science

Magic constant

Magic constant 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 constant rather than just read about it. In short: The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15.

Key takeaways

Magic constant 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 constant to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Magic constant from memory before moving on to harder problems.

Reference excerpt

The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a normal magic square of order n – that is, a magic square which contains the numbers 1, 2, ..., n2 – the magic constant is M = n ⋅ n 2 + 1 2 {\displaystyle M=n\cdot {\frac {n^{2}+1}{2}}} .

For normal magic squares of orders n = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175, and 260 (sequence A006003 in the OEIS). For example, a normal 8 × 8 square will always equate to 260 for each row, column, or diagonal. The normal magic constant of order n is ⁠n3 + n/2⁠. The largest magic constant of normal magic square which is also a:

triangular number is 15 (solve the Diophantine equation x2 = y3 + 16y + 16, where y is divisible by 4); square number is 1 (solve the Diophantine equation x2 = y3 + 4y, where y is even); generalized pentagonal number is 171535 (solve the Diophantine equation x2 = y3 + 144y + 144, where y is divisible by 12); tetrahedral number is 2925. Note that 0 and 1 are the only normal magic constants of rational order which are also rational squares. However, there are infinitely many rational triangular numbers, rational generalized pentagonal numbers and rational tetrahedral numbers which are also magic constants of rational order. The term magic constant or magic sum is similarly applied to other "magic" figures such as magic stars and magic cubes. Number shapes on a triangular grid divided into equal polyiamond areas containing equal sums give polyiamond magic constant.

Magic stars The magic constant of an n-pointed normal magic star is M = 4 n + 2 {\displaystyle M=4n+2} .

Magic series In 2013 Dirk Kinnaes found the magic series polytope. The number of unique sequences that form the magic constant is now known up to n = 1000 {\displaystyle n=1000} .

Moment of inertia In the mass model, the value in each cell specifies the mass for that cell. This model has two notable properties. First it demonstrates the balanced nature of all magic squares. If such a model is suspended from the central cell the structure balances. (consider the magic sums of the rows/columns .. equal mass at an equal distance balance). The second property that can be calculated is the moment of inertia. Summing the individual moments of inertia (distance squared from the center × the cell value) gives the moment of inertia for the magic square, which depends solely on the order of the square.

Illustrations

Worked examples

Example 1 — a first encounter with Magic constant

Start with the simplest possible case. Write down what Magic constant 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 constant 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 constant 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 constant

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

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

Frequently asked questions

What is Magic constant in simple terms?

The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15.

Why does Magic constant 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 constant?

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 constant.