h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Entry: MAGIC SQUARES - A17949694
Author: gavinorr - U3235914

A bit of numbers fun!

I haven't read it yet because I got stuck at your first example:
17 24 01 08 15
23 05 07 14 15
04 06 13 20 22
10 12 19 21 03
11 18 25 02 09


There's a mistake in there, the last digit in the second row should be 16.

Interesting subject.

Could you add something explaining why magic squares are so called - ie has any significance ever been attached to them beyond the mathematical properties you describe?

I'm happy with the manual method (although I haven't tested it yet), but I believe you could simplify it somewhat.

The Excel method leaves me a bit cold. I haven't time to test it, but it's quite long and involved, and it looks a bit 'trial and error' from some of the comments you have made, eg:

<...If it is not, then something is wrong and you need to identify and correct this>
etc

Is this a method you devised, or is it something you have researched elsewhere? If the latter, then I would seriously consider linking to it rather than reproducing it.

It's also Excel-centric, and we don't all use Microsoft Office products. Is there a way to simplify and generalise it?

Icy

I've fixed the typo in the example.

I'll research the "magic" reference.

The Excel method is my own invention, and is not trial and error. If you do it correctly it will work every time. Perhaps I should reword the "if" statements - basically if you don't get the expected result then you have done something wrong.

Similarly since I don't know if every release of Excel pops-up circular references I said "if". I'll re-word it to be more positive.

I might add a simple explaination at the start to say what you are doing - basically you fill up everything except one row and one column with one formula, fill the leftmost column with one formula, fill the bottom row with one formula, then put in the "special formula" for the "step down" cells.

O.K.
Modifications made , research and update re the magical properties of magic squares, plus a find of a cross reference between magic squares and h2g2 (bonus!).

A few years ago, somebody found a rather large "magic cube"... Maybe you could mention this too?

I was expecting a lot more about magic squares, and far less on methods for making them Lo Shu, Durer, Panmagic squares, antimagic squares, Magic constant, Magic cubes, nuumerology (A3094229), John Horton Conway (A2982468), Templar Magic square, with an aside on Latin Squares and Sudoku (A3865953), and how they differ from Magic squares.

While your methods for creating a Magic Square would be helpful to those wanting to make their own I think the entry would be better without them.

Possibly you could consider splitting the entry into the background Magic Squares entry, and a linked entry restricting itself to how to create Magic squares (on which you have already done quite a bit of work ).

A link to http://mathworld.wolfram.com/MagicSquare.html may be useful. The wolfram site is dauntingly good at providing explanations of mathematical concepts (frequently with diagrams, so valuable to grasping ideas), but it shouldn't stop h2g2 entries being written.

I am only really interested in the maths (or should that be arithmetic) of magic squares, and the purpose of this entry is to instruct people on how to make them (so I won't be taking that bit out
). For the same reason I don't want to add too many entries on the magic side of it, although the wolfram site sounds interesting.

In that case I'd retitle your entry to what it is actually about: "How to construct a magic square"

Most of the "magic" suggestions I made were to do with the mathematical properties of magic squares, not with magic/superstition, and would be wholly relevant to an introductory article about magic squares (ok, skip the Templar square and numerology ).

Note the wolfram entry provides several other methods of producing magic squares that in a well-researched h2g2 entry on creating magic squares would be mentioned eg the LUX method.

It is sometimes frustrating that in writing an entry one realises how much work is required to do the title justice . Then one must either do the work and learn about the subject in greater detail, or reduce the scope of the entry, by changing the title, to what one actually knows about. I think I've made clear that for me your entry doesn't live up to my expectations of the title.

If you are interested in the mathematicss of magic squares I'm sure you could do a good job of expanding this entry to describe the various ways that squares can be mathematically magic. Methods of creating magic squares can reasonably be budded off to be a linked entry for those who are interested, but shouldn't be the overwhelming content of this entry.

My suggestions boild down to:
> change the title and stick with one entry.
> expand the entry to live up the present title (might get a bit unwieldy)
> split the entry into two parts - general magic square details, and methods to create magic squares.

O.K. Title changed to limit the scope of the entry, and a link added to the Wolfram site.

I never realised that you could start with "1" in any column, not just the middle one; but I'm not sure how to incorporate that into the entry. as it would neccesitate removing the "checkpoint formulae".

First thought: FEWER CAPITALS. Please - only the first letters of important words in the title, headers and subheaders should be capitalised. So your entry should be called "How to Construct a Magic Square of (Almost) Any Size". Or how about "How to Construct a Magic Square of Any Size (So Long as it's Odd)"? I think that better fits what you actually describe.

I do like this entry - it's a neat way to make a magic square. However, I think your descriptions of going off the right or top could be simplified. Rather than having it as a separate instruction, say that we imagine the leftmost column to be just to the right of the rightmost, and similarly for the top and bottom rows. Then

I think you should also use and tags for the instructions. See <./>GuideML-OL</.>.

Do you know why this method works? To me that would be more interesting than the long list of Excel instructions. Could that be shortened? It takes up too much of the entry for my liking.

I treid to change the headers from upper to lower case, but either I didn't save it or there is an override int here somewhere (either that or it is too late in the evening and I have had too much red cordial).

As to the reason why this method works, I have no idea. I learned how to do this many years ago and can't remember where I got it from, or whether the instructions included any sort of rationale, but I'm happy for this entry to be a "How To" rather than a "Why"

The same method is listed in the linked web page, which uses graphics to describe how to construct the square which saves a lot of space. It also points out that you don't have to start in the middle column, which was news to me.

There shouldn't be a problem with changing the headers.

Mathworld's claim you can start anywhere doesn't hold up - imagine you start anywhere on the top-right/bottom-left diagonal (is there a name for it?). That diagonal will then contain 1, 2, 3, 4 and 5, which add up to significantly less than 65.

I'm going to have to work out why for myself, aren't I? Well, there's a symmetry to the square, in that any two numbers in opposite positions add up to 26 (with 13 in the central square). That takes care of the lines of symmetry (diagonals, central row and column), but I'm not sure about the other rows and columns.

Yes if you don't start with "1" in the middle column, the diagonals don't both add up to the right value (I only tested it with a 5 x 5 square, but one failure denies the proof).

The symmetry of the diagonals is also true, and if you rotate around the centre point, the opposite cells on the outer "ring" also add up to 26 (again in a 5x5) but I can't see why each number should go where the method puts them enough to make a rule

Hokay, here's what I reckon:

If we look at the n by n square as wrapping around, torus-like, there are n positive diagonals. The numbers that go on the diagonals are 1--n, n+1--2n, 2n+1--3n, ... and (n-1)n+1--n^2, which can be written as 1--n, n+1--n+n, 2n+1--2n+n, (n-1)n+1--(n-1)n+n. Take any row. A member of each of these groups falls in the row, say in+j_i, where i is between 0 and n-1 and j_i is between 1 and n. Because in+1 falls in a different row for each i, the j_i must all be different, ie they are 1,..,n in some order. The sum of the row is (summing over i):

\sum (in+j_i) = n*\sum i + \sum j_i
= n*n(n-1)/2 + n(n+1)/2
= n(n^2+1)/2

So all the rows sum to the same number. Similarly, so do the columns.

QED

Makes sense to me (just about) - thanks.

I'm afraid that might be a bit dry for inclusion in the article, though.

How's this coming along?

Not at all - I don't think it's really worth an edited entry - happy for someone else to use it as the basis for one if they feel so inclined.

I know I had certain problems with the article, but I think it's worth a place in the Edited Guide. Admittedly there's a few things to iron out first.