science

Multimagic square

Multimagic square 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 Multimagic square rather than just read about it. In short: In mathematics, a P-multimagic square (also known as a satanic square) is a magic square that remains magic even if all its numbers are replaced by their kth powers for 1 ≤ k ≤ P. 2-multimagic squares are called bimagic, 3-multimagic squares are called trimagic, 4-multimagic squares tetramagic, and 5-multimagic squares pentamagic. Constants for normal squares If the squares are normal, the constant for the power-squ…

Key takeaways

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

Reference excerpt

In mathematics, a P-multimagic square (also known as a satanic square) is a magic square that remains magic even if all its numbers are replaced by their kth powers for 1 ≤ k ≤ P. 2-multimagic squares are called bimagic, 3-multimagic squares are called trimagic, 4-multimagic squares tetramagic, and 5-multimagic squares pentamagic.

Constants for normal squares If the squares are normal, the constant for the power-squares can be determined as follows: Bimagic series totals for bimagic squares are also linked to the square-pyramidal number sequence is as follows :- Squares 0, 1, 4, 9, 16, 25, 36, 49, .... (sequence A000290 in the OEIS) Sum of Squares 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, ... (sequence A000330 in the OEIS) )number of units in a square-based pyramid) The bimagic series is the 1st, 4th, 9th in this series (divided by 1, 2, 3, n) etc. so values for the rows and columns in order-1, order-2, order-3 Bimagic squares would be 1, 15, 95, 374, 1105, 2701, 5775, 11180, ... (sequence A052459 in the OEIS) The trimagic series would be related in the same way to the hyper-pyramidal sequence of nested cubes. Cubes 0, 1, 8, 27, 64, 125, 216, ... (sequence A000578 in the OEIS) Sum of Cubes 0, 1, 9, 36, 100, ... (sequence A000537 in the OEIS) Value for Trimagic squares 1, 50, 675, 4624, ... (sequence A052460 in the OEIS) Similarly the tetramagic sequence 4-Power 0, 1, 16, 81, 256, 625, 1296, ... (sequence A000583 in the OEIS) Sum of 4-Power 0, 1, 17, 98, 354, 979, 2275, ... (sequence A000538 in the OEIS) Sums for Tetramagic squares 0, 1, 177, ... (sequence A052461 in the OEIS)

Bimagic square A bimagic square is a magic square that remains magic when all of its numbers are replaced by their squares. The first known bimagic square has order 8 and magic constant 260 and a bimagic constant of 11180. It has been conjectured by Bensen and Jacoby that no nontrivial bimagic squares of order less than 8 exist. This was shown for magic squares containing the elements 1 to n2 by Boyer and Trump. However, J. R. Hendricks was able to show in 1998 that no bimagic square of order 3 exists, save for the trivial bimagic square containing the same number nine times. The proof is fairly simple: let the following be our bimagic square.

It is well known that a property of magic squares is that a + i = 2 e {\displaystyle a+i=2e} . Similarly, a 2 + i 2 = 2 e 2 {\displaystyle a^{2}+i^{2}=2e^{2}} . Therefore,

( a − i ) 2 = 2 ( a 2 + i 2 ) − ( a + i ) 2 = 4 e 2 − 4 e 2 = 0 {\displaystyle (a-i)^{2}=2(a^{2}+i^{2})-(a+i)^{2}=4e^{2}-4e^{2}=0} . It follows that a = e = i {\displaystyle a=e=i} . The same holds for all lines going through the center. For 4 × 4 squares, Luke Pebody was able to show by similar methods that the only 4 × 4 bimagic squares (up to symmetry) are of the form

An 8 × 8 bimagic square.

Nontrivial bimagic squares are now (2010) known for any order from eight to 64. Li Wen of China created the first known bimagic squares of orders 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62 filling the gaps of the last unknown orders. In 2006 Jaroslaw Wroblewski built a non-normal bimagic square of order 6. Non-normal means that it uses non-consecutive integers. Also in 2006 Lee Morgenstern built several non-normal bimagic squares of order 7.

Trimagic square A trimagic square is a magic square that remains magic when all of its numbers are replaced by their cubes. Trimagic squares of orders 12, 32, 64, 81 and 128 have been discovered so far; the only known trimagic square of order 12, given below, was found in June 2002 by German mathematician Walter Trump.

Higher order The first 4-magic square was constructed by Charles Devimeux in 1983 and was a 256-order square. A 4-magic square of order 512 was constructed in May 2001 by André Viricel and Christian Boyer. The first 5-magic square, of order 1024 arrived about one month later, in June 2001 again by Viricel and Boyer. They also presented a smaller 4-magic square of order 256 in January 2003. Another 5-magic square, of order 729, was constructed in June 2003 by Li Wen.

See also Diabolic square Magic cube Magic square Multimagic cube

References

Weisstein, Eric W. "Bimagic Square". MathWorld. Weisstein, Eric W. "Trimagic Square". MathWorld. Weisstein, Eric W. "Tetramagic Square". MathWorld. Weisstein, Eric W. "Pentamagic Square". MathWorld. Weisstein, Eric W. "Multimagic Square". MathWorld.

External links multimagie.com puzzled.nl

Worked examples

Example 1 — a first encounter with Multimagic square

Start with the simplest possible case. Write down what Multimagic square 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 Multimagic square 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 Multimagic square 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 Multimagic square

In research: Multimagic square 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 Multimagic square 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: Multimagic square 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 Multimagic square 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 Multimagic square in 20 minutes

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

Frequently asked questions

What is Multimagic square in simple terms?

Why does Multimagic square 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 Multimagic square?

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 Multimagic square.