Gardner, Martin;
Aha! Insight
W.H. Freeman & Company 1978-04
ISBN-13: 9780716710172 / 071671017X paperback
topics: | math | puzzle
Martin Gardner's two "Aha!" books are both amazing - every page full of delightful insight. See also Aha! Gotcha
from http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Letters/BroomeBridge.html In 1835, at the age of 30, [Hamilton] had discovered how to treat complex numbers as pairs of real numbers. Fascinated by the relation between complex numbers and 2-dimensional geometry, he tried for many years to invent a bigger algebra that would play a similar role in 3-dimensional geometry. In modern language, it seems he was looking for a 3-dimensional normed division algebra. His quest built to its climax in October 1843. He later wrote to his son, ref Archibald H Hamilton, August 5, 1865: Every morning in the early part of the above-cited month, on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me: "Well, Papa, can you multiply triplets?" Whereto I was always obliged to reply, with a sad shake of the head: `No, I can only add and subtract them". The problem was that there exists no 3-dimensional normed division algebra. He really needed a 4-dimensional algebra. Finally, on the 16th of October, 1843, while walking with his wife along the Royal Canal to a meeting of the Royal Irish Academy in Dublin, he made his momentous discovery: (4) But on the 16th day of the same month - which happened to be a Monday, and a Council day of the Royal Irish Academy - I was walking in to attend and preside, and your mother was walking with me, along the Royal Canal, to which she had perhaps driven; and although she talked with me now and then, yet an under-current of thought was going on in my mind, which gave at last a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely directed thought and work, by myself if spared, and at all events on the part of others, if I should even be allowed to live long enough distinctly to communicate the discovery. Nor could I resist the impulse - unphilosophical as it may have been - to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula with the symbols, i, j, k; namely, i2 = j2 = k2 = ijk = -1 which contains the Solution of the Problem. but of course, as an inscription, this has long since mouldered away. A more durable notice remains, however, on the Council Books of the Academy for that day (October 16th, 1843), which records the fact, that I then asked for and obtained leave to read a Paper on Quaternions, at the First General Meeting of the session: which reading took place accordingly, on Monday the 13th of the November following. Hence this famous act of mathematical vandalism, where he carved these equations into the stone of the Brougham Bridge: i2 = j2 = k2 = ijk = -1 [the Broome Bridge, [Brougham is pron Broome], crosses the canal in Cabra. A plaque marks the spot today. - see pictures and how to reach the bridge, at http://math.ucr.edu/home/baez/octonions/node24.html In a separate letter to mathematician friend Peter G Tait, October 15, 1858, the carving on stone is replaced by writing on pocketbook: P.S. - To-morrow will be the 15th birthday of the Quaternions. They started into life, or light, full grown, on [Monday] the 16th of October, 1843, as I was walking with Lady Hamilton to Dublin, and came up to Brougham Bridge, which my boys have since called the Quaternion Bridge. That is to say, I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i, j, k; exactly such as I have used them ever since. I pulled out on the spot a pocket-book, which still exists, and made an entry, on which, at the very moment, I felt that it might be worth my while to expend the labour of at least ten (or it might be fifteen) years to come. But then it is fair to say that this was because I felt a problem to have been at that moment solved - an intellectual want relieved - which had haunted me for at least fifteen years before. Less than an hour elapsed before I had asked and obtained leave of the Council of the Royal Irish Academy, of which Society I was, at that time, the President - to read at the next General Meeting a Paper on Quaternions; which I accordingly did, on November 13, 1843. - Source: Life of Sir William Rowan Hamilton by Robert P. Graves, Volume II, Chapter XXVIII -- [quaternion.com] One of the first things Hamilton did was get rid of the fourth dimension, setting it equal to zero, and calling the result a "proper quaternion." He spent the rest of his life trying to find a use for quaternions. By the end of the nineteenth century, quaternions were viewed as an oversold novelty. In the early years of this century, Prof. Gibbs of Yale found a use for proper quaternions by reducing the extra fluid surrounding Hamilton's work and adding key ingredients from Rodrigues concerning the application to the rotation of spheres. He ended up with the vector dot product and cross product we know today. This was a useful and potent brew. Our investment in vectors is enormous, eclipsing their place of birth (Harvard had >1000 references under "vector", about 20 under "quaternions", most of those written before the turn of the century).
Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, p. 87
There are 28800 different 5x5 panmagic squares.
1979.
KNIGHTS MOVE method:
squares of 6k+-1:
ordinary vector (2, 1) and break vector (-1, -2). e.g. 5x5 ; break vector was (1,-1)
1 15 24 8 17
23 7 16 5 14
20 4 13 22 6
12 21 10 19 3
9 18 2 11 25
[You can start anywhere, since it is panmagic!]
ALTERNATE: based on base pan-magic squares :
generate a square with knight's move, and take transpose:
A b c d e A d b e c
d e A b c b e c A d
b c d e A c A d b e
e A b c d d b e c A
c d e A b e c A d b
5x one add to other (combine in base-5 notation) gives the magic square:
00 13 21 34 42 BASE-5: 0 8 11 19 22
31 44 02 10 23 16 24 2 5 13
12 20 33 41 04 7 10 18 21 4
43 01 14 22 30 23 1 9 12 15
24 32 40 03 11 14 17 20 3 6
The 3 Regular 4x4 Pan-Magic Squares
0 7 9 14
11 12 2 5
6 1 15 8
13 10 4 3
0 7 10 13
11 12 1 6
5 2 15 8
14 9 4 3
0 7 12 11
13 10 1 6
3 4 15 8
14 9 2 5
INDIA
A well known early magic square in India is found in Khajuraho in the
Parshvanath Jain temple. It dates from the 10th century [3].
7 12 1 14
2 13 8 11
16 3 10 5
9 6 15 4
This is referred to as the Chautisa Yantra, since each sub-square sums to 34.
---
reported by Henry Dudeney in his book which was first published in 1917;
he claims that these results have been ". . . verified over and over again". On
the next page Dudeney goes on to provide an analysis and classification which he
attributes to "Mr. Frost" - one of the earlier students of magic squares. He
describes "Nasik" squares - the only type which he describes as having
properties we now call "Pan-Magic". ("Nasik" was the name of the town in India
in which Mr. Frost lived.)
Boyer on his Multimagic pages.
On this page I will present all of Frost's cubes. They will appear in the
same order and the same orientation, exactly as they are in his papers.
Following are the references to the papers I have been referring to. There
must be a preceding paper, which I do not have. In [3] page 49, Frost says
"By the method adopted in No. XXV, 1865, of this journal...", and again in
[4] page 118, he says "The method employed for squares of the form 4n in
No. 25, 1865, of this Journal...". That paper presumably deals with nasik
magic squares, but I have not been able to locate it.
About the term Nasik
Dr. Frost coined the term Nasik for a magic square in which both diagonals
and all parallel diagonal pairs sum to the magic constant. This type of
magic square would later be referred to as Pandiagonal or Perfect. In fact
it is a Perfect Magic Hypercube of dimension 2. Later, Dr. C. Planck would
extend the definition of Nasik to Perfect Magic Hypercubes of any
dimension. [8]
The term Nasik is derived from the town Nassick, in Western India, where
Frost was stationed from 1855 to 1867 as a missionary.
[1] A. H. Frost, Invention of Magic Cubes. Quarterly Journal of Mathematics, 7, 1866, pp 92-103
He describes a method of constructing magic cubes and shows an order 7
pandiagonal and an order 8 pantriagonal magic cube. Frost was presumably still
in India at this time, because the paper was submitted by his brother,
Rev. Percival Frost
Albrecht Durer 4x4:
16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1
CONSTRUCTING 4X4 magic squares: The average number is a = 17/2 (2a=n²+1) and
the row-sum is a*n = 34; (n=4).
consider the square:
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
cut two diagonals through the square. They go over the numbers 1, 6, 11, 16 and
4, 7, 10, 11; note that these sum to 4a. Mirror image the numbers along these
diagonals to get your magic square:
16 2 3 13
5 11 10 8
9 7 6 12
4 14 15 1
PROOF: each row now has a mirror and an original side by side. The pair of
terms in a row always add up to 2a+1 and 2a-1, and along the columns alternate
pairs are 2a+n and 2a-n, thus each column and row add up to 4a. Since each
original diagonal added up to 4a, they still work.
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