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Colloquium - Combinatorial Games

Combinatorial Games. 2 Player Games Players take defined turns Not a game of chance Game ends at defined winning/losing positions Examples chess, checkers, tictactoe, NIM, etc. Progressively Finite Games Which end in a finite number of moves. What we'll learn !. Theory behind imparti

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Colloquium - Combinatorial Games

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    1. Colloquium - Combinatorial Games Arpit Goel 2007CS10162

    2. Combinatorial Games 2 Player Games Players take defined turns Not a game of chance Game ends at defined winning/losing positions Examples – chess, checkers, tictactoe, NIM, etc

    3. What we’ll learn ! Theory behind impartial games Possible techniques to tackle puzzles Helpful for job interviews / CAT

    4. Example – Restricted Takeaway Game 13 Objects – 2 players Remove 1/2/3 objects at a time Last one to pick loses

    5. Kernels – The Wining Positions Formulate the problem as a digraph Kernels – A set of ‘good’ vertices (winning positions) No edge joining any 2 vertices in the kernel Every non kernel vertex ? Some kernel vertex

    6. Restricted Takeaway Contd

    7. General Strategy for Winning Given kernel set K 1st player tries to move to a kernel vertex every time If 1st vertex is in kernel – 2nd player has a winning strategy

    8. Unique Winning Strategy Theorem - Every progressively finite game has a unique winning strategy. That is, the graph of every progressively finite game has a unique kernel. Organize the vertices into levels based on distances from winning vertices Find a unique kernel set for a level k, given a unique set for all levels upto k – Induction on number of levels

    9. Grundy Function For each vertex x in the directed graph, g(x) is the smallest nonnegative integer not assigned to any of x’s successors Theorem – The graph of a progressively finite game has a unique Grundy function. Further, the vertices with Grundy number 0 are the vertices in the kernel

    10. Restricted Takeaway Game Contd

    11. The Game of Nim

    12. 12 Winning Strategy

    13. 13

    14. 14

    15. 15 Example : The Game of Nim

    16. 16 Example : The Game of Nim

    17. Why Does this Work ?

    18. References

    19. A Problem ?

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