1 / 10

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:

wyatt-garner
Télécharger la présentation

GBK Geometry

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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!

More Related