opening choices
GAME MATHEMATICS
How large is Cubic Chess?
Reproducible engine counts are separated from cautious estimates. One branch is a completed ply: movement plus the player’s optional one-cube rebuild.
vs chess opening
sampled mean
tree envelope
What was counted
The production move generator enforces cube conservation, composition limits, opposite-coloured bishops, king safety and feasibility, promotion to bishop, rook or queen only, and an automatic draw on the third occurrence of the complete position.
Opening calculation
16 pawn movements × 17 endings = 272. Eight pawns can move one or two squares; after moving, the player may finish unchanged or add one cube to any of 16 foundations.
B₀=16×(1+16)=272Measured branching
Deterministic spread sample of 1,000 distinct engine positions per ply after the complete opening frontier. Values depend on the position.
| Ply | Positions | Min | Mean | Max |
|---|---|---|---|---|
| 0 | 1 | 272 | 272 | 272 |
| 1 | 272 | 272 | 272 | 272 |
| 2 | 1,000 | 142 | 203.6 | 289 |
| 3 | 1,000 | 131 | 202.3 | 306 |
| 4 | 1,000 | 65 | 156.2 | 486 |
| 5 | 1,000 | 65 | 153.2 | 486 |
| 6 | 1,000 | 65 | 203.8 | 517 |
| 7 | 1,000 | 65 | 199.7 | 500 |
Comparison without false precision
| Game | Branching | State space | Game tree |
|---|---|---|---|
| Checkers / Шашки | ≈7–8 | ≈5×10²⁰ | ≈10³¹ |
| Chess / Шахматы | 20; ≈35 typical | ≈10⁴⁴ | ≈10¹²⁰ |
| Cubic Chess | 272; ≈150–203 sampled | ≤7.22×10⁶⁸ ceiling | ≈10¹⁴⁰–10¹⁷⁰ |
| Go 19×19 / ГО | 361 | ≈2.08×10¹⁷⁰ | ≈10³⁶⁰ |
With observed branching around 150–200 and 66 completed plies, log₁₀(T)=d·log₁₀(b) gives roughly 10¹⁴⁴–10¹⁵². A 60–75-ply sensitivity band gives about 10¹³¹–10¹⁷³. The displayed 10¹⁴⁰–10¹⁷⁰ is an illustrative envelope, not a proof.
A loose encoding ceiling is 2 × 9² × 11⁶⁴ ≈ 7.22×10⁶⁸. Reachability, conservation and the rules remove most encodings; this is not a count of legal positions.