mathematics

Hexapawn

Hexapawn 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 Hexapawn rather than just read about it. In short: Hexapawn is a deterministic two-player game invented by Martin Gardner. It is played on a rectangular board of variable size, for example on a 3×3 board or on a regular chessboard.

Key takeaways

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

Reference excerpt

Hexapawn is a deterministic two-player game invented by Martin Gardner. It is played on a rectangular board of variable size, for example on a 3×3 board or on a regular chessboard. On a board of size n×m, each player begins with m pawns, one for each square in the row closest to them. The goal of each player is to either advance a pawn to the opposite end of the board or leave the other player with no legal moves (akin to stalemate), including by capturing all of their pawns. Hexapawn on the 3×3 board is a solved game; with perfect play, White will always lose in 3 moves (1.b2 axb2 2.cxb2 c2 3.a2 c1#). Indeed, Gardner specifically constructed it as a game with a small game tree in order to demonstrate how it could be played by a heuristic AI implemented by a mechanical computer based on Donald Michie's Matchbox Educable Noughts and Crosses Engine (MENACE). A variant of this game is octopawn, which is played on a 4×4 board with 4 pawns on each side. It is a forced win for White in 5.5 moves with optimal play (beginning with 1.a2 or 1.d2). Only 24 matchboxes are required for a hexapawn version of Matchbox Educable Noughts and Crosses Engine.

Rules As in chess, a pawn may be moved in two different ways: it may be moved one square vertically forward, or it may capture a pawn one square diagonally ahead of it. A pawn may not be moved forward if there is a pawn in the next square. Unlike chess, the first move of a pawn may not advance it by two spaces. A player loses if they have no legal moves or one of the other player's pawns reaches the end of the board.

Octopawn Octopawn is a variant of hexapawn played on a 4×4 grid with the same rules as hexapawn. It is mentioned by Gardner in "The Unexpected hanging and Other Mathematical Diversions":

Richard L. Sites, at M.I.T., wrote a FORTRAN program for an IBM 1620 so that it would learn to play Octapawn, a 4 x 4 version of hexapawn that begins with four white pawns on the first row and four black pawns on the fourth row. He reports that the first player has a sure win with a corner opening. In recent years, advances in computing has allowed the entire game tree of Octopawn to be computed using an average personal computer. An analysis of the game tree reveals that there are 11,420 unique game states and 2,330,356 unique games. The shortest game is 5 turns long(e.g. 1.d2 b3 2.d3 b2 3.dxc4#), and the longest game is 17 turns long(e.g. 1.d2 b3 2.d3 cxd3 3.c2 d2 4.c3 b2 5.axb2 dxc3 6.b3 c2 7.bxc2 axb3 8.c3 b2 9.c4#). The distribution of game lengths approximately follows a normal distribution curve with a center of 13 turns.

Dawson's chess Whenever a player advances a pawn to the penultimate rank and attacks an opposing pawn, there is a threat to proceed to the final rank by capture. The opponent's only sensible responses, therefore, are to either capture the advanced pawn or advance the threatened one, the latter only being sensible in the case that there is one threatened pawn rather than two. If one restricts 3×N hexapawn with the additional rule that capturing is always compulsory, the result is the game Dawson's chess. The game was invented by Thomas Rayner Dawson in 1935. Dawson's chess reduces to the impartial game denoted .137 in Conway's notation. This means that it is equivalent to a Nim-like game in which:

on a turn, the player may remove one to three objects from a heap, removing just one object is a legal move only if the removed object is the only object in the heap, and when removing three objects from a heap of five or more, the player may also split the remainder into two heaps. The initial position is a single heap of size N. The nim-sequence for this game is

0.1120311033224052233011302110452740 1120311033224455233011302110453748 1120311033224455933011302110453748 1120311033224455933011302110453748 1120311033224455933011302110453748 ...,

where bold entries indicate the values that differ from the eventual periodic behavior of the sequence.

References

Sources Mathematical Games, Scientific American, March 1962, reprinted in The Unexpected Hanging and Other Mathematical Diversions, by Martin Gardner, pp. 93ff

External links Hexapawn - an article by Robert Price. Hexapawn java applet - source code included. Hexapawn game for IOS Play Hexapawn - Play Online

Worked examples

Example 1 — a first encounter with Hexapawn

Start with the simplest possible case. Write down what Hexapawn 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 Hexapawn 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 Hexapawn 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 Hexapawn

In research: Hexapawn 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 Hexapawn 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: Hexapawn is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1962 in chess, Board games introduced in 1962, Chess variants, so understanding it makes those chapters shorter.
In everyday life: Look for Hexapawn 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Hexapawn” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Hexapawn in 20 minutes

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

Frequently asked questions

What is Hexapawn in simple terms?

Hexapawn is a deterministic two-player game invented by Martin Gardner. It is played on a rectangular board of variable size, for example on a 3×3 board or on a regular chessboard.

Why does Hexapawn 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 Hexapawn?

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 Hexapawn.