An isomorphism: Tic-Tac-Toe on Magic Square

Discussion in 'Physics & Math' started by Dinosaur, Apr 10, 2013.

  1. Dinosaur Rational Skeptic Valued Senior Member

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    4,885
    The concept of isomorphism is used in dealing with physics, mathematics, & other intellectual disciplines. It is often used without the user being aware of the term.

    The following is a simple example of an isomorhhism.

    A bright 4-5 year old can easily learn to play Tic-Tac-Toe. He/she might not be able to play a game requiring facility with simple arithmetic. The game is as follows.

    Take the Ace through the nine of Spades (or another suit) & lay them face up on the table.

    Players take turns picking a card.

    The winner is the first to have a set of three cards totaling 15. 6 & 9 are not considered a winning holding, being a pair of cards rather than a triplet.


    Consider the 3*3 Magic Square.

    8 1 6
    3 5 7
    4 9 2


    Make a Tic-Tac-Toe form with the above Magic Square written in the 9 boxes.

    Tell the 4-5 year old to pretend he is playing Tic-Tac-Toe & use that pretense to guide his choice of cards from 1-9 while playing the arithmetic game with cards.

    A mathematician would say that the card game is isomorphic to Tic-Tac-Toe using the Magic Square.

    Tic-Tac-Toe is a model. It is not the reality of the game with cards.
     
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  3. eram Sciengineer Valued Senior Member

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    That's interesting.

    We could say that the card game is like TTT, but we would not intuitively know which card to pick unless we wrote them in a magic square. So that's isomorphism.

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  5. arfa brane call me arf Valued Senior Member

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    7,832
    You could use (paths in) graphs to represent the different games, and show there is a map from one graph to the other that preserves the number of vertices and their degree.
    The most abstract mathematical object I've studied recently is a graph. Once you have a set of vertices and you connect (some of) them, you have a structure which can be represented in an arbitrary number of ways. Determining if two graphs are isomorphic is a problem we looked at yesterday.

    That seems to be what abstract algebra is about: structure, and 'algebraicity'. Morphisms generally preserve structure (or maybe 'shape', but it doesn't have to be geometric), so you have structural invariance.
     
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  7. Cheezle Hab SoSlI' Quch! Registered Senior Member

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    Interesting. Isomorphism implies that the mapping from one system is bijective, 1 to 1 and onto the other system. It is a one way functional mapping. So while the card game maps onto the magic square, the transformation is one way. Meaning that the magic square is not the same as the card game. If it were then it would be a bijective endomorphism (aka a automorphism). I have not looked at the problem but it does sound interesting.
     
  8. Cheezle Hab SoSlI' Quch! Registered Senior Member

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    745
    I woke up this morning thinking about this subject and it has been stuck in my head ever since. It seems to me that the general game is to take a 3x3 square and apply labels to each cell, [1,2,3,4,5,6,7,8,9]. In tick tac toe the goal is to achieve 3 in a row. In the card game it is also to get 3 in a row but with the additional rule to have the labels add up to 15. So the base problem is one of a permutation group (S9). So the card game is really just a subgroup of the main permutation group. I am not familiar with magic squares but I suspect that there are only 8 3x3 of them (4 rotation symmetries and their mirrors), so the subgroup is very small compared to the 9! permutations of the main group. The rotations and mirror flips are the generators of the subgroup. And the subgroup is an interesting group where the rotations and mirror flips are the generators of the group.

    But the relationship between the main (tic-tac-toe) group and the subgroup (magic square card game) is not isomorphic because the mapping of the subgroup into the main group is not bijective. It is 1-to-1 but not onto because the main group is huge and the subgroup is small. So the mapping is injective. I think the correct term is monomorphism not isomorphism. Or maybe an endomorphism (not sure).

    Another game that might be fun to try is to take a large grid and label each cell with a number and then randomly permute the grid. So that each cell has a unique random number. Or maybe each cell just has a random number from 1 to 5. Then play the 5 in a row game on the grid but rather just win or lose there is a score where the numbers of the 5 in a row are added up. So instead of just going for a 5 in a row you also go for a high score. Probably a boring game but better than the normal version.
     
  9. Dinosaur Rational Skeptic Valued Senior Member

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    Cheezle: The two games are isomorphic.

    While there are 9! permutations of 9 integers, that ignores the rules of the card game. The card game rules specify subsets of 3 integers whose sum is 15. There are only 8 such subsets, corresponding to the rows, columns, & diagonals of the Magic Square.

    Furthermore, the 3 by 3 Magic Square is unique. none of the transformations change its basic stucture, which is the basis of the isomorphism. Every transformation which preserves the row, column, & diagonal sums is a 3 by 3 Magic Square isomorphic to the card game.

    An isomorphic relationship is not defined as an arbitrary mapping of one set to another. It is a mapping or correspondence based on a set of rules.
     

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