1. JENGA Uri Zwick Tel Aviv University SODA’02, Jan. 6-8, San Francisco

2. JENGA A real-life game with a surprisingly simple analysis. We consider, of course, an idealized version of the game. Purely of recreational value.

4. JENGA JENGA is a very popular game!

5. JENGA The rules of the game JENGA – The rules of the game The game starts with an n-story tower of wooden blocks, three at each level. In the real-life game, n=18.

6. JENGA The rules of the game JENGA – The rules of the game Each player, in her turn, removes a block from anywhere below the highest completed level and stacks it on top. The player that topples the tower loses.

7. Who wins? How?

8. Instability Everything else is stable!

9. Implications Top most level, or the level just below it, is always full. The tower is stable, unless it contains the forbidden level: Two towers that differ only in the order of the levels are equivalent!

10. Possible Moves

11. Configurations (x,y,z) x - # of full levels y - # of levels with two adjacent blocks z - # of blocks on top. x=3y=6z=2 x≥1 y≥0 0≤z<3

12. Possible Moves (x,y,z) (x-1,y,z+1) (x,y,z) (x-1,y+1,z+1) (x,y-1,z+1) I-I -II -I- (x,y,3) → (x+1,y,0)

13. Analysis

14. Solution

15. Optimal Moves

16. Conclusion JENGA JENGA is a win for the first player iff n  1,2(mod 3) and n≥2.

17. Open Problems JENGAJENGA Throw ’n Go. JENGAJENGA Truth or Dare. JENGA,JENGA k, for k >3, k odd.

19. JENGA JENGA - Truth or Dare

20. JENGA JENGA 2k is a win for the second player! A simple strategy stealing argument.

21. Who wins in JENGA Who wins in JENGA 5 ?