Last year I started writing a little essay about what editors do in their spare time, which was, of course, about what I do in my spare time, or to be more exact, that time around midnight when I'm not ready for bed, dare not make a noise, and don't feel like reading. The truth of the matter is that I don't do much at all; sometimes I just sit and think, and sometimes (I feel that someone has said this before) I just sit. I wasn't going to pretend that at such times I invariably think about prestressed concrete verse, although that is usually when I do think about it, but I was going to proceed to a minor exercise in the craft that might have amused you. The drafting of this little essay was interrupted, first by a bout of paying work, then by a prolonged period of personal upheaval and compulsive alliteration, and the upshot of this is that I didn't get back to what I had in mind last year until last night. My notes are on the back of an envelope somewhere, but I don't need them: the thesis is simple enough to reconstruct.
This is the thesis. It is possible to take the numbers 1 to 32 and arrange them in 20 lines of 8 numbers in such a way that every pair of numbers in 32 (496 in all, from 1 and 2 to 31 and 32) occurs at least once, and -- this is the bit that really makes it fun -- that every line adds up to 132.
I hesitate to affront your intelligence by setting out the solution here, but will do so for the sake of readers who had trouble with their 11-times table at school and are still unsure about Fibonacci numbers, factorials and things like that.

01  14  07  12  32  19  26  21
15  04  09  06  18  29  24  27
10  05  16  03  23  28  17  30
08  11  02  13  25  22  31  20


01  15  10  08  32  18  23  25
14  04  05  11  19  29  28  22
07  09  16  02  26  24  17  31
12  06  03  13  21  27  30  20


01  04  16  13  32  29  17  20
14  15  03  02  19  18  30  31
07  06  10  11  26  27  23  22
12  09  05  08  21  24  28  25


01  09  03  11  32  24  30  22
14  06  16  08  19  27  17  25
07  15  05  13  26  18  28  20
12  04  10  02  21  29  23  31


01  06  05  02  32  27  28  31
14  09  10  13  19  24  23  20
07  04  03  08  26  29  30  25
12  15  16  11  21  18  17  22
That's the prestressed part of this PCV. Now I allocate a letter to each number, sometimes the same letter to several numbers, and this is what happens.

Q  B  S  M  A  K  L  I
E  I  R  U  A  I  N  U
U  I  S  B  E  L  E  U
E  S  N  C  S  N  T  R


Q  E  U  E  A  A  E  S
B  I  U  S  K  I  L  N
S  R  S  I  L  N  E  T
M  N  B  C  I  U  U  R


Q  I  S  C  A  I  E  R
B  E  B  N  K  A  U  T
S  U  U  S  L  U  E  N
M  R  I  E  I  N  L  S


Q  R  B  S  A  N  U  N
B  U  S  E  K  U  E  S
S  E  I  C  L  A  L  R
M  I  U  N  I  I  E  T


Q  U  I  N  A  U  L  T
B  R  U  C  K  N  E  R
S  I  B  E  L  I  U  S
M  E  S  S  I  A  E  N
Pretty basic, isn't it? -- not much to show for the effort involved -- just a short list of composers in chronological order, no rhyme and little reason. But what fascinated me when I did this the first time (on paper, last night) was the second last block of letters. This is what I saw first.

Q   R   B   S   A   N   U N
B   U   S   E   K   U   E   S
S   E   I   C   L   A   L   R
M   I   U   N   I   I   E   T

I wasn't expecting that. I looked further in this block, and of course Quinault, Bruckner and Sibelius are in there, too -- Quinault on the diagonal down from top left, Bruckner starting on line 2 and -- yes, I'm sure you can see that. What makes this interesting is that the letters originally had numerical values, and we know that each horizontal line adds up to 132, so here we have diagonals that must add up to 132.
Now, you may wonder why I started this whole sequence of numbers with 1 14 7 12 instead of 1 2 3 4. This will require a little digression.

A little digression
The first issue of Meanjin for 1991 explored something called "Language Poetry". A lot of it looked like any other sort of modern verse, some of it was fascinating, and some of it, to be charitable, was less interesting than a short list of composers in chronological order. One of these Language Poets seemed to be doing something similar to what I have called prestressed concrete verse, so when he came into the office I got chatting to him in a tactful way about what I was doing. (I was usually courteous to contributors, and this one was dead serious about something that PCV was originally intended to poke fun at.) "Magic squares," he said, totally uninterested, "yeah, it's all been done." Obnoxious man! I suddenly found something urgent to do in another room. I had never heard of magic squares, and wondered what they were, but decided I would much rather ask someone else. No-one I asked could tell me.
Early this year, looking for something else in my 1965 Britannica, I stumbled over its long article on the subject. Magic squares have been puzzled over since antiquity and their unique properties applied to matters military and statistical, among others. (Utterly fascinating. You can learn a lot from old encyclopedias.) One of the basic magic squares is four lines of four numbers, 1 to 16. It looks like this.

01  14  07  12
15  04  09  06
10  05  16  03
08  11  02  13
No matter how you add up the numbers, across, down, diagonally in any direction, you always get 34. So I thought I would use that this time, instead of the rather prosaic sequential square I started with last year.

This is how Messiaen looks when the original numbers are restored.

01  09  03  11  32  24  30  22
14  06  16  08  19  27  17  25
07  15  05  13  26  18  28  20
12  04  10  02  21  29  23  31
Note that, as I said above, the numbers add up to 132. Not only there: each set of eight numbers, across, down (two columns, separated by three others), and diagonally in any direction, adds up to 132. So what we have here is a magic rectangle.
A little frenzied calculation reveals that what we have here is in fact five magic rectangles. A little fitful contemplation explains why this is so -- and why I didn't need to start with a magic square to create magic rectangles. I could have started with the numbers 1 to 16 in any of the 20,922,789,890,000 possible ways of arranging them and would have achieved the same result. (If you don't mind, I won't attempt to demonstrate this here.) The secret, such as it is, is in the way I manipulated the numbers.
Now that I see what I have done, it is so obvious that I'm not sure why I bothered. Maybe this is what the Language Poet meant. In a curious way the whole exercise puts me in mind of Oscar Wilde's story "The Portrait of Mr. W.H." . . . You haven't read it? Oh, you must! Stop counting those numbers and do it now!

The Society of Editors Newsletter, 1996





John Bangsund
Melbourne, Australia



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