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