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GBK Geometry

GBK Geometry. Jordan Johnson. Today’s plan. Greeting Hand in Week 9 Journal The Game of Hex Clean-up. Hex. Invented by Piet Hein in 1942… …and by John Nash (independently) in 1947. Goal: Connect Opposite sides. Play Hex for 10 minutes. Hex: Follow-up. In pairs, discuss:

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GBK Geometry

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  1. GBK Geometry Jordan Johnson

  2. Today’s plan • Greeting • Hand in Week 9 Journal • The Game of Hex • Clean-up

  3. Hex Invented by Piet Heinin 1942… …and by John Nash (independently)in 1947.

  4. Goal: Connect Opposite sides

  5. Play Hex for 10 minutes

  6. Hex: Follow-up • In pairs, discuss: • What questions about Hex occurred to you as you played the game? • What have you noticed about how the game works?

  7. Hex • Basic theorems: • The game must end eventually. • Proof sketch:There are 112 = 121 empty cells at the beginning, and each turn subtracts one empty cell. • Ties are impossible. • Proof idea:If there is no path connecting one pair of opposite sides, there must be a path connecting the other pair of opposite sides.

  8. Hex • Theorem: • The first player of a Hex gamealways has a winning strategy. • Proof sketch (by John Nash): • Either the first or second player musthave a winning strategy; ties areimpossible and the game must end. • Extra stones on the board for a player always improve that player’s position. • Suppose the second player has a winning strategy. • The first player can play anywhere,and then use the second player’s winning strategy. • If the strategy calls for playing on a cell the player already played,they can play anywhere on that turn. • This way, the first player will win. There can be no winning strategy for the second player. • The second player doesn’t have a winning strategy,so the first player must.

  9. Assignments • For Monday: • 20 min on Khan Academy over the weekend • For Tuesday:Asg #20

  10. Clean-up / Reminders • Pick up all trash / items. • Push in chairs (at front and back tables). • See you tomorrow!

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