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Magic cube

Magic cube is a science 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 Magic cube rather than just read about it. In short: In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a collection of integers arranged in an n × n × n pattern such that the sums of the numbers on each row, on each column, on each pillar and on each of the four main space diagonals are equal, the so-called magic constant of the cube, denoted M3(n). If a magic cube consists of the numbers 1, 2, ..., n3, then it has magic constant…

Magic cube — main illustration
Magic cube — illustration

Key takeaways

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

Reference excerpt

In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a collection of integers arranged in an n × n × n pattern such that the sums of the numbers on each row, on each column, on each pillar and on each of the four main space diagonals are equal, the so-called magic constant of the cube, denoted M3(n). If a magic cube consists of the numbers 1, 2, ..., n3, then it has magic constant (sequence A027441 in the OEIS)

M 3 ( n ) = n ( n 3 + 1 ) 2 . {\displaystyle M_{3}(n)={\frac {n(n^{3}+1)}{2}}.}

If, in addition, the numbers on every cross section diagonal also sum up to the cube's magic number, the cube is called a perfect magic cube; otherwise, it is called a semiperfect magic cube. The number n is called the order of the magic cube. If the sums of numbers on a magic cube's broken space diagonals also equal the cube's magic number, the cube is called a pandiagonal magic cube.

Alternative definition In recent years, an alternative definition for the perfect magic cube has gradually come into use. It is based on the fact that a pandiagonal magic square has traditionally been called "perfect", because all possible lines sum correctly. That is not the case with the above definition for the cube.

Multimagic cubes

As in the case of magic squares, a bimagic cube has the additional property of remaining a magic cube when all of the entries are squared, a trimagic cube remains a magic cube under both the operations of squaring the entries and of cubing the entries (Only two of these are known, as of 2005.) A tetramagic cube remains a magic cube when the entries are squared, cubed, or raised to the fourth power. John R. Hendricks of Canada (1929–2007) has listed four bimagic cubes, two trimagic cubes, and two tetramagic cubes. Two more bimagic cubes (of the same order as those of Hendricks, but differently arranged) were found by Zhong Ming, a mathematics teacher in China. Several of these are perfect magic cubes, and remain perfect after taking powers. A mod-9 symmetric semiperfect tetramagic cube was found by Emlyn Ellis Addison in 2011.

Magic cubes based on Dürer's and Gaudi Magic squares A magic cube can be built with the constraint of a given magic square appearing on one of its faces, such as with the magic square of Dürer or the magic square of Gaudi.

See also John R. Hendricks Magic cube classes Magic hypercube Magic series Magic square Multimagic cube Nasik magic hypercube Perfect magic cube Semiperfect magic cube

References

Andrews, William Symes (1960), "Chapter II: Magic Cubes", Magic Squares and Cubes (PDF) (2nd ed.), New York: Dover Publications, pp. 64–88, doi:10.2307/3603128, ISBN 9780486206585, JSTOR 3603128, MR 0114763, OCLC 1136401, S2CID 121770908, Zbl 1003.05500 {{citation}}: ISBN / Date incompatibility (help)

External links Harvey Heinz, All about Magic Cubes Marian Trenkler, Magic p-dimensional cubes Marian Trenkler, An algorithm for making magic cubes Marian Trenkler, On additive and multiplicative magic cubes Archived 2012-03-21 at the Wayback Machine Ali Skalli's magic squares and magic cubes

Illustrations

Magic cube: An example of a 3 × 3 × 3 magic cube. In this example, no slice is a magic square. In this case, the cube is classed as a simple magic cube.
An example of a 3 × 3 × 3 magic cube. In this example, no slice is a magic square. In this case, the cube is classed as a simple magic cube.

Worked examples

Example 1 — a first encounter with Magic cube

Start with the simplest possible case. Write down what Magic cube claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Magic cube 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 Magic cube 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 Magic cube

In research
Magic cube appears in science 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 Magic cube 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
Magic cube is common in secondary-school and first-year university syllabi. It links to neighbouring topics Magic squares, so understanding it makes those chapters shorter.
In everyday life
Look for Magic cube 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 Magic cube in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Magic cube means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Magic cube out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Magic cube in simple terms?

In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a collection of integers arranged in an n × n × n pattern such that the sums of the numbers on each row, on each column, on each pillar and on each of the four main space diagonals are equal, the so-called magi…

Why does Magic cube matter?

Because it connects several science 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 Magic cube?

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 Magic cube.

Tags

  • Magic squares

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