1. Hex: a Game of Connecting Faces

2. Player 1 Player 2 Players take turns placing blue chips (player 1) and red chips (player 2). Player 1 plays first. Player 1 attempts to connect the top and bottom of the board with blue chips. Player 2 attempts to connect the left and right of the board with red chips. The Object of Hex

3. Blue wins! Hex on a Sample 3x3 Board

4. 4x4 Board and Dual Graph Superimposed

5. shift Shift of Slanted Dual to Square Dual

6. face adjacent faces vertex adjacent vertices vertex adjacent vertices face adjacent faces edge bounding two faces two adjacent faces edge joining two vertices two adjacent vertices Dual Graph Correspondences

7. Hex Sample Game Simultaneously on 3x3 Board and Dual Board

8. Hex Never Ends in a Draw: Setting up the Game Board 1. Start with a completed Hex game. Either there is a winner, or it is a draw. In the case of a draw, assume that play continues until all hexes are colored. Below illustrates the situation of the “draw”. (there are no draws, but that is what we are trying to prove) 2. Add boundary rows. On the top and bottom, add blue rows. On the left and right, add red rows.

9. Hex Never Ends in a Draw: Walking around the Game Board 3. Label the corners NW, NE, SW, and SE. NW NE SE SW 4. Walk around the board starting from the NW corner, keeping blue on the left side. A partial path is shown here. It is easy to see that the path will exit at the SW corner.

10. Hex Never Ends in a Draw: Keeping Blue on the Left Is it possible to keep blue on the left?

11. Hex Never Ends in a Draw: Keeping Blue on the Left Is it possible to keep blue on the left? Yes, by induction. At the start, at the NW corner, blue is on the left and red is on the right. NW

12. Hex Never Ends in a Draw: Keeping Blue on the Left Is it possible to keep blue on the left? Yes, by induction. At the start, at the NW corner, blue is on the left and red is on the right. NW Suppose at the nth step, blue is still on the left, and red on the right.

13. Hex Never Ends in a Draw: Keeping Blue on the Left Is it possible to keep blue on the left? Yes, by induction. At the start, at the NW corner, blue is on the left and red is on the right. NW Suppose at the nth step, blue is still on the left, and red on the right. Case 1: next hex is blue Path turns to the right. Blue is still on the left side, and red is still on the right side.

14. Hex Never Ends in a Draw: Keeping Blue on the Left Is it possible to keep blue on the left? Yes, by induction. At the start, at the NW corner, blue is on the left and red is on the right. NW Suppose at the nth step, blue is still on the left, and red on the right. Case 1: next hex is blue Path turns to the right. Blue is still on the left side, and red is still on the right side. Case 2: next hex is red Path turns to the left. Blue is still on the left side, and red is still on the right side.

15. Hex Never Ends in a Draw: Keeping Blue on the Left Is it possible to keep blue on the left? Yes, by induction. At the start, at the NW corner, blue is on the left and red is on the right. NW Suppose at the nth step, blue is still on the left, and red on the right. Case 1: next hex is blue Path turns to the right. Blue is still on the left side, and red is still on the right side. Case 2: next hex is red Path turns to the left. Blue is still on the left side, and red is still on the right side. By induction, we can walk around always keeping blue on the left and red on the right.

16. Hex Never Ends in a Draw: The Path Does not Intersect Itself Now that we know the path keeps blue on the left and red on the right, we know that if it is going to exit the board, it must do so either at the NE corner or the SW corner. The SE corner is out, because there blue is on the right and red is on the left. NW NE SE SW Enter Exit 1 Exit 2

17. Hex Never Ends in a Draw: The Path Does not Intersect Itself Now that we know the path keeps blue on the left and red on the right, we know that if it is going to exit the board, it must do so either at the NE corner or the SW corner. The SE corner is out, because there blue is on the right and red is on the left. NW NE SE SW Enter Exit 1 Exit 2 But how do we know that we can make the path exit the board? It might start going in circles.

18. Hex Never Ends in a Draw: The Path Does not Intersect Itself Now that we know the path keeps blue on the left and red on the right, we know that if it is going to exit the board, it must do so either at the NE corner or the SW corner. The SE corner is out, because there blue is on the right and red is on the left. NW NE SE SW Enter Exit 1 Exit 2 But how do we know that we can make the path exit the board? It might start going in circles. We prove by induction that the path cannot intersect itself, and therefore cannot go in circles and must eventually exit.

19. Hex Never Ends in a Draw: The Path Does not Intersect Itself, II The path doesn’t intersect itself. Proof by induction. At the start, the path doesn’t intersect itself. NW

20. Hex Never Ends in a Draw: The Path Does not Intersect Itself, II The path doesn’t intersect itself. Proof by induction. At the start, the path doesn’t intersect itself. NW Suppose after n steps, the path doesn’t intersect itself.

21. Hex Never Ends in a Draw: The Path Does not Intersect Itself, II The path doesn’t intersect itself. Proof by induction. At the start, the path doesn’t intersect itself. NW Suppose after n steps, the path doesn’t intersect itself. At the n+1 step, assume the path turns left (by symmetry).

22. Hex Never Ends in a Draw: The Path Does not Intersect Itself, II The path doesn’t intersect itself. Proof by induction. At the start, the path doesn’t intersect itself. NW Suppose after n steps, the path doesn’t intersect itself. At the n+1 step, assume the path turns left (by symmetry). Assume to the contrary that the path intersects itself at this added step. There are two cases. Case 1. Red is on the left side of the arrow that causes the intersection. Contradiction!

23. Hex Never Ends in a Draw: The Path Does not Intersect Itself, II The path doesn’t intersect itself. Proof by induction. At the start, the path doesn’t intersect itself. NW Suppose after n steps, the path doesn’t intersect itself. At the n+1 step, assume the path turns left (by symmetry). Case 2. Blue is on the right side of the arrow that causes the intersection. Contradiction! Assume to the contrary that the path intersects itself at this added step. There are two cases. Case 1. Red is on the left side of the arrow that causes the intersection. Contradiction! Thus the path doesn’t intersect itself.

24. Hex Never Ends in a Draw (Summary) NW NE SE SW Enter Exit 1 Exit 2 We know now that the path must enter at NW and leave at either NE or SW. What is the interpretation?

25. Hex Never Ends in a Draw (Summary) NW NE SE SW Enter Exit 1 Exit 2 We know now that the path must enter at NW and leave at either NE or SW. What is the interpretation? Case 1. If the path leaves at the NE, then red wins – to the right side of the path are red hexes.

26. Hex Never Ends in a Draw (Summary) NW NE SE SW Enter Exit 1 Exit 2 We know now that the path must enter at NW and leave at either NE or SW. What is the interpretation? Case 1. If the path leaves at the NE, then red wins – to the right side of the path are red hexes. Case 2. If the path leaves at the SW, then blue wins – to the left side of the path are blue hexes.

27. Bridge: A Variant of Hex

28. Players take turns placing blue squares (player 1) and red squares (player 2). Player 1 plays first. Player 1 attempts to connect the left and right of the board with a blue line. Player 2 attempts to connect the top and bottom of the board with a red line.

29. Bridge: Sample Play

30. Bridge: Sample Play II Blue wins!

31. Bridge: Trial Conversion to Hex Form