mathematics

Shannon number

Shannon number is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Shannon number rather than just read about it. In short: The Shannon number, named after the American mathematician Claude Shannon, is a conservative lower bound of the game-tree complexity of chess of 10120, based on an average of about 103 possibilities for a pair of moves consisting of a move for White followed by a move for Black, and a typical game lasting about 40 such pairs of moves. Shannon's calculation Shannon showed a calculation for the lower bound of the game…

Key takeaways

Shannon number belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Shannon number to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Shannon number from memory before moving on to harder problems.

Reference excerpt

The Shannon number, named after the American mathematician Claude Shannon, is a conservative lower bound of the game-tree complexity of chess of 10120, based on an average of about 103 possibilities for a pair of moves consisting of a move for White followed by a move for Black, and a typical game lasting about 40 such pairs of moves.

Shannon's calculation Shannon showed a calculation for the lower bound of the game-tree complexity of chess, resulting in about 10120 possible games, to demonstrate the impracticality of solving chess by brute force, in his 1950 paper "Programming a Computer for Playing Chess". (This influential paper introduced the field of computer chess.) Shannon also estimated the number of possible positions, of the general order of 6331 (8!)−2 (where the ! represents the factorial and the underlined superscript represents a falling factorial), or roughly 3.7×1034. This includes some illegal positions (e.g., pawns on the first rank, both kings in check) and excludes legal positions following captures and promotions.

After each player has moved a piece 5 times each (10 ply) there are 69,352,859,712,417 possible games that could have been played.

Tighter bounds

Upper, positions Taking Shannon's numbers into account, Victor Allis calculated an upper bound of 5×1052 for the number of positions, and estimated the true number to be about 1050. Later work proved an upper bound of 8.7×1045, and showed an upper bound 4×1037 in the absence of promotions.

Accurate, positions John Tromp and Peter Österlund estimated the number of legal chess positions with a 95% confidence level at (4.822±0.028)×1044, based on an efficiently computable bijection between integers and chess positions.

Lower, complexity Allis also estimated the game-tree complexity to be at least 10123, "based on an average branching factor of 35 and an average game length of 80". As a comparison, the number of atoms in the observable universe, to which it is often compared, is roughly estimated to be 1080.

Number of sensible chess games As a comparison to the Shannon number, if chess is analyzed for the number of "sensible" games that can be played (i.e., not counting ridiculous or obvious game-losing moves such as moving a queen to be immediately captured by a pawn without compensation), then the result is closer to around 1040 games. This is based on having a choice of about three sensible moves at each ply (half-move), and a game length of 80 plies (or, equivalently, 40 moves).

Illustrations

Shannon number: Claude Shannon — Claude Shannon

Worked examples

Example 1 — a first encounter with Shannon number

Start with the simplest possible case. Write down what Shannon number claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Shannon number before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Shannon number ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Shannon number

In research: Shannon number appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Shannon number in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Shannon number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Claude Shannon, Combinatorial game theory, Computer chess, so understanding it makes those chapters shorter.
In everyday life: Look for Shannon number outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Shannon number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Shannon number means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Shannon number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Shannon number in simple terms?

Why does Shannon number matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Shannon number?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Shannon number.