How Many Possible Chess Positions Exist? The Math Behind It

Chess, the ancient and elegant game of kings, queens, and strategy, has fascinated mathematicians and computer scientists for centuries. With only 64 squares and 32 pieces, it might seem that the number of possible chess positions would be relatively small. But nothing could be further from the truth. Beneath the surface of this seemingly simple game lies a universe of unimaginable complexity.

In this article, we’ll explore the math behind the question: How many possible chess positions exist? We’ll delve into the history of position estimation, key concepts like the Shannon number, legal vs. illegal positions, and the implications for computing and artificial intelligence. Spoiler alert: the number is astronomically large.

1. Definitions: What Do We Mean by “Position”?

Before we count, we need to define what a “chess position” actually is. In chess, a position refers to a specific arrangement of pieces on the board, along with other state information such as:

• Which side is to move (White or Black),

• Castling rights,

• En passant possibilities,

• Half-move clock for the fifty-move rule,

• Full move number.

These aspects combine to form what is technically called a chess game state or FEN (Forsyth–Edwards Notation) representation.

For the purpose of most estimates, however, we focus on the number of legal positions—those that could be reached in a real game obeying the rules of chess.

2. Legal vs. Illegal Positions

There’s a massive difference between all possible configurations of chess pieces and legal chess positions.

• Illegal positions include setups with too many pawns, kings in check simultaneously, or multiple kings of one color.

• Legal positions follow all the rules: only one king per color, no pawns on the first or eighth ranks unless promoted, correct turn order, etc.

While computers can generate all configurations of pieces, most of them don’t make sense in real chess. So we must narrow it down.

3. The Shannon Number: A Historic Estimate

In 1950, Claude Shannon, the father of information theory, made one of the first serious estimates of chess complexity. He calculated:

Estimated number of possible positions: ~10^43
Estimated number of possible games: ~10^120

The second number (game trees) is much larger because it includes different move sequences. The first—10^43—is his estimate of all reasonable legal positions.

Let’s break this down further.

4. Upper Bound: How Big Can It Get?

Let’s try to understand the upper bound of chess positions using combinatorics.

Piece Placement:

Each square can either be empty or occupied by a piece. Since there are 64 squares and 32 pieces:

• Number of ways to choose 32 squares from 64: C(64,32) ≈ 1.83 × 10^18

• Each piece has a type and color (pawn, knight, bishop, etc.):

  • For each arrangement, the pieces can be permuted in different ways depending on type and count.

That gives us somewhere around 10^40 to 10^50 different piece arrangements.

Adding Game State Info:

Once you add castling rights (16 possibilities), en passant (up to 8 possibilities), and side to move (2), that multiplies the total even more.

Hence, upper estimates reach into:

~10^47 total legal positions, as calculated by Victor Allis and others.

5. Recent Research: Exact Counts (or Close)

In 2016, a research paper by Marc Bourzutschky and Yakov Konoval estimated the total number of legal positions in standard chess as:

4.822 × 10^44 legal positions

This figure was derived through extensive computational analysis, eliminating impossible or illegal setups. While not exact, it’s one of the most accurate estimates available today.

6. Why Is the Number So Big?

Several factors contribute to the massive number:

1. Pawn Promotions:

Each pawn can promote to one of four pieces—queen, rook, bishop, knight—and there can be up to nine promotions per side (under unusual circumstances). This dramatically increases position counts.

2. Castling & En Passant:

Each of these rules introduces new position states that depend on history (e.g., whether a king or rook has moved before), which expands the number of distinct legal positions.

3. Move Order Complexity:

Different sequences of moves can lead to the same board setup, but from a rules perspective (like en passant or castling availability), those are different positions.

7. Game Tree Complexity

The number of possible chess positions is only part of the story. The game tree complexity—the number of distinct games or move sequences—is much larger.

Claude Shannon’s original estimate:

10^120 possible games

This number is often referred to as the Shannon Number, and it’s greater than the number of atoms in the observable universe (~10^80). This fact alone demonstrates why chess cannot be solved by brute-force methods alone.

8. AI and Chess Engines: Why Position Counts Matter

Modern chess engines like Stockfish and AlphaZero don’t try to calculate every possible position. Instead, they use heuristics, neural networks, and evaluation functions to focus only on likely paths.

Knowing the number of positions helps AI developers understand the search space and optimize algorithms like:

• Alpha-Beta Pruning (eliminates branches of the game tree),

• Monte Carlo Tree Search (used by AlphaZero),

• Neural Network evaluations (as in NNUE).

Even with massive computing power, covering the whole game tree is impractical. That’s why focusing on good positions—guided by machine learning—is the modern approach.

9. Chess Variants: Even More Possibilities

When we include chess variants like:

• Chess960 (Fischer Random) – 960 starting positions,

• Three-Check, Atomic Chess, or Bughouse,
  the total number of positions explodes even further.

Some estimates for variants reach into 10^50+ legal positions, depending on the rules.

10. Implications for Solving Chess

Because of the vast number of legal positions (~10^44) and even more possible move sequences, solving chess (i.e., proving a guaranteed win or draw from the starting position) remains out of reach.

For comparison:

• Checkers has ~5 × 10^20 positions and has been solved.

• Connect Four has been solved.

• Chess is exponentially more complex.

Even if Moore’s Law continues and quantum computers arrive, solving chess would require an astronomical level of computation.

11. Fun Fact: Endgame Tablebases

While full chess is unsolvable (so far), endgames with few pieces have been completely mapped using tablebases:

• All 7-piece endgames have been solved.

• These tablebases tell you the exact result (win/draw/loss) from any position with 7 or fewer pieces.

Even these databases are huge—more than 140 terabytes in total!

12. Conclusion: A Game of Infinite Wonder

To summarize:

Despite its limited board and set of rules, chess offers an almost limitless playground of creativity, strategy, and computation. The vast number of possible positions ensures that no two games need ever be the same—and that we’ll never run out of new things to learn, analyze, and enjoy.

Whether you’re a beginner or a grandmaster, part of the magic of chess lies in the knowledge that each move you make opens the door to an entire universe of possibilities.