Dialogue on magic square

Dialogue on Magic Square Armahedi Mahzar Š2010

Unomino Magic Square

Varomino Magic Square

Combino Magic Square

Combino Magic Tesseract

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Prologue on 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)

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

Magic 3x3 Square Ki Algo: Hi Suiti! What are in your hands?

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: Domino is two square containing dots.

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: How do you play it?

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?

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

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: I think we can make bigger and bigger Magic Square

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Magic 4x4 Square 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 D端rer's engraving " Melencolia", where the date of its creation, 1514 AD. See it under the bell.

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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: 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: Well. Because you're just easily solving the monomino puzzle. Next time I will bring other Emo's toy: varinoes.

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Ki Algo: Varinoes?

Ni Suiti: Yes. See you next time.

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

Varino Magic 4x4 Square 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: What is the puzzle around varinoes ?

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: That's another easy puzzle to be solved.

Get 16 equally sized square cards.

Now I will make another puzzle made of 16 cards

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

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: Since you've colored the letters, I see the similarity now Ki Algo: Here is the solution for Greco-Latin Square

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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

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: How come?

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

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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: I hate formulae.

Ki Algo: Sorry. But with the formula we can transform the Greco-Latin Square to the following Magic Square

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6 12 13

16 9 5

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4 14 11

10 15 1

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Ni Suiti: I hate numbers.

Ki Algo: It can easily be transformed to your Monomino Magic Square.

Here it is.

Ni Suiti: Yes. It is a Monomino Magic Square.

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

See if you can relate them to Combino Magic Square.

Ki Algo: Combino?

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

See you.

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Dialogue on Magic Square Part Three:Combino Three:Combino Magic Square http://integralist.multiply.com/journal/item/25/Dialogue_on_Magic_Square_3 Ni Suiti: Today, I bring you four colored combinoes.

Combino card contains all possible combination of four colored dot 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: 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: 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: 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.

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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: What is the correpondence?

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: OK. Let me associate Red Square in varino with Empty combino, Blue Square in varino with Red Dot in combino, Green Square in varino with Blue Dot in combino and Yellow Square in varino with combination of Red Blue Dots in combino and I will associate Red Dot in varino with Empty combino, Blue Dot in varino with Yellow Dot in combino, Green Dot in varino with Green Dot in combino and Yellow Dot in varino with combination of Yellow Green Dots in combino.

Let me choose this associations to built as correspondence rule 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: OK, let me try your suggestion. For example: Green Dot on Yellow Yellow Square varino is corresponded to Green Red Blue Dots combino.

Another example is Blue Dot on Green Square varino is corresponded to Yellow Green Dots combino.

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

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Ki Algo: After making the correspondence, you can transform the Greco-Latin Square into Combino Magic Square like this one.

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: 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: See you later!

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Dialogue on Magic Square Part Four: Magic 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: Can you solve it logically?!

Combino Magic 2x2x2 Cube 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.

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Catching the Hypercube Corners 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: 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: Yes. As I remember it, the Combino Magic Square is like this

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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: Yes. Yes. Yes it is.

Logic can reconstruct what the intuition see.

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