AI FIELD NOTES · 00 / START HERE
Behind the AIStart here: what OverGrid asks an AI to see.
This is the front door to the OverGrid AI Field Notes. The series follows the work in order: how we make a legal move set representative, when a reply is worth checking, what invalidated early results, and how a player’s concrete objection can improve the engine. It is a record of bounded, deterministic game-AI work—not a claim that any one bot has solved the game.
The game in one turn
OverGrid is a two-to-six-player territory game played with five-cube pieces. On every turn, a player must use all three pieces in the dealt hand: rotate each one through three dimensions, choose where it rests, and make the whole construction legal and connected to their territory. Covered ordinary squares become territory; surrounding territory captures it. Printed height squares ask the construction to reach an exact level before their one-time bonus can be banked. Black squares add another route-and-clear constraint.
The shared race is not to a fixed score. It is to resolve every initially neutral ordinary square on the selected map, then finish the fair final round. The highest score wins: an ordinary claim is 5 points, a capture is 10, and a printed height is worth 10, 20, or 30 more when its bonus is earned. Those point values do not create more legal placements; they make the placements that already exist disagree more sharply about what matters.
Why the small board is not a small turn
A board dimension tells only part of the story. Each dealt hand has three different five-cube shapes. Most have 12 geometrically distinct orientations; each orientation can have many board origins and elevations. Then the engine has to reject overlaps, illegal support, disconnected constructions, uncleared black squares, impossible height matches, and duplicates before it can compare a legal result.
Here are two concrete map facts from the current official two-player fixtures:
| Map | Shared territory target | Height scoring available at setup | What it means for the AI |
|---|---|---|---|
| Standard · 9 × 9 | 59 neutral ordinary squares | 15 height squares, 330 total bonus points | A compact board where capture and a few exact height routes can reorder a move. |
| Expert · 13 × 10 | 88 neutral ordinary squares | 52 height squares, 1,040 total bonus points | More places to rest a shape, more height obligations, and a much richer score trade-off. |
For one reproducible Standard opening (seed 101, hand U/W/T), the three pieces have 12 distinct orientations each. Before collisions, support, territory attachment, black-tile rules, and duplicate removal, their in-bounds placements combine into 374,360,000 raw triples. That is an illustration of the geometry, not a count of legal moves: the dealt hand and live board change it every turn.
ANIMATED EXPLAINER · ONE TURN, FOUR FILTERS
A legal move is the end of the sieve.
Tap a stage to watch the same opening turn shed impossible constructions. These are concrete Standard-opening measurements, not a promise that every turn has the same count.Put a number on the geometry
The opening number is not magic: it is 764 × 700 × 700 = 374,360,000. For a piece i on a board of width W and depth D, the engine takes each distinct orientation o, finds that orientation’s width wᵢₒ and depth dᵢₒ, then counts the origins that keep it on the board:
Pᵢ(W,D) = Σₒ max(0, W − wᵢₒ + 1) × max(0, D − dᵢₒ + 1)
For a three-piece hand, the unconstrained construction total is H = P₁ × P₂ × P₃. U has 764 such placements on a 9 × 9 board; W and T have 700 each. On the 13 × 10 Expert board, that same U/W/T hand becomes 1,292 × 1,208 × 1,208 = 1,885,369,088 raw constructions before a single game rule is applied.
That is still a deliberately loose ceiling: a player cannot begin a turn anywhere on the board. On this exact Standard opening, the terrain filter leaves 392 / 422 / 406 U/W/T placements with legal bases, but only 68 / 56 / 54 of those can touch Blue’s starting territory. If Bᵢ means a base-eligible placement and Aᵢ means a territory-touching one, then requiring at least one anchored piece gives the smaller union bound F ≤ A₁B₂B₃ + B₁A₂B₃ + B₁B₂A₃ = 29,495,984. Even that remains an upper bound: it can double-count a construction with two anchors and has not yet checked inter-piece contact, collisions, support, height, or duplicate outcomes.
The exact canonical census is the number that behaves like a game-AI legal branching factor: for seed 101’s Standard U/W/T opening, all rules reduce the position to 330,437 distinct legal full-turn constructions. That is not a universal property of a 9 × 9 board; it changes with hand, frontier, terrain, and game phase. It is nevertheless the right category to compare with Chess or Go’s legal moves. The raw geometry and frontier-aware bounds are here to explain why generating those legal moves needs care, not to claim that OverGrid has a fixed move count or a larger whole-game tree.
Adding players does not multiply the number of geometric choices on one player’s turn. It changes the intervening actors and the live position before the same player acts again. A frozen-board p-player round reference of Hᵖ is therefore only a combinatorial thought experiment, not a game-tree estimate. Likewise, the cell-label reference (p + 1)^(W × D) is a loose board-colouring universe that ignores terrain, height, scores, hands, and reachability.
INTERACTIVE MODEL · GEOMETRY BEFORE RULES
How quickly does the construction space grow?
Change the board, table size, or hand. The geometry is exact; official presets also show the measured opening frontier, clearly separated from a legal-game count.Show your work For each piece i, Pᵢ(W,D) = Σₒ max(0, W − wᵢₒ + 1) × max(0, D − dᵢₒ + 1), where w and d are that orientation’s board footprint. The hand total is H = P₁ × P₂ × P₃.
For an official opening, let Bᵢ be placements that avoid uncleared black bases and Aᵢ be the smaller subset that touches the player’s territory. The connected-turn union bound is F ≤ A₁B₂B₃ + B₁A₂B₃ + B₁B₂A₃. It can count some constructions more than once, so the complete rules census below it is the meaningful branching-factor measurement.
| Game | Useful scale | What it does—and does not—mean |
|---|---|---|
| OverGrid | 330,437 legal moves in this opening | Compare legal branching factor to legal moves in Chess or Go. The larger raw and frontier-aware figures explain generation cost; they are not directly comparable move counts. |
| Chess | ~35 legal moves in a typical position; Shannon’s game-tree estimate ~10¹²⁰ | Legal moves and a whole-game tree, not comparable one-for-one with an unfiltered OverGrid hand. |
| Go (19 × 19) | 361 opening placements; 3³⁶¹ ≈ 1.7 × 10¹⁷² coloured boards | A simple empty/black/white board-label universe; legal Go positions are a smaller, separately counted set. |
Scoring is intentionally absent from the formula. Height bonuses, captures, and the shared territory target do not create an x/z/orientation placement; they make the surviving legal constructions much harder to rank. That is why OverGrid uses bounded, representative search instead of presenting this raw number as work done every turn.
The important distinction is between possibilities and work actually performed. A full Standard-map scan of another representative hand (X/U/L) checked 159,156 complete constructions and found 75,195 distinct legal ones. That is a census of one hand, not a budget a responsive opponent can blindly repeat; a depth-two “my turn, then your turn” tree multiplies the problem again.
A vocabulary for arguing about a move
Words such as “pressure” and “smart” are too vague to make an AI claim checkable. The guide below gives the series’ terms a concrete meaning, then introduces the eight public opponents without converting a promising experiment into a retroactive strength claim.
INTERACTIVE FIELD GUIDE · TERMS AND OPPONENTS
Speak the game before judging the bot.
Working definitions for this series, separating visible game facts, bounded search, and the cautious research history of the eight public AI personalities.AUC (score AUC)
Area under a player’s score-margin curve at recorded checkpoints. It rewards being ahead earlier and for longer; it is not a natural win.
Why it matters: Two policies can finish near the same score while one spends the whole game under pressure.Promotion
A policy change earning its way into the live roster through a locked, replayable evaluation protocol—not simply looking good in one arena run.
Why it matters: A playable checkpoint, research lead, and formal promotion are different states.Pressure
A public threat that changes what an opponent must answer: a capture, leader-gap reduction, route denial, or territory race—not merely a dramatic-looking placement.
Why it matters: Good pressure is directed at the player or route that can decide the result.Candidate budget
The fixed number of complete constructions a responsive AI is allowed to inspect. Live play uses 800 in a duel and 250 with three to six players.
Why it matters: It bounds response time and makes representative candidate supply an engineering problem.Horizon
The measured point in the future where an analysis stops. A lead at that cap is a diagnostic observation, not automatically the winner.
Why it matters: A move can lead at a medium horizon and still lose after the territory race resolves.Flip war
An endgame whose apparent leader changes near completion, often after contested territory or final scoring changes hands.
Why it matters: Flip-war endings can make short-horizon gains look stronger than their fully resolved results.Frontier
The reachable edge of a player’s current territory: places a legal construction can connect to or extend from.
Why it matters: A broad frontier can create options; a trapped frontier can make a high-score move hollow.Reply-backed
A choice assessed with an opponent reply when the search can complete that second ply; otherwise the receipt must say it used one-ply public consequences.
Why it matters: It avoids overstating tactical foresight when a responsive budget runs out.THE EIGHT AIs
Select an opponent to see its public intent and the status of its research story.
GENERIC REFERENCE OPPONENT
Balanced
IntentBalance score, height routes, pressure, and room to grow.HistoryThe dependable reference personality: a competence baseline rather than a narrow doctrine.The contract behind the rest of this series
The live AI does not pretend to inspect every raw triple. It takes a fixed, deterministic, representative set of complete legal constructions, scores the public consequences, and keeps a legal fallback. The responsive budget is 800 complete constructions in a duel and 250 at three to six players. In two-player positions, Ender can deliberately widen that public-state census to 800–6,400 constructions when the score, remaining territory, or recent public swing warrants it.
That gives the series four recurring questions:
- Did the candidate generator give the move family a fair chance to appear?
- What visible consequence—territory, capture, height, route, or denial—does a candidate actually create?
- Is a shallow opponent reply needed to tell two good moves apart?
- Can the result be replayed and checked without giving the AI hidden future hands or RNG state?
The next note begins with the first of those questions: why a deterministic 400-candidate search still needed to be representative before it could be trusted.
Continue to 01 — Why a small board can still have an enormous turn →