Alphamagic Squares
Ivars Peterson
Magic squares have fascinated people for thousands of years. They consist of
a set of whole numbers arranged in a square so that the sum of the numbers is
the same in each row, in each column, and along each diagonal.
Some magic squares have special properties, such as using only consecutive
numbers. In ancient China, a three-by-three square that uses all of the digits
from 1 to 9 was said to bring good luck. The numbers add up to 15 in all rows,
columns, and diagonals.
In 1986, puzzle enthusiast Lee Sallows of the University of Nijmegen in the
Netherlands introduced a new form of magic square. He found magic squares for
which the number of letters in the word for each number in a magic square
generates another magic square.
Here's an example. Start with the following magic square, in which each of
the rows, columns, and diagonals adds up to 45:
Spell out the English word for each of these numbers to obtain the following
array:
| five |
twenty-two |
eighteen |
| twenty-eight |
fifteen |
two |
| twelve |
eight |
twenty-five |
Count the number of letters in each word and enter that number in the
appropriate space of a blank three-by-three grid.
The result is another magic square?one in which the rows, columns, and
diagonals add up to 21. This square contains the consecutive digits from 3 to
11.
Here's another alphamagic square, in English:
It turns out that there is a surprisingly large number of alphamagic squares,
not only in English but also in many other languages. Indeed, Sallows worked out
procedures for readily constructing additional alphamagic squares.
In French, there is just one alphamagic square involving numbers up to 200.
However, if the size of the entries is increased to 300, an additional 255
alphamagic squares occur. For entries less than 100, none occurs in Danish or in
Latin, but 6 occur in Dutch, 13 in Finnish, and an incredible 221 in German.
Here's a German alphamagic square:
| f?nfundvierzig (45) |
zweiundsechzig (62) |
achtundf?nfzig (58) |
| achtundsechzig (68) |
f?nfundf?nfzig (55) |
zweiundvierzig (42) |
| zweiundf?nfzig (52) |
achtundvierzig (48) |
f?nfundsechzig (65) |
Indeed, if every number in a certain language happened to be written out with
the same number of letters, you could get as many magic squares as you would
care to imagine. However, squares with repeated entries are much less
interesting than those in which entry is different.
It's tempting to expand the search to include other possibilities. In
English, is there a three-by-three square from which a magic square can be
derived, which in turn yields a third magic square?a magic triplet?
Furthermore, are there any instances of four-by-four and five-by-five
language-dependent alphamagic squares?
A quick search turns up several examples. The following table of numerical
values is an example of a four-by-four alphamagic square in English:
| 26 |
37 |
48 |
59 |
| 49 |
58 |
27 |
36 |
| 57 |
46 |
39 |
28 |
| 38 |
29 |
56 |
47 |
There's much more to be learned and investigated in the realm of higher-order
squares.
"With the transition from order 3 to order 4, and higher, comes a concomitant
jump in the perplexities confronting our advance, since hindsight reveals order
3 as a special, unusually tractable case," Sallows wrote in an article
describing his discoveries. "The problems involved having largely resisted
solution so far, this higher ground has been barely surveyed, let alone
exhausted."
"As a result," he added, "it is no exaggeration to say that for programmers
and pencil-owners alike, there remain rich pickings to be had, given ingenuity
and the will to explore."
References:
Peterson, I. 1999. Magic tesseracts. Science News
Online (Oct. 16). Available at http://www.sciencenews.org/sn_arc99/10_16_99/mathland.htm.
______. 1996. More than magic squares. Science News
Online (Oct. 12). Available at http://www.sciencenews.org/sn_arch/10_12_96/mathland.htm.
______. 1990. Islands of Truth: A Mathematical Mystery
Cruise. New York: W.H. Freeman.
______. 1986. Games mathematicians play. Science News
130(Sept. 20):186-189.
Pickover, C.A. 2002. The Zen of Magic Squares, Circles,
and Stars: An Exhibition of Surprising Structures across Dimensions.
Princeton, N.J.: Princeton University Press.
Sallows, L.C.F. 1994. Alphamagic squares. In The Lighter
Side of Mathematics: Proceedings of the Eug?ne Strens Memorial Conference on
Recreational Mathematics and Its History, R.K. Guy and R.E. Woodrow, eds.
Washington, D.C.: Mathematical Association of America.
______. 1994. Alphamagic squares, part II. In The Lighter
Side of Mathematics: Proceedings of the Eug?ne Strens Memorial Conference on
Recreational Mathematics and Its History, R.K. Guy and R.E. Woodrow, eds.
Washington, D.C.: Mathematical Association of America.
______. 1987. Alphamagic squares, part II. Abacus
4(No. 2):20-29, 43.
______. 1986. Alphamagic squares. Abacus 4(No.
1):28-45.
Stewart, I. 1997. Alphamagic squares. Scientific
American (January):106-109.
**********
A collection of Ivars Peterson's early
MathTrek articles, updated and illustrated, is now available as the
Mathematical Association of America (MAA) book Mathematical Treks: From
Surreal Numbers to Magic Circles. See http://www.maa.org/pubs/books/mtr.html.
| Comments are welcome. Please send messages to Ivars
Peterson at ip@sciencenews.org.
 Ivars Peterson is the mathematics writer and online editor at
Science News. He is the author of The Mathematical Tourist, Islands of Truth, Newton's Clock, Fatal Defect, The Jungles of Randomness, and Fragments of Infinity. He also writes for the
children's magazine Muse (see MatheMUSEments at http://home.att.net/~mathtrek/). The
Mathematical Association of America has published a collection of his online
MathTrek
articles.
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