mathematics

Longest uncrossed knight's path

Longest uncrossed knight's path 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 Longest uncrossed knight's path rather than just read about it. In short: The longest uncrossed (or nonintersecting) knight's path is a mathematical problem involving a knight on the standard 8×8 chessboard or, more generally, on a square n×n board. The problem is to find the longest path the knight can take on the given board, such that the path does not intersect itself.

Key takeaways

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

Reference excerpt

The longest uncrossed (or nonintersecting) knight's path is a mathematical problem involving a knight on the standard 8×8 chessboard or, more generally, on a square n×n board. The problem is to find the longest path the knight can take on the given board, such that the path does not intersect itself. A further distinction can be made between a closed path, which ends on the same field as where it begins, and an open path, which ends on a different field from where it begins.

Known solutions The longest open paths on an n×n board are known only for n ≤ 9. Their lengths for n = 1, 2, ..., 9 are:

0, 0, 2, 5, 10, 17, 24, 35, 47 (sequence A003192 in the OEIS) The longest closed paths are known only for n ≤ 10. Their lengths for n = 1, 2, ..., 10 are:

0, 0, 0, 4, 8, 12, 24, 32, 42, 54 (sequence A157416 in the OEIS)

Generalizations The problem can be further generalized to rectangular m×n boards, or even to boards in the shape of any polyomino. A restricted form of the problem for m×n boards, where n≤8 and m might be very large, was given at 2018 ICPC World Finals. It may be solved by dint of dynamic programming, helped by the insight that the solution should exhibit a cyclic behaviour. Other standard chess pieces than the knight are less interesting, but fairy chess pieces like the camel ((3,1)-leaper), giraffe ((4,1)-leaper) and zebra ((3,2)-leaper) lead to problems of comparable complexity.

See also A knight's tour is a self-intersecting knight's path visiting all fields of the board. TwixT, a board game based on uncrossed knight's paths.

References

L. D. Yarbrough (1968). "Uncrossed knight's tours". Journal of Recreational Mathematics. 1 (3): 140–142. George Jelliss, Non-Intersecting Paths Non-crossing knight tours 2018 ICPC World Finals solutions (Problem J)

External links Uncrossed knight's tours

Illustrations

Worked examples

Example 1 — a first encounter with Longest uncrossed knight's path

Start with the simplest possible case. Write down what Longest uncrossed knight's path 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 Longest uncrossed knight's path 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 Longest uncrossed knight's path 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 Longest uncrossed knight's path

In research: Longest uncrossed knight's path 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 Longest uncrossed knight's path 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: Longest uncrossed knight's path is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational problems in graph theory, Mathematical chess problems, so understanding it makes those chapters shorter.
In everyday life: Look for Longest uncrossed knight's path 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 Longest uncrossed knight's path in 20 minutes

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

Frequently asked questions

What is Longest uncrossed knight's path in simple terms?

Why does Longest uncrossed knight's path 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 Longest uncrossed knight's path?

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 Longest uncrossed knight's path.