mathematics

Grundy's game

Grundy's game 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 Grundy's game rather than just read about it. In short: Grundy's game is a two-player mathematical game of strategy. The starting configuration is a single heap of objects, and the two players take turn splitting a single heap into two heaps of different sizes.

Key takeaways

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

Reference excerpt

Grundy's game is a two-player mathematical game of strategy. The starting configuration is a single heap of objects, and the two players take turn splitting a single heap into two heaps of different sizes. The game ends when only heaps of size two and smaller remain, none of which can be split unequally. The game is usually played as a normal play game, which means that the last person who can make an allowed move wins.

Illustration A normal play game starting with a single heap of 8 is a win for the first player provided they start by splitting the heap into heaps of 7 and 1:

player 1: 8 → 7+1

Player 2 now has three choices: splitting the 7-heap into 6 + 1, 5 + 2, or 4 + 3. In each of these cases, player 1 can ensure that on the next move he hands back to his opponent a heap of size 4 plus heaps of size 2 and smaller:

player 2: 7+1 → 6+1+1 player 2: 7+1 → 5+2+1 player 2: 7+1 → 4+3+1 player 1: 6+1+1 → 4+2+1+1 player 1: 5+2+1 → 4+1+2+1 player 1: 4+3+1 → 4+2+1+1

Now player 2 has to split the 4-heap into 3 + 1, and player 1 subsequently splits the 3-heap into 2 + 1:

player 2: 4+2+1+1 → 3+1+2+1+1 player 1: 3+1+2+1+1 → 2+1+1+2+1+1 player 2 has no moves left and loses

Mathematical theory The game can be analysed using the Sprague–Grundy theorem. This requires the heap sizes in the game to be mapped onto equivalent nim heap sizes. This mapping is captured in the On-Line Encyclopedia of Integer Sequences as OEIS: A002188:

Heap size : 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ... Equivalent Nim heap : 0 0 0 1 0 2 1 0 2 1 0 2 1 3 2 1 3 2 4 3 0 ...

Using this mapping, the strategy for playing the game Nim can also be used for Grundy's game. Whether the sequence of nim-values of Grundy's game ever becomes periodic is an unsolved problem. Elwyn Berlekamp, John Horton Conway and Richard Guy have conjectured that the sequence does become periodic eventually, but despite the calculation of the first 235 values by Achim Flammenkamp, the question has not been resolved.

See also Nim Sprague–Grundy theorem Wythoff's game Subtract a square

References

External links Grundy's game on MathWorld Sprague-Grundy values for Grundy's Game by A. Flammenkamp

Illustrations

Worked examples

Example 1 — a first encounter with Grundy's game

Start with the simplest possible case. Write down what Grundy's game 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 Grundy's game 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 Grundy's game 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 Grundy's game

In research: Grundy's game 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 Grundy's game 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: Grundy's game is common in secondary-school and first-year university syllabi. It links to neighbouring topics Combinatorial game theory, Mathematical games, so understanding it makes those chapters shorter.
In everyday life: Look for Grundy's game 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 Grundy's game in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Grundy's game 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 Grundy's game out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Grundy's game in simple terms?

Why does Grundy's game 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 Grundy's game?

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 Grundy's game.