1. An Analysis of Jenga Using Complex Systems Theory
Avalanches
Wooden Blocks
Spherical Cows
By John Bartholomew, Wonmin Song, Michael Stefszky and Sean Hodgman

2. Jenga – A Brief History Developed in 1970’s by Leslie Scott
Name from kujenga, Swahilli verb “to build”
Israel name Mapolet meaning “collapse”
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3. Jenga - The Game Game involves stacking wooden blocks
Tower collapse game over
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4. Jenga - A Complex System?
Why would Jenga be Complex?
Displays properties of Complex Systems
Tower collapse similar to previous work on Avalanche Theory
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5. Jenga - A Complex System?
Emergence
History
Self-Adaptation
Not completely predictable
Multi-Scale
Metastable States
Heterogeneity
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6. Ultimate Jenga Strategy
Motivation?
Ultimate Jenga Strategy
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7. Motivation Power Law Complex Systems Assignment 1:
Power Law
Frette et al. (1996)
Turcotte (1999)
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8. Self Organizing Criticality
Theory Proposed by Bak et al. (1987)
Dynamical systems naturally evolve into self organized critical states
Events which would otherwise be uncoupled become correlated
Frette et al. (1996)
Periods of quietness broken by
bursts of activity
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9. Sandpile model Avalanche size: 2
Minor perturbation can lead to local instability or global collapse – ‘avalanche’
Avalanche size: 2
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10. Sandpile model
Jenga cannot be modelled using the Sandpile Model because:
We have removed the memory affects
A more suitable model involves assigning a
‘fitness’ to each level which is altered dependant
on the removal of a block
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11. Cautious way forward… “Experimental results have been quite ambiguous”
Turcotte 1999
Quasi-periodic behaviour for large avalanches Evesque and Rajchenbach 1989, Jaeger et al 1989
Power law behaviour Rosendahl et al 1993, 1994, Frette et al 1996
Large: periodic
Small: power law
Bretz et al 1992
Small: periodic
Large: power law
Held et al 1990
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12. What We Did Played a LOT of games of Jenga ~400
From This
To This
Played a LOT of games of Jenga ~400
Chose 5 different strategies to play
Recorded 3 observables
Number of bricks that fell in “avalanche”
Last brick touched before “avalanche”
Distance from base of tower to furthest brick after the tower fell
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13. Strategies JENGA JENGA JENGA JENGA JENGA JENGA JENGA JENGA JENGA JENGA
Middles Out
ZigZag
JENGA
JENGA
JENGA
JENGA
Side 1
Side 2
JENGA
Middle Then Sides
JENGA
Side 1
JENGA
JENGA
AND FINALLY…
An optimal game strategy where we would start from the bottom and work our way up, pulling out any bricks which were loose enough to pull out easily
Side 1
Side 2
All Outside Bricks
JENGA
JENGA
JENGA
Side 1
Side 2
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14. Many Strategies So We Could …
Compare strategies to see if any patterns were emerging
Compare more ordered methods of pulling bricks out to the random optimal strategy
See if strategies used had a large impact on the data obtained.
Whoooooaaaaaaa!!!!!!!!
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15. What We Expected
We hoped to see at least some emerging signs of a complex system as more data was taken
We assumed the distance of blocks from base would be Gaussian to begin with but maybe tend towards a power law
Perhaps some patterns relating to strategies used and observables
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16. Results – Stability Regions
Analysed number of blocks before tower collapse
Separately for each strategy and combined
Results show stability regions for many strategies
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17. Complex Systems Assignment 1:

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19. Results – Different Strategies
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20. Maximum Distance of falling Block
Results
Maximum Distance of falling Block
Not Enough Data to definitively rule out one distribution, Gaussian and Cauchy-Lorentz look to fit data quite well
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21. Results – Step Size Blocks Removed
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22. Results – Step Size Blocks Remaining
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23. Results – Step Size Maximum Distance
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24. Results – Memory effects?
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25. Modeling – Another Spherical Cow?
Universality of network theory:
Topology of networks explains various kinds of networks.
Social networks, biological networks, WWW
Why not Jenga?
Look at Jenga layers as nodes of a network with: specified fitness values assigned to each layer, and each layer is connected to the layers above it.
This simplifies the picture for us to look at 18 layers, not at all 54 pieces!!
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26. Modified sandpile model
- As mentioned before, the sandpile model eliminates least fit cells of sand Selection law: life is tough for weak and poor!
- The whole system self-organizes itself to punctuated equilibriums due to the memory effect.
- Our case is a bit different.
Sand-pile model
Toy model
Attack the least fit cell
Attack the fittest layer
Neighbors to the least
fit cell attacked
subsequently
Layers above the
attacked layer are
attacked subsequently
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27. Fitness & The Magic Number
We describe stability of each layer by fitness
Fitness = 1 indicates stability, and fitness below a threshold value is unstable.
Algorithm
We tested values for: threshold fitness between strength of attack with randomness added i.e. human hands apply attack with uncertainty in strength value (shaky hands).
Each attack affects the layers above with decreasing attack power.
Repeat the attack until a layer appears with fitness lower than the threshold.
Stack a layer on the top for every 3 successions of attack.
Outcomes? Distributions for: Maximum height layer index number average fitness
Magic number!!
- There is always some magic number turn that you are almost guaranteed to have a safe pass at the turn!!!!
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31. Playing Jenga is a random walk process!!!!
Accordance with the data
No indication of power-law behavior because of the
absence of memory
Gaussian, and Poisson distributions emerge instead.
Playing Jenga is a random walk process!!!!
Real data analysis shows the random walk process by exhibiting Gaussian features in fluctuation plots.
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32. And the magic number emerged…..
In the case of the model:
Whoever takes the 7th turn is almost guaranteed a safe pass.
The Toy Model mimics the emergence of stability regions and gives an indication about the gross behavior of the ‘Jenga’ network.
Allows us to see the Jenga tower as a cascade network.
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33. Conclusions Randomness in all strategies
Step size structure due to artificial memory
Modified sandpile model: directed network
Model mimicking real situation: Emergence of stability regions
Complex structure identified but more data needed
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34. Bibliography
Bak et al., Self-organized Criticality, Phys. Rev. A. 31, 1 (1988)
Bak et al., Punctuated Equilibrium and Criticality in a simple model of evolution, Phys. Rev. Lett. 71, 24 (1993)
Bak et al., Complexity, Contingency, and Criticality, PNAS. 92 (1995)
Frette et al., Avalanche Dynamics in a pile of rice, Nature, 379 (1996)
“Jenga”, Available online at:
Turcotte, Self-organized Criticality, Rep. Prog. Phys. 62 (1999)
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