JENGA Uri Zwick Tel Aviv University SODA’02, Jan. 6-8, San Francisco.
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Transcript of JENGA Uri Zwick Tel Aviv University SODA’02, Jan. 6-8, San Francisco.
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JENGAJENGAUri Zwick
Tel Aviv University
SODA’02, Jan. 6-8, San Francisco
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JENGAJENGA
• A real-life game with a surprisingly simple analysis.
• We consider, of course, an idealized version of the game.
• Purely of recreational value.
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JENGA JENGA is a very popular game!
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JENGAJENGA – The rules of the gameThe 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.
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JENGAJENGA – The rules of the gameThe 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.
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Who wins?Who wins?
How?How?
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InstabilityInstability
Everything else is stable!Everything else is stable!
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ImplicationsImplications• 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!
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Possible MovesPossible Moves
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ConfigurationsConfigurations
(x,y,z)
x - # of full levels
y - # of levels with two adjacent blocks
z - # of blocks on top.
x=3 y=6 z=2x≥1 y≥0 0≤z<3
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Possible MovesPossible Moves(x,y,z) (x-1,y,z+1)
(x,y,z)
(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)
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AnalysisAnalysis( 1, , 1)
( , , ) 1 min ( 1, 1, 1) 1, 0
( , 1, 1)
( 1, , 1)( ,0, ) 1 min 1
( 1, 1, 1)
(1, , ) 1 ( , 1, 1) 0
(1,0, ) 0
( , , 3) ( 1, ,0)
V x y z
V x y z V x y z if x y
V x y z
V x y zV x z if x
V x y z
V y z V x y z if y
V z
V x y V x y
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SolutionSolution
0
0
1 1 1 0 1 1 0 0 1
1 0 1 1 1 1 1 1 0
0 1
( , , ) ( mod 3, mod 3, ) 1 0
(1, ,0) ( m
0 0 1 1 1 1 1
0 1 2
3
0 1 0
od )
V x y z v x y z if x z
V y
v
z z z
v
v y
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Optimal MovesOptimal Moves
0
-I- -I- -I-I-I -I- II-I-I II- I-II-I II- II- I-I II- -I- -I--I-
-I- I-III- I-I I-I II- II-01 2
I-I
m
zz z
m
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ConclusionConclusion
JENGA JENGA is a win forthe first player iff
n1,2(mod 3) and n≥2.
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Open ProblemsOpen Problems
•JENGAJENGA Throw ’n Go.
•JENGAJENGA Truth or Dare.
•JENGAJENGAk, , for k >3, k odd.
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JENGAJENGA - Truth or Dare
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JENGAJENGA2k is a win for the second player!
A simple strategy stealing argument.
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Who wins inWho wins in JENGA JENGA5?