell, somewhere between the center and an edge.
While Hex partitions the board perimeter into four contiguous, alternating color segments, my own fundamental connection games, Conect and Atoll, partition the perimeter into two colored segments and eight colored segments, respectively.
Conect
https://www.marksteeregames.com/Conect_rules.pdf
Atoll
https://www.marksteeregames.com/Atoll_rules.pdf
In Conect, you try to connect your one segment to itself with a path that wraps around (or occupies) the center cell.
The less perimeter segments a connection game has, the more susceptible it is to first move advantage. Simple pie alone is not sufficient to balance Conect. So it's played on the projection of a hexagonally tessellated cone, which depowers the central area. Hence the name.
Group isn't a connection game, but it's based on a geometric/topological principle. The main difference between the game and its founding hypothesis is the inclusion of an anti-blob rule, to deny Player 1 a runaway advantage. Pie alone might only switch the advantage to Player 2. There might not be an equitable first placement cell.
Group
https://www.marksteeregames.com/Group_rules.pdf
Maximum Minimum Group Hypothesis
https://www.marksteeregames.com/Maximum ... thesis.pdf
Matt Emerson identified an inconsistency in the MMGH, which has been corrected.
Thanks for the feedback, everyone. I had originally used the word "conjecture." Then I upgraded it to "theorem." Now I compromised with "hypothesis."