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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 JENGA is a very popular game!

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JENGA The rules of the 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.

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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.

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Who wins? How?

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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 0 0 *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=2y=6z=2 x≥0 y≥0 0≤z<3

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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)

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

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What next?

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JENGA JENGA - Truth or Dare

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Who wins in JENGA Who wins in JENGA k ? k=5

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JENGA JENGA 2k is a win for the second player! A simple symmetry argument.

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Some interesting JENGA 5 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 JENGA k towers? YES, for k=3,4 and 6. NO, otherwise.

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Unstable JENGA 5 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 JENGA k JENGA kRemove 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 JENGA k games.

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

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