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NEW ADVANCES WITH 4 X 4 MAGIC SQUARES
by Lee Sallows
Introduction
One of the best known results in the magic square canon is Bernard Frnicle de Bessy's enumeration of the 880 normal 4(4 squares that can be formed using the arithmetic progression 1,2,..,16. A natural question this suggests concerns non-normal squares: Is 880 the largest total attainable if any 16 distinct numbers are allowed?
The answer is no. A computer program that will generate every square constructable from any given set of integers has identified 1040 distinct squares using the almost arithmetic progression: 1,2,3,4,5,6,7,8,10,11,12,13,14,15,16,17. Note the doubled step from 8 to 10. Extensive trials with alternative sets make it virtually certain that 1040 is the maximum attainable (or 8 ( 1040 = 8320, if rotations and reflections are included), although an analytic proof of this assumption is lacking. A listing of the 1040 squares can be had on request via email at lee.sal@inter.nl.net.
But if 1040 is the maximum, what of lower totals? Can a set of numbers be found that will yield any desired total under 1040? If not, what totals are possible? In the absence of mathematical insight, trial and error was the only way to find out. The results of computer trials on thousands of different sets of small integers are presented below. Again, although proof is lacking, there are empirical grounds for supposing that every possible total has been identified. Conceivably, one or two may have escaped the net. In any case, the wealth of new data represented by these findings will provide a starting point for further researches in the field for years to come. Some preliminary comments will be helpful in understanding these results.
Fertility and set structure
We define the fertility, f, of a set of 16 distinct elements, usually integers, as the total number of 4(4 magic squares it can produce, rotations and reflections included, divided by 32. This is because there are magic-preserving rearrangements besides rotations and reflections that are applicable to any square such that it always has 31 variants that we may count as equivalent. That is, the group of magic-preserving transformations is of order 32. For example, the following transformations can be combined with rotations and reflections to produce all 32 squares in the group:
EMBED Equation.3 ( EMBED Equation.3 EMBED Equation.3 ( EMBED Equation.3
Thus, for the above-mentioned set, f = 8320 ( 32 = 260, the putative maximum, while for a set of 16 consecutive integers, f = 880 ( 8 ( 32 = 220. Besides these two cases, integer sets representing around one hundred different fertility values have been found, as listed in Tables 1 and 2 below.
Curiously, among these new finds is a second set yielding f = 220, but not using consecutive integers. In other words, for every one of the 880 normal squares, there exists a unique counterpart square using 16 numbers than do not form an arithmetic progression. An example is as follows:
EMBED Equation.3
In comparing different sets it is useful to look at their structure rather than the numbers occurring. This is because the fertility of a set is unaffected by adding a constant to each member, or by multiplying each member by a constant, so that different numbers need not imply different fertilities. Sets are thus better described by the sequence of 15 differences reduced to lowest terms occurring between adjacent members when ordered by magnitude. And since differences in the sets considered here are always less than ten, a single digit suffices for each. Thus the sets {-3,-1,1,3,5,7,9,13,15,17,19,21,23,25,27} and {1,2,3,4,5,6,7,8,10,11,12,13,14,15,16,17} are both interpreted as particular instances of the structure 111111121111111, which is our canonical representation of all sets for which f = 260. The two set structures yielding f = 220 are then 111111111111111, representing arithmetic progressions, and 111211111112111, as exampled in the magic square above.
The foregoing is an example of a palindromic or symmetric set, which is one composed of 8 complementary pairs of numbers having equal sums, such as 1 & 16, 2 & 15, .. 8 & 9 in normal squares. If we think of the numbers as points along the real number line, then complementary pairs are points equidistant about a common centre, which is half their sum. Asymmetric sets are non-palindromic, but since the fertility of a set is unchanged by multiplying each member by 1, the fertility of an asymmetric set is the same as that of a set with reversed structure.
These remarks are sufficient to explain Table 1, which lists 66 asymmetric sets yielding fertility values in the range 1 to 71. No asymmetric set with a fertility outside this range was discovered; the five gaps indicate cases for which no asymmetric sets were found. Asterisks indicate fertility values for which symmetric sets also exist. Table 1 extends beyond f = 71 to include the asterisk at f = 76, beyond which no further symmetric set is found until f = 132, as seen in Table 2. These are among many findings that have as yet no theoretical explanation.
The set listed in each case is usually the smallest, in the sense that the sum of its differences is least, but not always. For instance, 111112111212111 also yields f = 1, the sum of its differences being the same as the set structure listed for f = 1 in Table 1. In general, the lower the value, the more asymmetric sets with the same fertility exist. For instance, among all possible sets with structures in the range 111111111111111 to 222222222222222, we find 455 yielding f = 1; a few further examples being as follows:
f 1 3 5 7 9111315171921232527total455248117835319613331300
This is not difficult to understand. Consider a general formula that describes the structure of every 4(4 magic square:
EMBED Equation.3
The number of magic squares that can be formed using the 16 algebraic terms in the formula is clearly the same as the number of magic-preserving rearrangements applicable to every numerical magic square, and no more: 32. The fertility of this algebraic set is thus 1, but sets of greater fertility must represent particular instances of the above satisfying still more stringent conditions, and to that extent will be scarcer.
Table 2 records a total of 63 symmetric set structures, all of them yielding even fertility values in the range 4 to 260. How does it come about that the fertility of symmetric sets is always even?
A symmetric set is one comprising 8 complementary pairs. A magic square remains magic when each of its numbers is subtracted from a fixed total. But if that fixed total is equal to half the magic constant then the effect is the same as that of exchanging every pair of complementary numbers. For example, a square may use the numbers 1,2,..,16. Half the magic constant is 17. But 17 1 = 16, and 17 16 = 1, and so on for the other complementary pairs. Hence, for any magic square S formed from 8 complenetary pairs, there exists a second magic square S1 that is obtained by switching the complementary pairs in S.
Tempting as this explanation may seem, there remains a question to be answered, namely: Can we be sure that is S1 is distinct from S? The answer is no. In particular, S1 will be a 180 degree rotation of S when its complementary pairs are arranged as in H.E. Dudeneys Type III classification [see Amusements in Mathematics p.120], while S1 will be a reflection about a midline when S is of Type VI. The for every square there is a second produced by switching complementary pairs hypothesis thus fails to explain why the fertilities of symmetric sets are always even. I am frankly embarrassed to confess that an explanation of why the fertility of a symmetric set is always even still eludes me.
Returning to Table 2, note that sets having fecundities in the range 1 to 76 have been found, with about 10 gaps, those appearing in the right hand column all being multiples of 4. Beyond these there is a wide gap up to 132, followed by most even values from 132 to 220, again with about 10 gaps. No sets with 220 < f < 260 have been found. Both sets for f = 220 are recorded in the list.
The criteria used in selecting the particular set structure shown for each fertility value in the Tables differs from case to case. Frequently, there exist many alternative structures with the same fertility that could have been shown instead. Many of the results are due to Saleem al-Ashhab, a Jordanian email correspondent. Saleem was excited by my 1040 squares find, and soon had his PC running night and day in search of sets with new f values. In such cases I received only his results, but without details of the search procedure used. Still other results are due to programs of my own, one of which checked blocks of sets in systematic order, calculating the fertility value of each in turn and recording any new values found. I might mention that, thanks to insights due to Saleem, the fertility of a chosen set of 16 integers took our computer program less than 2 seconds to compute on a Pentium II machine. In another approach, sets composed of 16 random numbers were generated one after the other, their fertility calculated and the result stored. Haphazard as this may seem, following an initial flood, with the passage of time the number of fresh fertility values found slowed to a dribble and then finally dried up completely, even following whole days of running time. Hence my confidence in the completeness of the results here presented.
Lee Sallows Oct. 2000
* * * * *
The above is an extract from a more detailed report. Data reported in the two Tables shown below has been similarly reduced. In the fuller version, magic squares produced by a given set structure are analysed in terms of 9 basic classes (depending on the algebraic structure they reflect) and the fertility of each class recorded.
Table 1
Asymmetric set structures and their fertilities
f set structure f set structure
1 111111211112211 45 111211111113111
2 111111212112111 46 111111121114111
3 111211111311111 47 111111111114111
4 111111111131121* 48 111111111312131*
5 111111111112131 49 111211111211121
6 111111111121211 50 111111111111151
7 111111111211121 51 121111111112113
8 111111111212111* 52 111113131111131*
9 111111112122111 53 111111111112111
10 111111211121121 54 111111121112111
11 111111111111321 55 311111212211122
12 111111111121122* 56 111121211111212*
13 111121111111112 57 111111111115111
14 111111211111211 58 121111111111113
15 111111111122211 59 111111122111112
16 111111111111212* 60 111111141113111*
17 111111123111121 61 311111111123212
18 111112111112121* 62 111111112211122
19 111111111111131 63 111111111113111
20 111111111221221* 64 111111142111112*
21 111112111311121 65 no set found
22 111111111113115* 66 no set found
23 111111111111115 67 121111111112133
24 111112222121112* 68 *
25 111112111111311 69 111211111114111
26 111113111211311* 70 no set found
27 211111111111322 71 111111112111112
28 111111111112511* 72 *
29 111112111111222 73 no set found
30 111111121111321* 74 no set found
31 311111111111222 75 no set found
32 111111321112111* 76 *
33 no set found >76 no sets found
34 112111111111122*
35 no set found
36 311111111211321*
37 121111112111134
38 111212121112121*
39 121121241211212
40 111311111114111*
41 111211111111123
42 111112131111121*
43 111111111112212
44 111112111111121* * = symmetric set exists
Table 2
Symmetric set structures and their fertilities.
f set structure f set structure
2 none 4 11111223
6 none 8 11212121
10 none 12 11121221
14 none 16 11112221
18 22443111 20 11111213
22 11111331 24 22121211
26 11111131 28 11122211
30 11222111 32 11111121
34 22121111 36 11112111
38 11222112 40 21121111
42 22121131 44 13111231
46 12321111 48 41121122
50 none 52 12111132
54 none 56 11111221
58 none 60 13111212
62 none 64 11111212
66 none 68 12111112
70 none 72 21112112
74 none 76 11211122
78-130 none 132 23111327 134-142 none 144 23111325 146 23111323 148 11131117 150-4 none 156 12161211
158 12141212 160 13111315
162 21111124 164 11131113
166 none 168 31121132
170 12141211 172 22111224
174 21111123 176 42112111
178 11111115 180 13111312
182 22212111 184 21212122
186 11141111 188 11131111
190 31211111 192 23211112
194 11121113 196 21111121
198 11111113 200 13121111
200 12111211 202 13111111
204 22111111 206 11111114
208 11121112 210 none
212 32111111 218 21111111
220 11111111 220 11121111
260 11111112 >260 none
none = no set found
First half only of palindromic set structures are shown; i.e. 11111112 is short for 111111121111111.
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