# JENGA and other wooden block games

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JENGA and other wooden block games
Uri Zwick Tel Aviv University

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

JENGA is a very popular game!

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

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.

Who wins? How?

Everything else is stable!
Instability Everything else is stable!

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!

Possible Moves *2 *1

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

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

Analysis I

Analysis II

Solution

Nim values of JENGA

Optimal Moves

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

What next?

JENGA - Truth or Dare

Who wins in JENGAk? k=5

JENGA2k is a win for the second player!
A simple symmetry argument.

Some interesting JENGA5 positions
*15 *17

Which towers are stable?

“Simple” towers The center of gravity of each upper part of the tower should be above the area of contact between the upper and lower parts of the tower

Does this hold for more general towers?
Of course NOT!

Is this simple necessary condition sufficient for JENGAk towers?
YES, for k=3,4 and 6. NO, otherwise.

Unstable JENGA5 towers

Rigid body in equilibrium

Forces acting on towers

Equivalent systems of forces

Stability and linear programming
A tower is weakly stable if and only if its corresponding linear program is feasible. A tower is stable if and only if its corresponding linear has a strictly positive feasible point.

Simple Variations of JENGA
Remove a block from anywhere and put it anywhere on the top level. If the top level is full, then start a new level. Remove a block from anywhere and put it anywhere on top, or start a new level. If a block from the top level is removed, then it must start a new level.

More complicated variations of JENGA
Remove a block from anywhere, and put it anywhere higher. (Filling in gaps is allowed.) Remove a block, or slide it outward by a multiple of 1/k of the length of a block. If a block is completely removed, then put it anywhere on top.

Free Play JENGAk Remove a block from anywhere, and put it in an arbitrary position at the top level, or start a new level, not necessarily in one of the fixed k positions of standard JENGAk games.

More basic open problems
Which positions in JENGAk are: REACHABLE? CONSTRUCTIBLE? SCULPTUREABLE?