Magic series

Key takeaways

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

Reference excerpt

A magic series is a set of distinct positive integers which add up to the magic constant of a magic square and a magic cube, thus potentially making up lines in magic tesseracts. So, in an n × n magic square using the numbers from 1 to n2, a magic series is a set of n distinct numbers adding up to n(n2 + 1)/2. For n = 2, there are just two magic series, 1+4 and 2+3. The eight magic series when n = 3 all appear in the rows, columns and diagonals of a 3 × 3 magic square. Maurice Kraitchik gave the number of magic series up to n = 7 in Mathematical Recreations in 1942 (sequence A052456 in the OEIS). In 2002, Henry Bottomley extended this up to n = 36 and independently Walter Trump up to n = 32. In 2005, Trump extended this to n = 54 (over 2 × 10111) while Bottomley gave an experimental approximation for the numbers of magic series:

1 π ⋅ 3 e ⋅ ( e n ) n n 3 − 3 5 n 2 + 2 7 n {\displaystyle {\frac {1}{\pi }}\cdot {\sqrt {\frac {3}{e}}}\cdot {\frac {(en)^{n}}{n^{3}-{\frac {3}{5}}n^{2}+{\frac {2}{7}}n}}}

In July 2006, Robert Gerbicz extended this sequence up to n = 150. In 2013 Dirk Kinnaes was able to exploit his insight that the magic series could be related to the volume of a polytope. Trump used this new approach to extend the sequence up to n = 1000. Mike Quist showed that the exact second-order count has a multiplicative factor of 1 n 3 ( 1 + 3 5 n + 31 420 n 2 + ⋯ ) {\displaystyle {\tfrac {1}{n^{3}}}\!\left(1+{\tfrac {3}{5n}}+{\tfrac {31}{420n^{2}}}+\cdots \right)} equivalent to a denominator of n 3 − 3 5 n 2 + ( 2 7 + 1 2100 ) n + ⋯ . {\displaystyle n^{3}-{\tfrac {3}{5}}n^{2}+\left({\tfrac {2}{7}}+{\tfrac {1}{2100}}\right)\!n+\cdots .} Richard Schroeppel in 1973 published the complete enumeration of the order 5 magic squares at 275,305,224. This recent magic series work gives hope that the relationship between the magic series and the magic square may provide an exact count for order 6 or order 7 magic squares. Consider an intermediate structure that lies in complexity between the magic series and the magic square. It might be described as an amalgamation of 4 magic series that have only one unique common integer. This structure forms the two major diagonals and the central row and column for an odd order magic square. Building blocks such as these could be the way forward.

References

External links Walter Trump's pages on magic series Number of magic series up to order 150 De Loera, Jesús A.; Kim, Edward D. (2013), Combinatorics and Geometry of Transportation Polytopes: An Update, arXiv:1307.0124, Bibcode:2013arXiv1307.0124D

Worked examples

Example 1 — a first encounter with Magic series

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

In research: Magic series 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 series 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 series 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 series 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.

How to study Magic series in 20 minutes

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

Frequently asked questions

What is Magic series in simple terms?

A magic series is a set of distinct positive integers which add up to the magic constant of a magic square and a magic cube, thus potentially making up lines in magic tesseracts. So, in an n × n magic square using the numbers from 1 to n2, a magic series is a set of n distinct numbers adding up to…

Why does Magic series 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 series?

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 series.