In an early twentieth-century edition of The Monist, an article appeared entitled “Magics and Pythagorean Numbers” by Mr. C. A. Browne, Jr. Mr.
Browne was versed in neo-Pythagorean number lore and an astute cre-ator of magic squares. The article later found its way into the book, Magic Squares and Cubes, first published in 1917.30
In the article, Browne examines the origins of certain significant Pythagorean numbers, particularly the number 729. In Plato’s Republic, 729 represents the difference between a “kingly man” and a tyrant as well as the “number of the state.” Building upon his knowl-edge of ancient literature and numerology, Brown evolves an admit-tedly fanciful theory that supplied answers and produced an interesting magic square. According to Brown, the proportions of the Pythagorean Tetractys were embedded into many old number schemes. Plutarch assigned numbers to the planets using such a scheme: 729, (36), represented the Sun; 243, (35), Venus; 81, (34), Mercury; 27, (33), the Moon; 9, (32), Earth; and 3, (31), Antichathon, an imaginary anti-Earth. Plato, in his Timaeus, combined the proportions into one series: 1, 2, 3, 4, 9, 8, 27. In Pythagorean thinking, the num-ber 27 dominates Plato’s series because it is the sum of its predeces-sors: 1 + 2 + 3 + 4 + 9 + 8 = 27. Browne notes the significance of the number 27, particularly its association with the Moon—the Moon com-pletes its elliptical orbit about the Earth in 27 days. Further, 729 = 27
× 27, twenty-seven squared. Buttressed by this knowledge, he seeks to unravel his number mysteries by the use of a 27× 27 normal magic square comprised of the numbers 1 to 729. The square he devised is shown in figure 7.20. Its method of construction appears to be unique for its time.
FIGURE 7.19
5 22 18
28 15 2
12 8 25
FIGURE 7.20
Browne supplied no indication of his methods. Paul Carus, editor of the Monist, and himself a magic square investigator, analyzed the array and noted that the square was composite and that its construction could have been derived from the luoshu by a repeated use of the “Knight’s movement” technique. But no further unraveling of the technique took place. Indeed, the luoshu can be viewed as the building block for this mysterious square. As previously discussed (see page 16), the luoshu
FIGURE 7.21
7 4 1
8 5 2
9 6 3
Luoshu Miscellanea 137
can be constructed by a rearrangement of the numbers of a natural square of order three. See figure 7.21. This rearrangement or permuta-tion of the numbers is given by the yubu algorithm (see page 50), where 1 goes to 6, 2 goes to 1, etc. Mathematically, this permutation can be symbolized by µ, where
µ= .
Let the lattice of cells shown in figure 7.21 also designate a partition of the square, a dividing up, Pk, which operates upon any array of num-bers of order 3ksuch that it divides the configuration into subarrays of order 3k–1. Further, let partition Pk sequence the subarrays as indicated in figure 7.21; that is, the first subarray occupies cell 1 at the upper right, the second subarray occupies cell 2 directly below, and so on until the ninth subarray occupies cell 9. Then using the two operations of parti-tion Pk and permutation µ alternately, any natural number square of order 3n, N(3n), can be transformed into a magic square of order 3n, M(3n). Formula (1) summarizes this technique:
, (1)
where k = n, n – 1, …, 1 since partition Pkoperates on the largest num-ber square first.
Let us construct a ninth-order magic square by using this yubu technique.
Begin with a natural number square of order nine where a Chinese lex-ographical ordering is employed. Apply formula (1). First, P2partitions the square as shown in figure 7.22, and then µ operates on the parti-tioned sets of numbers, mapping them into different positions.
FIGURE 7.22
For simplicity, consider only the mapping of the subarrays or sets of numbers that lie on the diagonal extending from the upper right to the lower left: the set in cell 1 moves to cell 6, the set in cell 5 remains in place, and the set in cell 9 moves to cell 4. See figure 7.23.
Now P1operates on the set of numbers in each cell, followed again by µ, the yubu ordering. The final result of applying formula (1) to the diagonal elements of our original square is shown in figure 7.24.
Finally, when this transformation is completed on the whole square, N(32) → M(32), the ninth-order magic square shown in figure 7.25 is obtained. Marie Baldys, a mathematical games enthusiast in Harrisburg, PA, programmed this algorithm on her computer and applied it to N(33), obtaining Browne’s magic square. She also successfully constructed M(34) and M(35), a magic square of order 243!
This square was developed by applying the yubu technique at two independent levels or stages: first, it was used to form the three-by-three magic squares and second, to order those magic squares to form
FIGURE 7.23
Luoshu Miscellanea 139
larger nine-by-nine magic square. If we began with a 27× 27 natural number square, partitioned it into nine different nine-by-nine number squares, converted each one of those by yubu iteration into a magic square, and then applied the yubu technique again to order the result-ing nine magic squares, we arrive at Browne’s square as shown in fig-ure 7.20. This square is the result of a three-stage process based on the yubu movement.
In his quest to solve the number mysteries, Browne had somehow to arrive at the specific numbers, certainly at least the kingly number 729, and had to meet the ancient conditions regarding the numbers. A hint as to those conditions is related in a conversation from the Republic between Socrates and fellow philosopher Glaucon:
Socrates: And if a person tells the measure of the interval which separates the king from the tyrant in truth of pleasure, he will find him, when the multiplication is completed, living 729 times more pleasantly, and the tyrant more painfully by this same interval.
Glaucon: What a wonderful calculation.
Socrates: Yet a true calculation and a number which closely concerns human life, if human life is concerned with days and nights and months and years.31
Thus, in the calculations from which the number 729 emerges there must be a chronological connection with days, months, and years.
37 48 29 70 81 62 13 24 5
30 38 46 63 71 79 6 14 22
47 28 39 80 61 72 23 4 15
16 27 8 40 51 32 64 75 56
9 17 25 33 41 49 57 65 73
26 7 18 50 31 42 74 55 66
67 78 59 10 21 2 43 54 35
60 68 76 3 11 19 36 44 52
77 58 69 20 1 12 53 34 45
FIGURE 7.25
Incredibly, such a connection can be found in the large magic square:
the central number is 365 (the number of days in a year); the most cen-tral three-by-three magic square has a magic sum of 1,095 (the number of days in three years); next, the central nine-by-nine magic square pos-sesses the magic constant 3,285 (the number of days in nine years); and finally, the magic constant for the whole square is 9,855 (the number of days in 27 years). Viewed as a nested set of magic squares, the magic constants of the respective subsquares form a geometric progres-sion: 1, 3, 9, 27. Delighted with his find, Browne shaded alternate cells in his square to emphasize the yearly connection: 365 white cells corresponded to days and 364 dark cells corresponded to nights. With some further justifications, Browne concluded that the magic constant for his unusual square, 9,855, was the elusive “number of the State.”
Charles W. Trigg, an avid problem solver and student of magic squares, also constructed a 27× 27 magic square by building upon the luoshu. Trigg took the luoshu and augmented it repeatedly by adding nine to all entries. He then combined the resultant nine third-order magic squares into a composite ninth-order square; 134, 217, 728 such squares were possible. These ninth-order squares could in turn be manipulated and then combined to form a composite twenty-seventh-order square. Trigg calculated that (89)2 or 18, 014, 398, 509, 481, 984 such squares are possible. All of these squares are comprised of the first 729 integers and possess the magic constant 9,855.32