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Pasles2 12/12/03 2:52 PM Page 10 Take a trip and visit magic squares from India, Spain, Mars, and Jupiter. Some Magic Squares of Distinction Paul C. Pasles Villanova University U nless you’ve actually been living on Mars—and maybe In this magic square, the usual entry set {1, 2, …, 16} has even if you have—you should be familiar with the been altered just slightly; there’s no occurrence of 16, and we notion of a ‘magic square.’ One hasty definition, not have instead an extra 8. Let’s call this example the ‘crop altogether incorrect, is as follows: a magic square is a square square.’ (Magic squares were often engraved on circular plates matrix of nonnegative integers in which each row, column, and and used as talismans or charms. Obviously the process would diagonal adds up to the same constant. See Figure 1 for a clas- turn this one into a ‘crop circle.’) sic example. Like most first attempts at a definition, this one probably Genetically Modified admits more creatures than we had originally intended, for Crop Squares example an n ¥ n constant matrix. But that’s okay. Let’s keep But can the set of entries always be modified in this fashion? things fairly loose and permit the set of entries to be any prop- That is, can we drop the 16 and, say, double the 9? What if er subset of {0, 1, 2, 3, …, n2}, so that repeats are not prohib- instead I prefer to have two copies of my personal favorite ited. These will be our magic squares of distinction, for their number, 12? (Ever notice that people’s favorite numbers are entries are not necessarily distinct. (If this seems a little back- usually positive integers? What’s up with that? Why not –3 or ward, just remember what the number theorists say, “Two is p or e?) the oddest prime, because it is the only even one.”) There are two approaches to the question: trial and error, or Which brings us to an item for sale in the ‘New Age’ sec- actual thinking. The second approach is more fun, so let’s try it. tion of your local book superstore. The Complete Book of If a magic square has entries {1, 2, …, 16} then each row adds Magic and Witchcraft is an ambitious title for a small paper- up to 34. (Why?) In the crop square, the magic sum is 32, back. True to its name, this book is at least complete enough to because the 16 in our list of entries has been replaced by an 8, include a magic square of distinction, illustrated in Figure 2. and that decrease must be spread equally over all rows. You can (The author, Kathryn Paulsen, cites a 1905 publication of the see pretty quickly that some replacements will never work, and Anthropological Society of Bombay.) that in fact the only possibilities left are 4, 8 and 12; that is, you 11 24 7 20 3 1 8 9 14 4 12 25 8 16 17 5 13 21 9 11 12 3 6 10 18 1 14 22 7 2 15 8 23 6 19 2 15 13 10 5 4 Figure 1. The “Mars” magic square (after Heinrich Cornelius Figure 2. A magic square “for protection of crops (India),” from Agrippa, 1531). The Complete Book of Magic and Witchcraft. 10 FEBRUARY 2004 + ☺ Pasles2 12/12/03 2:52 PM Page 11 MATH HORIZONS Figure 3. Magic square found in a Figure 4. The “Jupiter” square, vari- Figure 5. The Jupiter square after one Barcelona Cathedral. Photo by Gloria ation from Albrecht Dürer’s Melan- half-rotation. Fluxà. cholia I. can only replace 16 in the set by another multiple of four. magic square of distinction. Clearly, this is indicative of a How about a general theorem? Complete this sentence: If an general principle: the sum or difference of magic squares is n ¥ n magic square includes the entries {1, 2, 3, …, n2 } \ {t}, again magic. and j appears twice in the square, where j and t are both integers 1 14 15 4 1 14 14 4 between 1 and n2, then… (Answer: j ≡ t (mod n). The proof is similar to the argument just given.) 12 7 6 9 11 7 6 9 – Now, that’s certainly not to say that 4, 8, and 12 are all actu- 8 11 10 5 8 10 10 5 ally possible — just that nothing else is. We can get away with 13 2 3 16 13 2 3 15 double 8’s, since that’s the very example that started this dis- cussion. But can we double the 12? How about the 4? These 0 0 1 0 two questions turn out to be equivalent, and we’ll answer them 1 0 0 0 soon. But first: instead of doubling just one entry, why not go = . 0 1 0 0 for “two pair”? 0 0 0 1 Moving On, Now let’s try to find a magic permutation matrix that gives From Mars to Jupiter an answer with two 12s instead. That means we want a 4 ¥ 4 Regular readers of Math Horizons will recognize the matrix of zeros and ones, where each row, column and diago- Barcelona magic square from the inside back cover of the Sep- nal sums to 1. If the nonzero entries happen to appear in the tember 2002 issue. The mathematical portion is enlarged in same spots as the 13, 14, 15, and 16 in some traditional magic Figure 3. Notice that the integers 12 and 16 are missing, while square, then subtraction will give us the desired solution. (By 10 and 14 appear twice each. Thus the ‘magic sum’ in this ‘traditional,’ I mean that the entries are 1, 2, 3, º, 16.) square is not 34, but rather 33, signifying the number of years So, do there exist traditional magic squares wherein 13, 14, Jesus lived. (To underscore the point, a nearby statue depicting 15, and 16 occupy such convenient positions? It turns out that, the Judas kiss can be seen in the original photograph.) not only is the answer yes, but these are fairly common; pick a A look at the Barcelona square will help us solve our other 4 ¥ 4 magic square at random, and there is one chance in five puzzle. First, note that it bears no small similarity to the that it will satisfy this condition. Unfortunately, our friend famous magic square given in Figure 4. This particular depic- Jupiter isn’t one of them, but two of its close relatives are: tion is taken from Albrecht Dürer’s Melancholia I of 1514 (though like the Mars square it is really much older than that). 16 3 9 6 1 14 11 8 In case you don’t see the resemblance, rotate it by 180 degrees 5 10 4 15 12 7 2 13 as illustrated in Figure 5. 2 13 7 12 6 9 16 3 A quick matrix subtraction summarizes the discrepancies between Barcelona and Jupiter. The difference is a nice little 11 8 14 1 15 4 5 10 permutation matrix, so called because if you multiply it by a vector in 4-space, you just permute the coordinates. It is also a Subtract off the appropriate 0,1 matrix from each + WWW.MAA.ORG 11 Pasles2 12/12/03 2:52 PM Page 12 MATH HORIZONS 1 0 0 0 0 1 0 0 a b c d a a b c d 0 0 0 1 0 0 0 1 b b a d c 0 1 0 0 0 0 1 0 c c d a b 0 0 1 0 1 0 0 0 d d c b a and you get just two of the many solutions to our original ques- Drop the headings for this or any other group table and you tion. These are shown below. (Notice the peculiar affinity always get a Latin square: every row and every column fea- between all of the permutation matrices covered so far.) tures the same set of entries. For consistency’s sake, let’s rename the letters as numbers: 15 3 9 6 1 13 11 8 0 1 2 3 5 10 4 14 12 7 2 12 1 0 3 2 2 12 7 12 6 9 15 3 2 3 0 1 11 8 13 1 14 4 5 10 3 2 1 0 That takes care of 12. What about 4? Easy: replace every entry k by its “complement” 16 – k, and your double 12s You can convince yourself pretty quickly that this Latin square become double 4s! And now it’s time to do four sets of four is almost magic, but not quite. Those lousy diagonals don’t repetitions each. work. No matter; just swap the first and last rows, and all is well. Voila! A magic square of distinction. Four of a Kind: 3 2 1 0 Some Latin Squares are Magic Squares of Distinction 1 0 3 2 2 3 0 1 Warning. Some material in this section may be unsuitable for those viewers who have yet to take a course in abstract algebra. 0 1 2 3 We’ve seen a magic square with exactly two pairs of repeats. Can we get four sets of quadruplets? Moral: Every group table is a Latin square, but only some There are two non-isomorphic groups of order four. One of Latin squares are magic squares of distinction. ■ them, the Klein Vier-Group, has the group table that follows. (‘Vier,’ German for 4, is pronounced almost like the English ‘fear;’ the vowel sound is long ‘e,’ not ‘i.’ As my high school Conclusion German teacher used to say, “When two vowels go walking, That was quite an itinerary. We’ve visited India, Spain, Mars the second one does the talking.”) and Jupiter. See where you can go, if only you believe in magic? PEANUTS reprinted by permission of United Feature Syndicate, Inc. A little bird experiments with constant matrices. 12 FEBRUARY 2004 r

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