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A Retrograde Approximation Algorithm for One Player Can’t Stop
James Glenn Loyola College in Maryland
Haw-ren Fang University of Maryland, College Park
Clyde Kruskal University of Maryland, College Park
Can’t Stop Equipment 2 to 4 players 4 6-sided dice Board with
columns 2-12 Colored markers
for each column 3 neutral markers Goal: advance to
top of 3 columns78
910
1112
65
43
2
Game Play
Roll 4 dice Split into 2 pairs Advance neutral markers in columns
for pair totals Roll again or stop Turn ends if no way to use pair totals
Example
78
910
1112
65
43
2
Position Graph
Similar to Backgammon or Yahtzee Bipartite (V1, V2, E) (v2, v1) in E correspond to player
choices (v1, v2) in E correspond to outcomes of
random event
Can’t Stop Graph
Comparison to Other Games
Solitaire Yahtzee No cycles
Backgammon General: long cycles Bearing off: only short cycles
Can’t Stop Cycles long but only within one turn components form a DAG
Can’t Stop Graph
anchor
anchors of other components
Retrograde Analysis Topologically sort graph Compute position value for each
vertex Start with final states Work back towards start state Player choice: compute max/min over
outgoing edges Random event: weighted average Can’t Stop: retrograde analysis on
components
Symbolic Analysis
f(B)=f(E) f(C)=min(f(E), f(F)) f(D)=f(A) f(A)=p1f(E)
+p2•min(f(E),f(F))
+p3f(A)A
B C D
E F
p1p2
p3
Symbolic Analysis for Can’t Stop f(E), f(F) given f(H)=f(F) f(G)=f(A) f(D)=p3f(G)+p4f(H) f(C)=min(f(E),f(F)) f(B)=min(f(D),f(E)) f(A)=1+p1f(B)+p2f(C)=
1+p1min(p3f(A)+p4f(F),f(E))
+p2min(f(E), f(F))A
B C
E F
D
G H
p1 p2
p3 p4
Numerical Analysis Make a copy of anchor component is now a DAG Guess value of f(A’) f(A) is a function of f(A’) Want fixed point
Function is piecewise linear and continuous
Fast convergence from Newton’s method
A
B C
E F
D
G H
p1 p2
p3 p4
A’
Retrograde Analysis for Can’t Stop
f(v) = 0 for all final states v For each non-final anchor in reverse
order of topological sort Make a copy of the anchor Topologically sort the anchor’s
component Apply Newton’s method to find value of
anchor; using retrograde analysis on each iteration
Results
Official game: estimated 3000 years
Dice Size Graph Size Time to Solve Optimal Turns
2 1 225 0.166 sec 1.298
2 2 1,936 0.405 sec 1.347
2 3 9,025 0.601 sec 1.400
3 1 64,372 1.70 sec 1.480
3 2 787,600 5.05 sec 1.722
3 3 4,934,006 23.3 sec 1.890
4 1 20,802,843 5 min 2.187
4 2 289,091,584 59 min 2.454
4 3 2,104,663,011 6 hr 2.700
5 1 7,105,015,062 2.8 days 2.791
Future Work
Optimizations to Algorithm Better initial estimate Shortcuts when evaluating components
Distributed Algorithm Analysis of 2-player Can’t Stop