mathematics

Steinhaus chessboard theorem

Steinhaus chessboard theorem 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 Steinhaus chessboard theorem rather than just read about it. In short: In combinatorial topology, the Steinhaus chessboard theorem is the following theorem, due to Hugo Steinhaus:Consider a chessboard on which some cells contain landmines. Then, either the king can cross the board from left to right without meeting a mined square, or the rook can cross the board from top to bottom moving only on mined squares.

Key takeaways

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

Reference excerpt

In combinatorial topology, the Steinhaus chessboard theorem is the following theorem, due to Hugo Steinhaus:Consider a chessboard on which some cells contain landmines. Then, either the king can cross the board from left to right without meeting a mined square, or the rook can cross the board from top to bottom moving only on mined squares.

Two-dimensional variants David Gale proved a variant of the theorem in which the tiles on the chessboard are hexagons, as in the game of Hex. In this variant, there is no difference between king moves and rook moves. Kulpa, Socha and Turzanski prove a generalized variant of the chessboard theorem, in which the board can be partitioned into arbitrary polygons, rather than just squares. They also give an algorithm for finding either a king route or a rook route.

n-dimensional variants Tkacz and Turzanski generalize the chessboard theorem to an n-dimensional board:Consider a grid of n-dimensional cubes. Color each cube with one of n colors 1,...,n. Then, there exists a set of cubes all colored i, which connect the opposite grid sides in dimension i.Ahlbach present the proof of Tkacz and Turzanski to the n-dimensional chessboard theorem, and use it to prove the Poincaré-Miranda theorem. The intuitive idea is as follows. Suppose by contradiction that an n-dimensional function f, satisfying the conditions to Miranda's theorem does not have a zero. In other words, for each point x, there is at least one coordinate i for which fi(x) is nonzero. Let us color each point x with some color i for which fi(x) is nonzero. By the Steinhaus chessboard theorem, there exists some i for which there is a path of points colored i connecting the two opposite sides on dimension i. By the Poincaré-Miranda conditions, fi(x)<0 at the start of the path and fi(x)>0 at the end of the path, and the function is continuous along the path. Therefore, there must be a point on the path on which fi(x)=0 - a contradiction.

See also A different theorem of Steinhaus, related to arranging rooks on a chessboard, that can be proved using Hall's marriage theorem.

References

Illustrations

Worked examples

Example 1 — a first encounter with Steinhaus chessboard theorem

Start with the simplest possible case. Write down what Steinhaus chessboard theorem 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 Steinhaus chessboard theorem 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 Steinhaus chessboard theorem 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 Steinhaus chessboard theorem

In research: Steinhaus chessboard theorem 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 Steinhaus chessboard theorem 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: Steinhaus chessboard theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Fixed-point theorems, so understanding it makes those chapters shorter.
In everyday life: Look for Steinhaus chessboard theorem 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 Steinhaus chessboard theorem in 20 minutes

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

Frequently asked questions

What is Steinhaus chessboard theorem in simple terms?

Why does Steinhaus chessboard theorem 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 Steinhaus chessboard theorem?

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 Steinhaus chessboard theorem.