Dialogue on magic square

Dialogue DDDiiiaaallloooggguuueee oooonnnn MMMMaaaaggggiiiicccc SSSSqqqquuuuaaaarrrreeee
Armahedi Mahzar ©2010
Unomino Magic Square
Varomino Magic Square
Combino Magic Square Combino Magic Tesseract
1

PPPPrrrroooolllloooogggguuuueeee oooonnnn tttthhhheeee ddddiiiiaaaalllloooogggguuuueeee
I invented the varino and combino cards in the 70-s to teach my preschooler kids
the set theory. Then I realized that varinoes are essentially color representations
of base 2 numbers and combinoes are essentially color representation of base 4
numbers, so they can also be arranged to form magic square. So I published my
discoveries it in my campus student magazine Scientiae.
In the 90-s our campus had an internet connection. In one of the website in it, I
found out that 2x2x2x2 hypercube can be projected into a 4x4 square. So any
special kind magic 4x4 square can be transformed into a magic 2x2x2x2
hypercube replacing columns and rows with square faces. My solution of
Combino Magic Square can be transformed into Combino Magic Hypercube
Those Magic Square puzzles are actually equivalent to each other. It is so
amazing, that make me wonder: if all those various forms of Magic Square are
just projections in our minds of a general Geometric Formation of Numbers in a
Mathworld outside our mind, outside our physical world, or a general Formation
of Combinatoric Variations or Combinations, out there in the World of
Mathematical Objects: the Mathworld. So in the Mathworld the Geometry and
Algebra, Arithmetic and Combinatorics are unified . This my vision of TOF or
Theory of Every Forms for mathematics. TOE of Physics will be only subset of
TOF.
The forms discovered in by scientists in natural world will sometime also
discovered by mathematicians or artists in their mind such as the aperiodic
symmetry of quasicrystals as it is discussed in the dialogues of Ki Algo and Ni
Suiti on the Integralism Symbol. In fact mathematicians later on are proving that
aperiodic quasicrystallographic pattern in our physical space is just a projection
of the periodic crystallographic pattern in a higher multidimensional hyperspace
to our lowly physical 3 dimensional space. Do you have other explanation of the
phenomenon without using the objective Platonic Mathworld? What is the
structure of the Mathworld?
Nowadays, there are many attempts to unify all mathematics to one theory. The
latest one is the hierarchical N-category Theory. This N-category theory will
include the logistic theory, the formalistic theory and the intuitionistic theory as
sub-theories of category theory. It is a powerful theory, but I think the functors of
category theory must be generalized to relators so we have a web structure of the
Mathworld rather than the ladder structure of category theory. But I am not a
mathematician, I can't develop such web of relators concept into a working theory.
Mathematician called such relator theory as theory of allegory. However the
following dialogues are no need of such exotic math. So, please enjoy it it as
recreational math (armahedi@yahoo.com)
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PPPPaaaarrrrtttt OOOOnnnneeee:::: UUUUnnnnoooommmmiiiinnnnoooo MMMMaaaaggggiiiicccc SSSSqqqquuuuaaaarrrreeee
http://integralist.multiply.com/journal/item/21/Dialogue_on_Magic_Square_1
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Magic 3x3 Square
KKKKiiii AAAAllllggggoooo::::
Hi Suiti! What are in your hands?
NNNNiiii SSSSuuuuiiiittttiiii::::
Oh! This is the toy of my grandson Si Emo It is like dominoes, but is it made of of one
square.
Some of the pattern is similar to the pattern found in dominoes.
Domino is two square containing dots.
Yes, but an unomino, as my grandson called it, is a half of domino. Each unomino is
containing dots like dominos.
How do you play it?
It is a kind of puzzle. For example you can arrange the 9 unominos in a 3x3 checker
board sequentially like this.
Now, can you rearrange the little black square places so each column, row and diagonal
is containing exactly the same numbers of dots?

That's too easy, because the puzzle is similar to the problem of Magic Square. Here is the
answer.
That's why I called it Unomino Magic Square. Yes it is too easy. The solution known as
Lo Shu was discovered thousand years ago in the back of mythical turtle by Fuh-Shi, the
mythical founder of Chinese civilisation in around 2400 BC.
Before they invented the zero numeral, the Arabs used alphabets as the written symbols
of numbers. Here is the 3x3 Magic Square
I think we can make bigger and bigger Magic Square
4

5
Magic 4x4 Square
Yes, the earliest 4X4 Magic Square is discovered in Khajuraho India dating
from the eleventh or twelfth century.
The following 4X4 Magic Square can be found in Albert Dürer's engraving "
Melencolia", where the date of its creation, 1514 AD. See it under the bell.

The nine monominoes is only part of larger set of monominoes containing dots from 1 up
to 16.
Can you rearrange the unominoes such that each column, row, diagonal and little 2x2
square is containing exactly the same number of dots? This is is the 4 x 4 Monomino
Magic Square Puzzle.
Well, Well. To me it seems that this monomino magic square problem is nothing but a
different guise of 4X4 ordinary Magic Square. One of the solution of the puzzle can be
gotten by exchanging the diagonal monominos symmetrically based on the center point.
Here it is.
This solution is wonderful. Because all diagonals are always summed to 34. The numbers
in the center 2x2 square are also added up to 34. The numbers in the corner 2x2 squares
are also added to 34 .
This is only one solution of the Puzzle. The French mathematician Frenicle de Bessy in
1693 enumerated the number of all possible 4x4 Magic Square and get the number 880.
Well. Because you're just easily solving the monomino puzzle. Next time I will bring
other Emo's toy: varinoes.
6

7
Varinoes?
Yes. See you next time.

DDDDiiiiaaaalllloooogggguuuueeee oooonnnn MMMMaaaaggggiiiicccc SSSSqqqquuuuaaaarrrreeee
PPPPaaaarrrrtttt TTTTwwwwoooo::::VVVVaaaarrrroooommmmiiiinnnnoooo MMMMaaaaggggiiiicccc SSSSqqqquuuuaaaarrrreeee
8
Varino Magic 4x4 Square
Varino is single dot monomino
with the the color of the dot
and the the color of the background
is varied in four color
What is the puzzle around varinoes ?
The following picture is a 4x4 checkerboard
with each little square containing
one varino
The color variations are red, blue, green and yellow.
Can you rearrange the varinoes in the little squares
so each column, row and diagonal is containing
different colored squares and different colored dots?
That's another easy puzzle to be solved.
Get 16 equally sized square cards.
Now I will make another puzzle made of 16 cards

similar to varinoes. Instead of drawing and coloring
the cards, I will write two letters in each card.
Each card containing one Greek letter and one Latin letter.
The Lattin letters are a, b, c and d.
The Grrek letters are α, β, γ and δ.
The two letters cards can be arranged in 4x4 checkerboard
such that every row, column and diagonal contains exactly
one of the 8 letters. Such 4x4 square called the Greco-Latin Square.
I do not like letters. I prefer colors and forms.
If you look to the solution, then you will probably
realize that the Varomino Magic Square is also
a disguise of the famous Leonhard Euler Greco-Latin Square.
You can get 4x4 Greco-Latin Square from this ordered Letter square
αa βa γa δa
αb βb γb δb
αc βc γc δc
αd βd γd δd
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in which
• Any letter, Greek or Latin, occurs once in any row, column
• Any letter, Greek or Latin, occurs once in any diagonal
It is equivalent to your ordered varino square.
Since you've colored the letters, I see the similarity now
Here is the solution for Greco-Latin Square

10
αc βb γd δa
δd γa βc αb
βa αd δb γc
γb δc αa βd
It can be transformed into this Varino Magic Square
The interesting fact is that the combinatorial Greco-Latin Square of Euler is
actually similar (or isomorphic) to the ordinary arithmetical 4x4 Magic Square.
How come?
Let the greek letters alpha, beta, gamma and delta are representing
the numbers 0, 1, 2 and 3 respectively and
let the Latin a, b, c and d are also representing
the numbers 0, 1, 2 and 3 respectively.
Mathematically this representation is a function Number,
such that
Number(α) = Number(a) = 0
Number(β) = Number(b) = 1
Number(γ) = Number(c) = 2
Number(δ) = Number(d) = 3

Now replace the combination of Greek and Latin letters
with the number following this formula
Number(Greek Latin) = 4 x Number(Greek) + Number(Latin) + 1
in the little squares of Greco-Latin Square,
then automatically the Greco-Latin Square is transformed
to an arithmetic Magic Square Solution
11
I hate formulae.
Sorry. But with the formula we can transform the Greco-Latin Square to the following
Magic Square
3 6 12 13
16 9 7 2
5 4 14 11
10 15 1 8
I hate numbers.
It can easily be transformed to your Monomino Magic Square.
Here it is.
Yes. It is a Monomino Magic Square.

You've really connect the Varino Magic Square and Monomino Magic Square.
See if you can relate them to Combino Magic Square.
12
Combino?
Yes, I will bring the combinoes, just another toy of Si Emo, later.
See you.

PPPPaaaarrrrtttt TTTThhhhrrrreeeeeeee::::CCCCoooommmmbbbbiiiinnnnoooo MMMMaaaaggggiiiicccc SSSSqqqquuuuaaaarrrreeee
13
Today, I bring you four colored combinoes.
Combino card contains all possible combination of four colored dot
Ok. There are exactly 16 four colored combinoes if we include the empty combination.
But, what is the puzzle you promised me last time..
The following picture is a 4x4 checkerboard with each cell is containing a combino of 4
colors. The colors are red, blue, yellow and green.
I see that each combino is placed randomly into each small square. So the numbers of
colored dots in each column, row or diagonal are different. Now, once again, what is the
puzzle?
Here is the puzzle. Can you rearrange the combino's places to get the Combino Magic
Square where each column, row and diagonal is containing exactly two of each colored
dots?
To me, it's so difficult.

No. It is not too difficult. In fact, if only you realize there is one to one correspondence
between the combinoes and the varinoes. You can change Varino Magic Square to
Combino Magic Square.
14
What is the correpondence?
I think you can associate the four colored squares in varino with the four possible
ombinations of any two colors in combino, and then you associate the four colored dots
in varino with the four possible combinations of the other two colors in combino.
OK. Let me associate
RRRReeeedddd SSSSqqqquuuuaaaarrrreeee in varino with Empty combino,
BBBBlllluuuueeee SSSSqqqquuuuaaaarrrreeee in varino with Red Dot in combino,
GGGGrrrreeeeeeeennnn SSSSqqqquuuuaaaarrrreeee in varino with Blue Dot in combino and
YYYYeeeelllllllloooowwww SSSSqqqquuuuaaaarrrreeee in varino with combination of Red Blue Dots in combino
and I will associate
RRRReeeedddd Dot in varino with Empty combino,
BBBBlllluuuueeee Dot in varino with YYYYeeeelllllllloooowwww Dot in combino,
GGGGrrrreeeeeeeennnn Dot in varino with GGGGrrrreeeeeeeennnn Dot in combino and
YYYYeeeelllllllloooowwww Dot in varino with combination of YYYYeeeelllllllloooowwww GGGGrrrreeeeeeeennnn Dots in combino.
Let me choose this associations to built as correspondence rule
That's a good choice. Now you can associate any varino with one combino by
associating any varino with colored dot on colored square with the combination of
combino colored dots associated to the varino colored dot and colored square into one
combino.
OK, let me try your suggestion. For example:
GGGGrrrreeeeeeeennnn Dot on YYYYeeeelllllllloooowwww SSSSqqqquuuuaaaarrrreeee varino is corresponded to
GGGGrrrreeeeeeeennnn Red Blue Dots combino.
Another example is BBBBlllluuuueeee Dot on GGGGrrrreeeeeeeennnn SSSSqqqquuuuaaaarrrreeee varino is corresponded to
YYYYeeeelllllllloooowwww GGGGrrrreeeeeeeennnn Dots combino.
OK. I see I can correspond the 16 combinoes to the 16 varinoes one by one.

After making the correspondence, you can transform the Greco-Latin Square
into Combino Magic Square like this one.
You've done it once more.
But do you see that this Combino Magic Square is just a projection of a Combino Magic
2x2x2x2. Hypercube in which each one of its faces is containing exactly two colored
dots?
What I know is the projection of the four dimensional hypercube is like this
But I have to make my logical mind think out what your intuitive eyes see.
15
See you later!

PPPPaaaarrrrtttt FFFFoooouuuurrrr:::: MMMMaaaaggggiiiicccc HHHHyyyyppppeeeerrrrccccuuuubbbbeeee
Ki Algo:
In our last meeting you asked if we can put the 16 four-colored combinoes in the corners
of an hypercube so that any colored dots occurs twice in each its square face? In fact you
see the answer with your intuition's eyes.
16
Can you solve it logically?!
Combino Magic 2x2x2 Cube
It's a pity I can't, but I try to solve the easier puzzle: can we put the 8 three-colored
combinoes in the corners of a cube such that each square face contains exactly two dots.
It turns out to be an easy puzzle.

Catching the Hypercube Corners
I think before I solve the higher dimensional Combino Magic Hypercube, I will catch the
hypercube corners with a 4x4 checkerboard like this
Oh my goodness! You really caught the hypercube in a checkerboard.
You did it by rotating four square faces a bit and stretching the horizontal and vertical
edges of the hypercube,
Yes. As I remember it, the Combino Magic Square is like this
17

Combino Magic 2x2x2x2 Hypercube
By overlaying the Combino Magic Square to Hypercube-Caught-in-Checkerboard I got
this
I think this is the projection of Magic Hypercube that you see in the 4-d space.
I see it follows all the Combino Magic Hypercube rules.
Yes. Yes. Yes it is.
Logic can reconstruct what the intuition see.
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Dialogue on magic square

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Dialogue on magic square