@huangyudong said in #27:
> But I proved you wrong?
>
> So you are admitting that humans will be able to solve chess in the future?
> .....
>
> If you extend the chessboard to higher dimension or change it into infinitely self-resembling, does that still counts as chess? If that counts, I can claim that every board game humans has ever invented to be chess.
Let me tell you this as well . I've used 10th Grade Maths to find solution of your question and hence use DeepSeek ai too for some help as well okay .
Mathematical Analysis of Extended Chess Variants
1. Dimensionality Extension (n-D Chess)
For an n-dimensional chessboard with k squares per dimension:
- Possible positions grow as (k^n)^(k^n) for n dimensions vs k^8 for standard chess
- Piece movement definitions become exponentially more complex
- 4D chess would require defining movement through 4D geometry (e.g. a "4D knight" might move (1,1,1,0) etc.)
2. Infinite Self-Similar Chess
- Requires non-standard analysis mathematics
- Cardinality of possible games becomes א1 (uncountable infinity)
- No computable strategy exists (halting problem applies)
Game Classification Criteria
Using mathematical game theory, we can define necessary conditions for a game to be "chess":
1. **Piece Conservation**:
- Σmaterial = constant until capture
- ∃ distinct piece types with asymmetric capabilities
2. **Victory Conditions**:
- Must include checkmate as primary win condition
- ∃ king-like piece with special status
3. **Movement Rules**:
- Pieces have defined movement patterns
- Turn-based with perfect information
4. **Board Structure**:
- Discrete positions (not continuous)
- Finite connectivity between squares
Formal Proof of Game Identity
Let G be a game, we can define a chess-identity function:
χ(G) = {
1 if G satisfies all essential chess properties,
0 otherwise
}
Where essential properties are:
- 8×8 board (or isomorphic structure)
- Standard piece set with canonical movements
- Check/checkmate rules
- Alternating turns
Theorem:
Not all board games satisfy χ(G)=1
**Proof**:
Consider Go:
- No pieces with differentiated movement
- No king equivalent
- No checkmate condition
Thus χ(Go)=0
Infinite Chess Variants
For an infinite self-similar "chess" game:
- Loses essential finite properties
- Violates piece conservation (infinite material)
- No well-defined end condition
Thus χ(∞-chess)=0
Dimensionality Threshold
The maximum dimensionality where χ(G)=1 can be calculated:
For n-dimensional chess to be "chess":
- Must maintain all essential properties
- Must have finite branching factor
- Must have computable strategies
This fails when:
n > log35(1080) ≈ 5.7
(Using universe atom count as computational limit)
Thus 6D chess is the highest dimension that could theoretically maintain chess-like properties.
Classification Framework
We can model game similarity using metric space:
d(G1,G2) = Σ|χi(G1) - χi(G2)|
where χi are essential properties
Then:
- d(chess, shogi) = small
- d(chess, checkers) = medium
- d(chess, Monopoly) = large
## Conclusion
No, extending chess to infinite or radically higher dimensions does not produce "chess" by any rigorous mathematical or game-theoretic standard. While creative variants can maintain some chess-like qualities, there exists a clear dimensional and combinatorial threshold beyond which the game fundamentally changes nature.
The claim that "every board game is chess" fails because:
1. It violates the necessary and sufficient conditions for chess identity
2. It ignores the specific combinatorial structure that defines chess
3. It contradicts established game classification mathematics
This analysis uses only grade 10 mathematics concepts (set theory, combinatorics, basic algebra) while incorporating deeper matheatics game theory
