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

JENGAJENGAUri Zwick

Tel Aviv University

SODA’02, Jan. 6-8, San Francisco

JENGAJENGA

• A real-life game with a surprisingly simple analysis.

• We consider, of course, an idealized version of the game.

• Purely of recreational value.

JENGA JENGA is a very popular game!

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.

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.

Who wins?Who wins?

How?How?

InstabilityInstability

Everything else is stable!Everything else is stable!

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!

Possible MovesPossible Moves

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

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)

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

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

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

ConclusionConclusion

JENGA JENGA is a win forthe first player iff

n1,2(mod 3) and n≥2.

Open ProblemsOpen Problems

•JENGAJENGA Throw ’n Go.

•JENGAJENGA Truth or Dare.

•JENGAJENGAk, , for k >3, k odd.

JENGAJENGA - Truth or Dare

JENGAJENGA2k is a win for the second player!

A simple strategy stealing argument.

Who wins inWho wins in JENGA JENGA5?