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**JENGA and other wooden block games**

Uri Zwick Tel Aviv University

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**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.

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**JENGA is a very popular game!**

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**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.

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**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.

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Who wins? How?

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**Everything else is stable!**

Instability Everything else is stable!

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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!

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Possible Moves *2 *1

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**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

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**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)

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Analysis I

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Analysis II

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Solution

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Nim values of JENGA

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Optimal Moves

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**JENGA is a win for the first player iff n1,2(mod 3) and n≥2.**

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What next?

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JENGA - Truth or Dare

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Who wins in JENGAk? k=5

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**JENGA2k is a win for the second player!**

A simple symmetry argument.

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**Some interesting JENGA5 positions**

*15 *17

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**Which towers are stable?**

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“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

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**Does this hold for more general towers?**

Of course NOT!

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**Is this simple necessary condition sufficient for JENGAk towers?**

YES, for k=3,4 and 6. NO, otherwise.

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Unstable JENGA5 towers

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**Rigid body in equilibrium**

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**Forces acting on towers**

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**Equivalent systems of forces**

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**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.

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**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.

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**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.

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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.

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**More basic open problems**

Which positions in JENGAk are: REACHABLE? CONSTRUCTIBLE? SCULPTUREABLE?

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