1

Dialogue on Magic SquareDialogue on Magic SquareDialogue on Magic SquareDialogue on Magic Square Armahedi Mahzar ©2010

Unomino Magic Square

Varomino Magic Square

Combino Magic Square Combino Magic Tesseract

2

Prologue Prologue Prologue Prologue on the dialogueon the dialogueon the dialogueon the dialogue

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)

3

Dialogue on Magic SquareDialogue on Magic SquareDialogue on Magic SquareDialogue on Magic Square Part One: Unomino Magic SquarePart One: Unomino Magic SquarePart One: Unomino Magic SquarePart One: Unomino Magic Square http://integralist.multiply.com/journal/item/21/Dialogue_on_Magic_Square_1

Magic 3x3 Square

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

Hi Suiti! What are in your hands?

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

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.

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

Domino is two square containing dots.

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

Yes, but an unomino, as my grandson called it, is a half of domino. Each unomino is

containing dots like dominos.

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

How do you play it?

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

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?

4

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

That's too easy, because the puzzle is similar to the problem of Magic Square. Here is the

answer.

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

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

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

I think we can make bigger and bigger Magic Square

5

Magic 4x4 Square

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

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.

6

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

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.

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

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.

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

Well. Because you're just easily solving the monomino puzzle. Next time I will bring

other Emo's toy: varinoes.

7

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

Varinoes?

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

Yes. See you next time.

8

Dialogue on MDialogue on MDialogue on MDialogue on Magic Squareagic Squareagic Squareagic Square Part Two:Varomino Magic SquarePart Two:Varomino Magic SquarePart Two:Varomino Magic SquarePart Two:Varomino Magic Square http://integralist.multiply.com/journal/item/24/Dialogue_on_Magic_Square_2

Varino Magic 4x4 Square

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

Varino is single dot monomino

with the the color of the dot

and the the color of the background

is varied in four color

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

What is the puzzle around varinoes ?

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

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?

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

That's another easy puzzle to be solved.

Get 16 equally sized square cards.

Now I will make another puzzle made of 16 cards

9

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.

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

I do not like letters. I prefer colors and forms.

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

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

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.

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

Since you've colored the letters, I see the similarity now

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

Here is the solution for Greco-Latin Square

10

αc βb γdddd δa

δd γa βc αb

βa αd δb γc

γb δc αa βd

It can be transformed into this Varino Magic Square

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

The interesting fact is that the combinatorial Greco-Latin Square of Euler is

actually similar (or isomorphic) to the ordinary arithmetical 4x4 Magic Square.

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

How come?

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

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

11

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

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

I hate formulae.

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

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

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

I hate numbers.

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

It can easily be transformed to your Monomino Magic Square.

Here it is.

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

Yes. It is a Monomino Magic Square.

12

You've really connect the Varino Magic Square and Monomino Magic Square.

See if you can relate them to Combino Magic Square.

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

Combino?

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

Yes, I will bring the combinoes, just another toy of Si Emo, later.

See you.

13

Dialogue on Magic SquareDialogue on Magic SquareDialogue on Magic SquareDialogue on Magic Square Part Three:ComPart Three:ComPart Three:ComPart Three:Combino Magic Squarebino Magic Squarebino Magic Squarebino Magic Square http://integralist.multiply.com/journal/item/25/Dialogue_on_Magic_Square_3

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

Today, I bring you four colored combinoes.

Combino card contains all possible combination of four colored dot

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

Ok. There are exactly 16 four colored combinoes if we include the empty combination.

But, what is the puzzle you promised me last time..

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

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.

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

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?

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

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.

14

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

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.

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

What is the correpondence?

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

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.

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

OK. Let me associate

Red SquareRed SquareRed SquareRed Square in varino with Empty combino,

Blue SquareBlue SquareBlue SquareBlue Square in varino with Red Dot in combino,

Green SquareGreen SquareGreen SquareGreen Square in varino with Blue Dot in combino and

Yellow SquareYellow SquareYellow SquareYellow Square in varino with combination of Red Blue Dots in combino

and I will associate

RedRedRedRed Dot in varino with Empty combino,

BlueBlueBlueBlue Dot in varino with YellowYellowYellowYellow Dot in combino,

GreenGreenGreenGreen Dot in varino with GreenGreenGreenGreen Dot in combino and

YellowYellowYellowYellow Dot in varino with combination of YellowYellowYellowYellow GreenGreenGreenGreen Dots in combino.

Let me choose this associations to built as correspondence rule

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

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.

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

OK, let me try your suggestion. For example:

GreenGreenGreenGreen Dot on YelloYelloYelloYellow Square w Square w Square w Square varino is corresponded to

GreenGreenGreenGreen Red Blue Dots combino.

Another example is BlueBlueBlueBlue Dot on Green SquareGreen SquareGreen SquareGreen Square varino is corresponded to

YellowYellowYellowYellow GreenGreenGreenGreen Dots combino.

OK. I see I can correspond the 16 combinoes to the 16 varinoes one by one.

15

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

After making the correspondence, you can transform the Greco-Latin Square

into Combino Magic Square like this one.

Ni Suiti: Ni Suiti: Ni Suiti: Ni Suiti:

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?

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

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.

Ni Suiti: Ni Suiti: Ni Suiti: Ni Suiti:

See you later!

16

Dialogue on Magic SquareDialogue on Magic SquareDialogue on Magic SquareDialogue on Magic Square Part Four: Part Four: Part Four: Part Four: Magic HypercubeMagic HypercubeMagic HypercubeMagic Hypercube http://integralist.multiply.com/journal/item/26/Dialogue_on_Magic_Square_4

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.

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

Can you solve it logically?!

Combino Magic 2x2x2 Cube

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

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.

17

Catching the Hypercube Corners

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

I think before I solve the higher dimensional Combino Magic Hypercube, I will catch the

hypercube corners with a 4x4 checkerboard like this

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

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,

Ki Algo:Ki Algo:Ki Algo:Ki Algo:

Yes. As I remember it, the Combino Magic Square is like this

18

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.

Ni Suiti:Ni Suiti:Ni Suiti:Ni Suiti:

Yes. Yes. Yes it is.

Logic can reconstruct what the intuition see.