How to Generate Theorems for an Abstract System like Sudoku?

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In summary, the conversation is about axiomatizing sudoku and generating theorems for the newly defined system. The definition of sudoku in mathematical terms is given, along with the definition of a solved puzzle. The use of an operation * to represent a sudoku puzzle and its properties are also discussed. Suggestions for changes to the definition are welcomed.
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TylerH
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For an exercise, I want to axiomatize sudoku.

I've came up with the definition of the sudoku puzzle in mathematical terms, as well as the definition of a solved puzzle. I'm have trouble going from there to draw theorems from my newly defined system. How does one generate theorems about an abstract system like this?

My definition:

Generalizing for an n^2 x n^2 sudoku puzzle:

Let there be a matrix, A, of deminsions, n x n, such that each element is a matrix of deminsions, n x n.
Let there be a set, S = {x exists in N, 1 <= x <= n^2}.

The puzzle is solved iff every element of A contains exactly one of each member of S and for every i from 1 to n, the set of elements in A(i, j) for j from 1 to n contains exactly one of each member of S.

Any suggestions on changes to the definition are also appreciated.
 
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With a sudoku puzzle, we can associate a operation * as:

[tex]*:\{1,...,n\}\times\{1,...,n\}\rightarrow \{1,...,n\}[/tex]

such that k*l is the number in row k, column l. Thus the Cayley multiplication table of this operation is our Sudoku puzzle.

An interesting thing to find out is what properties our * satisfies. For example, you can show that * is cancelative, that is:

[tex]a*b=a*c~\Rightarrow~b=c[/tex]

That's a theorem you can prove. Maybe you can search some other things that hold for our *...
 

FAQ: How to Generate Theorems for an Abstract System like Sudoku?

What does it mean to axiomatize sudoku?

Axiomatizing sudoku means to create a set of axioms or rules that define the game and its properties. These axioms can then be used to prove or disprove statements about sudoku.

Why would someone want to axiomatize sudoku?

Axiomatizing sudoku can provide a deeper understanding of the game and its mathematical properties. It can also allow for the creation of new variations or extensions of the game.

What are the main challenges in axiomatizing sudoku?

The main challenge in axiomatizing sudoku is determining the appropriate set of axioms that accurately describe the game while also being comprehensive and concise. Additionally, different variations of sudoku may require different axioms.

How does axiomatizing sudoku relate to mathematics?

Axiomatizing sudoku is a mathematical process that involves creating a set of logical rules to describe the game. It can also involve using mathematical concepts and techniques to prove or disprove statements about sudoku.

Can axiomatizing sudoku lead to any real-world applications?

While the main purpose of axiomatizing sudoku may be for theoretical study and exploration, it can also have practical applications. For example, the techniques used in axiomatizing sudoku can be applied to other games or puzzles, and the logical reasoning skills developed can be useful in various fields such as computer science and artificial intelligence.

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