mathematics

N-dimensional sequential move puzzle

N-dimensional sequential move puzzle 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 N-dimensional sequential move puzzle rather than just read about it. In short: The Rubik’s Cube is the original and best-known of the three-dimensional sequential move puzzles. Over the years, many virtual implementations of this puzzle have been created in software.

Key takeaways

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

Reference excerpt

The Rubik’s Cube is the original and best-known of the three-dimensional sequential move puzzles. Over the years, many virtual implementations of this puzzle have been created in software. Extending these ideas naturally leads to the development of sequential move puzzles in four or more dimensions. The rules governing their operation are rigorously defined mathematically and are closely related to principles of three-dimensional geometry. As a result, they can be simulated in software, and some have even been realized as physical puzzles. As with their three-dimensional counterparts, there are records for solvers, although there is not yet the same level of competitive organisation.

Glossary Vertex. A zero-dimensional point at which higher-dimension figures meet. Edge. A one-dimensional figure at which higher-dimension figures meet. Face. A two-dimensional figure at which (for objects of dimension greater than three) higher-dimension figures meet. Cell. A three-dimensional figure at which (for objects of dimension greater than four) higher-dimension figures meet. n-Polytope. A n-dimensional figure continuing as above. A specific geometric shape may replace polytope where this is appropriate, such as 4-cube to mean the tesseract. n-cell. A higher-dimension figure containing n cells. Piece. A single moveable part of the puzzle having the same dimensionality as the whole puzzle. Cubie. In the solving community this is the term generally used for a 'piece'. Sticker. The coloured labels on the puzzle which identify the state of the puzzle. For instance, the corner cubies of a Rubik's cube are a single piece but each has three stickers. The stickers in higher-dimensional puzzles will have a dimensionality greater than two. For instance, in the 4-cube, the stickers are three-dimensional solids. For comparison purposes, the data relating to the standard 33 Rubik's cube is as follows;

Number of achievable combinations = 12 ! ⋅ 8 ! 2 ⋅ 2 12 2 ⋅ 3 8 3 ∼ 10 20 {\displaystyle ={\frac {12!\cdot 8!}{2}}\cdot {\frac {2^{12}}{2}}\cdot {\frac {3^{8}}{3}}\sim 10^{20}}

There is some debate over whether the face-centre cubies should be counted as separate pieces as they cannot be moved relative to each other. A different number of pieces may be given in different sources. In this article the face-centre cubies are counted as this makes the arithmetical sequences more consistent and they can certainly be rotated, a solution of which requires algorithms. However, the cubie right in the middle is not counted because it has no visible stickers and hence requires no solution. Arithmetically we should have

P = V + E + F + C {\displaystyle P=V+E+F+C\,\!}

But P is always one short of this (or the n-dimensional extension of this formula) in the figures given in this article because C (or the corresponding highest-dimension polytope, for higher dimensions) is not being counted.

Magic 4D Cube

Geometric shape: Tesseract The Superliminal MagicCube4D software implements many twisty puzzle versions of 4D polytopes including N4 cubes. The UI allows for 4D twists and rotations plus control of 4D viewing parameters such as the projection into 3D, cubie size and spacing, and sticker size. Superliminal Software maintains a Hall of Fame for record breaking solvers of this puzzle.

34 4-cube

Achievable combinations:

= 24 ! ⋅ 32 ! 2 ⋅ 16 ! 2 ⋅ 2 23 ⋅ ( 3 ! ) 31 ⋅ 3 ⋅ ( 4 ! 2 ) 15 ⋅ 4 {\displaystyle ={\frac {24!\cdot 32!}{2}}\cdot {\frac {16!}{2}}\cdot 2^{23}\cdot (3!)^{31}\cdot 3\cdot {\left({\frac {4!}{2}}\right)}^{15}\cdot 4}

= 2 84 ⋅ 3 47 ⋅ 16 ! ⋅ 24 ! ⋅ 32 ! {\displaystyle =2^{84}\cdot 3^{47}\cdot 16!\cdot 24!\cdot 32!}

∼ 10 120 {\displaystyle \sim 10^{120}\,\!}

24 4-cube

Achievable combinations:

= 15 ! 2 ⋅ ( 4 ! 2 ) 14 ⋅ 4 {\displaystyle {}={\frac {15!}{2}}\cdot {\left({\frac {4!}{2}}\right)}^{14}\cdot 4}

= 2 29 ⋅ 3 14 ⋅ 15 ! {\displaystyle {}=2^{29}\cdot 3^{14}\cdot 15!}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with N-dimensional sequential move puzzle

Start with the simplest possible case. Write down what N-dimensional sequential move puzzle 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 N-dimensional sequential move puzzle 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 N-dimensional sequential move puzzle 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 N-dimensional sequential move puzzle

In research: N-dimensional sequential move puzzle 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 N-dimensional sequential move puzzle 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: N-dimensional sequential move puzzle is common in secondary-school and first-year university syllabi. It links to neighbouring topics Combination puzzles, Multi-dimensional geometry, so understanding it makes those chapters shorter.
In everyday life: Look for N-dimensional sequential move puzzle 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 N-dimensional sequential move puzzle in 20 minutes

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

Frequently asked questions

What is N-dimensional sequential move puzzle in simple terms?

The Rubik’s Cube is the original and best-known of the three-dimensional sequential move puzzles. Over the years, many virtual implementations of this puzzle have been created in software.

Why does N-dimensional sequential move puzzle 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 N-dimensional sequential move puzzle?

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 N-dimensional sequential move puzzle.