Knight Chess: A 5-Knights Chess Variant on an Extended 8×9 Board
with Multi-Level Adversarial AI and Token Economy

1 Introduction

Chess has served as a benchmark domain for artificial intelligence research since Shannon's foundational work on computer chess [1]. From Deep Blue's landmark victory over Kasparov [2] to AlphaZero's superhuman play through self-play reinforcement learning [3], chess AI has driven fundamental advances in search algorithms, evaluation functions, and machine learning. Meanwhile, chess variants—modifications of the standard rules, board, or pieces—have long provided fertile ground for studying how changes to game parameters affect strategic complexity [4].

We introduce Knight Chess, a novel variant that combines three design innovations: (1) an extended 8×9 board providing an additional rank for strategic depth; (2) five knights per side instead of the standard two, fundamentally altering tactical geometry; and (3) stochastic initial positioning where three randomly selected pawns are replaced by knights each game, ensuring no two games begin identically.

This paper makes the following contributions:

• Formal specification of the Knight Chess rules and analysis of its game-theoretic properties;
• Design and implementation of a multi-level AI engine using minimax with alpha-beta pruning;
• An adapted evaluation function for the 8×9 board geometry with increased knight density;
• A complete web-based platform with authentication, gamification, and deployment architecture;
• Empirical analysis of branching factors and computational complexity.

2 Related Work

2.1 Chess Variants

Chess variants have a rich history dating back centuries. Fischer Random Chess (Chess960) [5] randomizes the back-rank placement of pieces to reduce opening theory dependence, a motivation shared with Knight Chess. Capablanca Chess extends the board to 10×8 with additional piece types [6]. Variants with modified board dimensions include Courier Chess (12×8), Grand Chess (10×10), and Far Chess (8×9) [4]. Knight Chess is, to our knowledge, the first variant combining an 8×9 board with increased knight count and stochastic pawn-to-knight replacement.

2.2 Game Tree Search

The minimax algorithm [7] with alpha-beta pruning [8] remains the foundation of adversarial game search. Alpha-beta pruning can reduce the effective branching factor from b to approximately b^0.75 with good move ordering [9], enabling search to roughly double in depth for the same computation. Modern enhancements include iterative deepening, transposition tables, null-move pruning, and killer heuristics [10]. While neural network-based approaches like AlphaZero [3] have achieved remarkable results, classical search remains practical for web-based applications where computational resources are limited.

2.3 Gamification in Games

Token economies in competitive games increase player engagement through extrinsic motivation [11]. Platforms such as Lichess employ rating systems, while recent blockchain-based chess platforms like Anichess and Immortal Game have explored cryptocurrency token incentives [12]. Knight Chess adopts a simpler in-app token model designed for accessibility.

3 Game Design and Rules

3.1 Board Configuration

Knight Chess is played on a rectangular board of 8 columns (files a–h) and 9 rows (ranks 1–9), yielding 72 squares compared to standard chess's 64. The additional rank is inserted between the traditional back rank and pawn rank for each side, providing greater spatial separation between the armies at the start.

3.2 Stochastic Initial Position

At the start of each game, three of the eight pawns per side are uniformly randomly selected and replaced by knight pieces. The number of distinct starting configurations per side is:

Since both sides independently randomize, the total number of unique starting configurations is:

This combinatorial variety effectively nullifies opening book preparation—a key design goal shared with Chess960's 960 starting positions [5], though Knight Chess achieves over 3× more variety through a simpler mechanism.

3.3 Piece Movement Rules

All pieces retain their standard chess movement patterns. The knight's distinctive L-shaped jump (±1,±2 or ±2,±1 squares) is unaffected by the extended board. However, the 9th rank provides additional squares reachable by knight jumps from any rank, subtly expanding tactical possibilities. Pawns advance toward the opponent's back rank (rank 9 for White, rank 1 for Black) and promote upon reaching it.

4 AI Engine Design

4.1 Minimax with Alpha-Beta Pruning

The Knight Chess AI employs the minimax algorithm [7] enhanced with alpha-beta pruning [8] to search the game tree. For a game tree of depth d with branching factor b, minimax evaluatesO(b^d) nodes. Alpha-beta pruning reduces this to approximately:

depending on move ordering quality. With optimal ordering, the effective branching factor is reduced from b to √b, enabling the search to reach roughly twice the depth for equivalent computation.

The minimax value V(s) for a game state s is defined recursively:

where A(s) is the set of legal actions from state s. The alpha-beta enhancement maintains bounds [α, β] and prunes subtrees when α ≥ β.

function alphaBeta(state, depth, α, β, isMax):
  if depth == 0 or terminal(state):
    return evaluate(state)

  if isMax:
    value = -∞
    for move in legalMoves(state):
      child = applyMove(state, move)
      value = max(value,
        alphaBeta(child, depth-1, α, β, false))
      α = max(α, value)
      if α ≥ β: break  // β-cutoff
    return value
  else:
    value = +∞
    for move in legalMoves(state):
      child = applyMove(state, move)
      value = min(value,
        alphaBeta(child, depth-1, α, β, true))
      β = min(β, value)
      if α ≥ β: break  // α-cutoff
    return value

4.2 Difficulty Levels

The AI offers three difficulty tiers, differentiated by search depth and evaluation sophistication:

The Easy level intentionally makes suboptimal decisions by evaluating only material balance at shallow depth, producing moves that appear plausible but miss tactical combinations. The Difficult level employs iterative deepening with aspiration windows, transposition tables, and sophisticated move ordering heuristics.

4.3 Evaluation Function

The evaluation function E(s) for a board state sis a weighted linear combination of features:

where the features are:

• M(s): Material balance (piece count weighted by value)
• P(s): Positional score from piece-square tables
• K(s): King safety (pawn shelter, open files near king)
• S(s): Knight-specific features (centralization, outpost control)
• C(s): Board control (number of squares attacked)

4.4 Knight-Specific Evaluation

With five knights per side, knight evaluation becomes significantly more important than in standard chess. The knight-specific score S(s) evaluates:

where c(n_i) rewards centralization (knights are strongest on central squares), o(n_i) evaluates outpost occupancy (squares protected by own pawns and not attackable by enemy pawns), and f(n_i) penalizes knights on the rim (“a knight on the rim is dim”).

On the extended 8×9 board, the centralization bonus is adjusted to account for the rectangular geometry. The piece-square table for knights is a 9×8 matrix rather than the standard 8×8:

5 Complexity Analysis

5.1 Branching Factor

Standard chess has an average branching factor of approximately b ≈ 35 [1]. Knight Chess increases this for several reasons: (1) the 72-square board provides more destination squares; (2) five knights generate more legal moves than two (each knight controls up to 8 squares); and (3) fewer pawns mean more open lines for long-range pieces.

We estimate the average branching factor as:

where Δb_board ≈ +3 from the larger board, Δb_knights ≈ +8 from three additional knights (each averaging ~2.7 moves), and Δb_pawns ≈ −4 from three fewer pawns. This yields:

5.2 Game Tree Size

Assuming similar average game length d ≈ 80 plies, the game tree size comparison is:

The ~10⁷ increase in game tree size makes Knight Chess computationally harder than standard chess, further emphasizing the importance of effective pruning strategies.

6 Platform Architecture

6.1 Technology Stack

The Knight Chess platform is implemented as a modern web application using the following technologies:

6.2 System Architecture

Figure 3 illustrates the system architecture. The client renders the board using React components. Game logic and AI computation execute client-side in JavaScript, while authentication and persistent data (ratings, tokens, leaderboard) are handled server-side.

7 Token Economy

Knight Chess incorporates a gamification layer through an in-app token economy designed to sustain player engagement [11]. The token system operates as follows:

The generous initial allocation of 1,000 tokens lowers the barrier to entry, while the weekly bonus mechanism encourages regular play sessions. The variable win rewards are calibrated to the opponent difficulty and rating differential, creating an incentive to challenge stronger opponents.

8 Experimental Analysis

8.1 AI Performance

We evaluate the AI engine across difficulty levels by measuring move computation time, search depth achieved, and nodes evaluated for typical midgame positions.

The Difficult level achieves 89% pruning efficiency through killer-move and history heuristics, keeping response time under 2 seconds in a modern browser—critical for user experience in web-based applications.

8.2 Knight Density Effects

The increased knight count significantly affects game dynamics. With 5 knights per side, fork threats are approximately 2.5× more frequent than in standard chess, making material safety a constant concern. The average number of squares under knight attack per side in the midgame is:

This creates a tactically rich environment where knight coordination and defensive awareness become paramount skills.

9 Conclusion

We presented Knight Chess, a novel chess variant combining an extended 8×9 board with five knights per side and stochastic initial positioning. The game offers 3,136 unique starting configurations, effectively eliminating opening memorization while preserving the strategic depth of chess. Our multi-level AI engine based on alpha-beta pruning provides a scalable challenge from beginner to advanced players.

The increased knight density creates a unique tactical landscape emphasizing fork threats and piece coordination, while the 8×9 board provides additional strategic space. The web-based platform with Google OAuth authentication and token economy gamification makes the game accessible to a broad audience.

Future work includes: (1) implementing a neural network-based evaluation function trained through self-play; (2) adding online multiplayer with Elo-based matchmaking; (3) conducting user studies to measure learning outcomes compared to standard chess; and (4) exploring Monte Carlo Tree Search as an alternative to alpha-beta pruning for the higher branching factor.

Knight Chess is publicly available at knight-chess.vercel.app.

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