First
"Russian Alphamagic Square"
discovered by ASU professor and student
Dr Croft (left) is Head of the Faculty of German, Romanian, and Slavic Languages; School of International Letters and Cultures. Samuel Comi (right) is a student in Croft's RUS-212 Russian Conversation course, a sophomore mathmatics major, and active in the Chess Club. Croft had been researching the problem for over 10 years, then Comi wrote a computer program that found the solution in one weekend.
Croft and Comi call their discovery the "Lee Sam," (a pun on Croft's Russian signature "Ли сам") and Sam's name. They gratiously advise everyone to keep a copy of it in their possession, so as to extend their life by 36 years. Magic Squares are ancient Chinese good luck charms.
The 3 x 3 array of numbers (74, 50, 92 / 90, 72, 54 / 52, 94, 70) has several unique properties:
- each row, column, and diagonal adds to 216 — definition of
a magic
square
- it's also a Russian alpha-magic square because the number
of letters in each Russian number creates
another a magic square in which each row, column, and diagonal adds to
36.
- this is the first
Cyrillic (Russian) and first non-Roman-alphabet alpha-magic square to
be
documented
|
||||||||||||||||
|
Dr Croft explains:
"An alpha-magic
square is a math puzzle in which the numbers of letters
needed to spell the numbers also form a magic square … so
that the
array
above adds to a constant sum of 216 on any row, column, or diagonal AND
the numbers of Cyrillic letters needed to spell the Russian names of
the numbers in this array, or precisely 15, 9, 12 / 9, 12, 15 / 12, 15,
9, also adds to a constant sum (36) on any row, column, or diagonal."
"The concept of alpha-magic squares is derived from a fifth-century Anglo-Saxon runic charm called by the discovering scholar Lee C.F. Sallows of Holland the 'Li Shu' (since the very first discovered magic square in China, circa 2300 B.C. is called the 'Lo Shu')."
"The alphamagic square represents a very rare confluence of 'magic' between the world of numbers and the world of letters. The runic original was reputedly devised by an anonymous wizard of Legendary King Mi (perhaps King Ida (550-616 AD)) to extend by its magic the King's life by the number of years of the secondary square's magic sum."
"The concept of alpha-magic squares is derived from a fifth-century Anglo-Saxon runic charm called by the discovering scholar Lee C.F. Sallows of Holland the 'Li Shu' (since the very first discovered magic square in China, circa 2300 B.C. is called the 'Lo Shu')."
"The alphamagic square represents a very rare confluence of 'magic' between the world of numbers and the world of letters. The runic original was reputedly devised by an anonymous wizard of Legendary King Mi (perhaps King Ida (550-616 AD)) to extend by its magic the King's life by the number of years of the secondary square's magic sum."
More than 10 years of research to find a solution
Dr. Croft was a former math major and has been fascinated by magic squares for a long time. He says, "I encountered the alphamagic square work of Lee Sallows about ten years ago. I made an initial attempt to find a Russian one and failed. I revived the effort inspired by math-puzzlist extraordinaire Martin Gardner who proved mathematically that the third-order magic cube is impossible."
Croft reports:
"But I had produced an earlier
'semi-magic' 3 x 3 x 3
cube by
triply applying the 'Siamese method' to adjacent two-dimensional
squares and had some partial success with a Russian magic square with a
semi-magic (one diagonal out) logorithmic square, and a semi-magic 4x4
square with a magic logorithmic."
"In 2007 last semester, I began to try to enlist people of computer savvy to apply Sallows' ALPHA.BAS variation of Pascal to a list of Russian logorithms. I asked a math professor, a math-major former student, even our department computer wizard. I think I failed to completly explain the problem and provide them with enough information. I got no where."
"Over Christrmas break I just about wore out a ream of paper trying to find the needed concentric constant-difference triples and came tantalizingly close, as it later turned out. But when I returned to ASU this 2008 semester I presented the idea to my RUS-212 class (a real conglomeration of young geniuses) in which was student Sam Comi, and he immediately proposed to do a complete rewrite of the problem in Javascript. So I armed him with Sallows' articles (there are actually two), his Pascal-based computer program, the Edouard Lucas formula for general 3x3 magic squares, and the list of Russian logorithms."
"In 2007 last semester, I began to try to enlist people of computer savvy to apply Sallows' ALPHA.BAS variation of Pascal to a list of Russian logorithms. I asked a math professor, a math-major former student, even our department computer wizard. I think I failed to completly explain the problem and provide them with enough information. I got no where."
"Over Christrmas break I just about wore out a ream of paper trying to find the needed concentric constant-difference triples and came tantalizingly close, as it later turned out. But when I returned to ASU this 2008 semester I presented the idea to my RUS-212 class (a real conglomeration of young geniuses) in which was student Sam Comi, and he immediately proposed to do a complete rewrite of the problem in Javascript. So I armed him with Sallows' articles (there are actually two), his Pascal-based computer program, the Edouard Lucas formula for general 3x3 magic squares, and the list of Russian logorithms."
"Sam Comi found the Russian magic
square over a single weekend!"
News of this discovery will be published in several mathmatical journals and presented at conferences. Dr. Croft did their first public presentation — “The Search for a Russian Alphamagic Square”(download PDF) — at the AATSEEL meeting, University of Arizona, Tucson on April 19. An article about this discovery entitled "Russian Alphamagic Squares" by Lee B. Croft and Samuel Comi has been accepted for publication in Word Ways: The Journal of Recreational Linguistics, and they plan to also submit to The Journal of Recreational Mathematics. More news about publications later.
— Submitted by Dr.
Lee Croft, April 17, 2008.
Updated April 22, 2008
Updated April 22, 2008
Update Jan 7, 2009:
- Croft, Lee B. and Comi, Samuel. (2008) “Russian Alphamagic Squares” in Word Ways: The Journal of Recreational Linguistics, Vol. 41, No. 2 (May 2008), pp. 95-100.
- Croft, Lee B. and Comi, Samuel. “A Fourth-Order, Digitally-Reversible, Polylingual, Bialphabetic Alphamagic Square.” Journal of Recreational Mathematics. Vol 34, No. 3 (2005-2006), pp. 247-257.
- “The Magic of Alphamagic Squares” by: Lee B. Croft in Croft, Lee B. (1946--), Andrew W. Abbott, Alicia C. Baehr, Jeremy Ecton, Jon Harris, Patrick J. Heuer, Vadim S. Kagan, Kyle M. Kucharski, Jaime R. Nielsen, Megan Plachecki, Shane C. Sarlo, Eric D. Strachan, and Shamella Tribble. NOT TO PERISH: The Articles of an American Professor of Russian. Capstone Publications, Phoenix, AZ 85044 USA, 2009, 299 pp. il. ISBN 978-0-578-00468-6.
The Lucky Magic Square
A 3 x 3 alphamagic square is a magic square for which the number of letters in the word for each number generates another magic square, for instance:
|
|
|
||||||||||||||||||
| = 45
(magic
sum) |
=
21 (logorithmic magic sum) |
This is the actual "Li Shu," the
first runic
charm containing it
discovered by Lee C. F. Sallows. King Mi was reputedly 45 years old
(the first magic sum) and the wizard who made it for him was likely
21-years old (the logorithmic magic sum).
Dr. Croft points out, "The listed lifespan of King Ida (whom I think is the King referred to in Sallows' source as "King Mi") is precisely 66 years (the sum of the Li Shu's primary constant of 45 and the logorithmic constant of 21, the anonymous wizard's age ... so that the wizard devised a charm that added his age to that of his King ... and the King lived so long)."
Dr. Croft points out, "The listed lifespan of King Ida (whom I think is the King referred to in Sallows' source as "King Mi") is precisely 66 years (the sum of the Li Shu's primary constant of 45 and the logorithmic constant of 21, the anonymous wizard's age ... so that the wizard devised a charm that added his age to that of his King ... and the King lived so long)."