sudoku1

# sudoku1 - The Mathematics of Sudoku Joshua Cooper...

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The Mathematics of Sudoku Joshua Cooper Department of Mathematics, USC

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Rules: Place the numbers 1 through 9 in the 81 boxes, but do not let any number appear twice in any row, column, or 3 3 “box”. Usually you start with a subset of the cells labeled, and try to finish it. 1 3 7 8 7 4 8 5 9 2 8 1 6 6 8 7 1 2 8 4 7 1 8 1 3 7 5 9 6 5 4 2 8 3 2 6 1 5 9 9 2 4 7 6 1 3 4 7 5 3 3 1 2 9 4 6 5 8 7 5 3 9 4 5 9 6 1 3 7 2 2 3 6 9 5 8 4 4 2 9 6
Seemingly innocent question: How many sudoku boards are there? The same? We could define a group of symmetries – flips, rotations, color permutations, etc. – and only count orbits. Let’s just say that two boards are the same if and only if they agree on every square. Recast the question as a “hypergraph” coloring problem.

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Graph : A set (called “vertices”) and a set of pairs of vertices (called “edges”). Example . V = {1,2,3,4,5}, E = {{1,2},{2,3},{3,4},{4,5},{1,5},{1,4},{2,4}}. 1 2 5 4 3 Hypergraph : A set (called “vertices”) and a set of sets of vertices (called “edges” or sometimes “hyperedges”). If all the edges have the same size k , then the hypergraph is said to be k -uniform. In particular, a 2-uniform hypergraph is just a graph.
Example of a 3-uniform hypergraph : The “Fano Plane”, V = {1,2,3,4,5,6,7} and E = {{1,2,4},{2,3,5},{3,4,6},{4,5,7},{5,6,1},{6,7,2},{7,1,3}}. 1 3 2 6 5 7 4 A k- coloring of a graph G is an assignment of one of k colors to the vertices of G so that no edge has two vertices of the same color. Alternatively: A k- coloring of a graph G is an assignment of one of k colors to the vertices of G so that no edge is monochromatic (i.e., has only one color on it).

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Typical Graph Coloring Questions : 1. Does there exists a coloring of G with k colors? 2. What is the fewest number of colors one can color G with? (“Chromatic Number”, denoted (G).) 3. How many colorings are there of G with k colors? (“Chromatic Polynomial”, often denoted P
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sudoku1 - The Mathematics of Sudoku Joshua Cooper...

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