Tuesday, 26 February 2013

The 3x3 Magic Square, More Magical than You Thought!

I just came across an older article from the Journal of Recreational Mathematics about the 3x3 Magic square that reminded me of some beautiful relations in the square, and showed me a few I had never seen. The article is by Owen O'Shea and is titled "SOME WORDS ON THE LO SHU". If you want to search out the whole thing (well worth the read) it is in Volume 35(1) starting on page 23.

The Lo Shu Square ( literally: Luo (River) Book/Scroll) is the unique normal magic square of order three. Except for rotations or reflections it is the only order three magic square that can be formed with the digits 1-9. Chinese legends concerning the pre-historic Emperor Yu tell of the Lo Shu: In ancient China there was a huge deluge: the people offered sacrifices to the god of one of the flooding rivers, the Luo river, to try to calm his anger. A magical turtle emerged from the water with the curious and decidedly unnatural (for a turtle shell) Lo Shu pattern on its shell: circular dots giving unary (base 1) representations of the integers one through nine are arranged in a three-by-three grid. The representation in the more common Arabic Numerals looks like this:

The odd and even numbers alternate in the periphery of the Lo Shu pattern; the 4 even numbers are at the four corners, and the 5 odd numbers (outnumbering the even numbers by one) form a cross in the center of the square. The sums in each of the 3 rows, in each of the 3 columns, and in both diagonals, are all 15 (the number of days in each of the 24 cycles of the Chinese solar year.

Beyond the basics of the magic square, O'Shea points out several other interesting relations.  First, the sum  squares of the numbers in the top and bottom row are equal.  42 + 92 + 22 = 82 + 12 + 62 = 101.  You can do the same thing with the two outside columns, 42 + 32 + 82 = 22 + 72 + 62 = 89.  Go ahead, try the two diagonals, you now you are dying to know. 

So what about the middle row and column?  Well, the middle column is special; Because north is placed at the bottom of maps in China, the 3x3 magic square having number 1 at the bottom and 9 at the top is used in preference to the other rotations/reflections. As seen in the "Later Heaven" arrangement, 1 and 9 correspond with ☵ Kǎn 水 "Water" and ☲ Lí 火 "Fire" respectively. In the "Early Heaven" arrangement, they would correspond with ☷ Kūn 地 "Earth" and ☰ Qián 天 "Heaven" respectively. The 951 does have a nice numerical representation in the number. If you  read the rows or columns as three digit numbers, you might notice that 492 – 357 + 816 = 951 and that 294 – 753 + 618 = 159. Kind of a transition from Heaven to Earth and back again.
An original O'Shea contribution is his discovery that, "Ignoring the middle column, form two-digit numbers with the other columns as follows: 42 + 37 + 86. These numbers sum to 165. Their sum of their
reversals, 68 + 73 + 24, is also 165. The same is true of 84 + 19 + 62 and their reversals, 26 + 91 + 48. Curiously, the sum of the squares of the odd digits, 1, 3, 5, 7, and 9, also equals 165."

If we go back to considering the rows as a three digit number, the square of each row numeral is the same as the square of their reversal:  4922 + 3572 + 8162 = 6182 + 7532 + 2942. Of course that would be really impressive if it worked with the columns too... I mean, awesome impressive... ahh go on, try it.

The article goes on with several dozen interesting numerical relations, and if that's your thing, you should seek it out.  I'll leave you with one last beauty:
There is a not too well know problem in math called the Tarry-Escott problem  which asks if there are sets of integers with the same order (same number in each set) so that the integers in each set have the same sum, the same sum of squares, etc.up to and including the same sum of kth powers.
Remarkably, the pattern in the lo shu gives a solution to the Tarry-Escott problem. Starting at the top left and reading around the outside you get the four three digit numbers, 492 ,276 , 618 , 834 .  Now read them going the other way round, 438, 816, 672, 294.    Now add up the numbers in each set.  Add up their squares..... their cubes?

I just became aware of, and have not yet read, a 2008 book about the 3x3 magic square by Frank Swetz.

Interesting looking chapters on the use as in fortune telling and Fengshui as well as a chapter on the 3x3 square in other cultures. 

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