Solving Rummikub with Heuristic Guided State Space Search
David Samuelson and Michael Seibert
226 Atanasoff Hall Ames, IA 50011-1041

Abstract

The Rummikub problem of placing the maximum number of tiles from your rack onto the table is difcult due to state space explosion. To the best of our knowledge, it has never been solved as a state space search problem. We demonstrate the feasibility of solving the problem as a standard state space search. We test the performance of several different heuristics functions with standard best rst search and predatory search (Linhares 1998), a modied best rst search technique. We compare our results to an integer linear programming solution (Den Hertog and Hulshof 2006) of the same problem and comment on modern approaches to state space search.
and adding a single tile to a run or group. States are checked for validity by enumerating over all the runs and groups and checking their validity. If the total collection of tiles from all invalid groups and runs is a subset of the hand, the state is valid.

Heuristics

We used three basic heuristics: the number of invalid tiles, the number of invalid sets, and the number of tiles needed to make all sets valid. We applied these on the entire game state (hand and board) and the board only (excluding penalties originating from the hand) for a total of six heuristics.

Introduction

Rummikub is a game for 24 players that can be played with either specialized Rummikub tiles or 2 standard playing card decks with 2 jokers total. Our objective was to create an AI agent capable of playing Rummikub at least as well as a competent player. To the best of our knowledge, the only known approach to solving the Rummikub problem is integer linear programming (Den Hertog and Hulshof 2006). The Rummikub problem (RP) involves moving tiles from a private rack onto the public, shared board. The board holds a collection of groups of size at least 3: sets (tiles of the same rank) and runs (tiles of the same suit or color, and consecutive rank. Jokers may be played as any tile, but retain their value once played. A player may replace a joker with the tile it is representing, but then must immediately play the joker with at least two other tiles from their hand to form a new group. If no move can be made, then a tile is drawn from the deck. The rst player to play all their tiles wins. In addition to research into Rummikub specically, there is also signicant research into state space exploration, especially constrained state space exploration. (Dwyer et al. 2007) (Den Hertog and Hulshof 2006) (Linhares 1998) (Paredis 1993) (Passino 1994) (Perny, Spanjaard, and Storme 2007)

Search Algorithms

We used two search algorithms: simple best rst search (BFS) and modied version. The modied version behaved exactly the same as BFS except each states raw heuristic was multiplied by a proximity function. The proximity function mapped depth of a node to a real value: () where is some constant measuring the strength of the predatory search and [0, 1] is some decay constant.

Results

All algorithms were implemented in Java. Tests were run on a lightly loaded dual core Intel machine (at 1.86 Ghz) with 3.5 Gb of RAM running Debian Linux. For each example, the six heuristics were tested with and without predatory search. Each test had a timeout of 1 second (the time required for the integer linear programming technique (Den Hertog and Hulshof 2006) to nd solutions).

Vs. Linear Approach

Two examples were given in the linear approach paper (Den Hertog and Hulshof 2006). Our results on the rst test were favorable. Two heuristics matched their performance (with and without predatory search). Unfortunately, their solution for the second test required different (and invalid) joker rules, so comparison was not meaningful.

Approach

We modeled each game state as a collection of runs and groups. We allow several types of transitions: removing a tile from either end of a run, removing any tile from a group,
c Copyright 2009, Association for the Advancement of Articial Intelligence (www.aaai.org). All rights reserved.

Absolute Testing

We generated ten tests of increasing complexity. All have solutions with all cards from hand placed onto the board.
The test setup and raw results are available in the appendix. All three heurstics which considered the entire board found all solutions (invalid cards (IC), invalid card collections (ICC), and distance to legal (DTL)). The three heuristics which did not consider the hand (invalid cards on board (ICOB), invalid card collections on board (ICCOB), and distance to legal on board (DTLOB)) had lower performance.

Appendix

Test #1 heuristic IC ICOB ICC ICCOB DTL DTLOB Test #2 heuristic IC ICOB ICC ICCOB DTL DTLOB Test #3 heuristic IC ICOB ICC ICCOB DTL DTLOB Test #4 heuristic IC ICOB ICC ICCOB DTL DTLOB Test #5 heuristic IC ICOB ICC ICCOB DTL DTLOB Test #6 heuristic IC ICOB ICC ICCOB DTL DTLOB Test #7 heuristic IC ICOB ICC ICCOB DTL DTLOB nodes nodes nodes nodes nodes nodes nodes BFS tiles left BFS tiles left BFS tiles left BFS tiles left BFS tiles left BFS tiles left BFS tiles left predator nodes tiles left predator nodes tiles left predator nodes tiles left predator nodes tiles left predator nodes tiles left predator nodes tiles left predator nodes tiles left 132 2

Conclusion

Implementation of our algorithm did not go as smoothly as we had hoped. Our initial heuristics were relatively crude, but we could not nd any that worked better. Jokers proved to be difcult to implement correctly, and caused issues when comparing to similar work by previous researchers. We now have a better understanding of how to implement a state-space oriented AI agent.

References

Den Hertog, D., and Hulshof, P. B. 2006. Solving rummikub problems by integer linear programming. Comput. J. 49(6):665669. Dwyer, M. B.; Elbaum, S.; Person, S.; and Purandare, R. 2007. Parallel randomized state-space search. In ICSE 07: Proceedings of the 29th international conference on Software Engineering, 312. Washington, DC, USA: IEEE Computer Society. Linhares, A. 1998. State-space search strategies gleaned from animal behavior: a traveling salesman experiment. Biological Cybernetics 78(3):167173. Paredis, J. 1993. Genetic state-space search for constrained optimization problems. In IJCAI93: Proceedings of the 13th international joint conference on Artical intelligence, 967972. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc. Passino, K.M., A. P. 1994. A metric space approach to the specication of the heuristic function for the a* algorithm. Systems, Man and Cybernetics, IEEE Transactions on 24(1):159166. Perny, P.; Spanjaard, O.; and Storme, L.-X. 2007. State space search for risk-averse agents. In IJCAI07: Proceedings of the 20th international joint conference on Artical intelligence, 23532358. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc.

test # 9 10

initial board state R7-13 R4-7 R8-13 RBKO4 R5-7 R8-13 RBK4 O4-7 R5-7 R8-13 O4-6 R5-7 R8-13 RKB4 RBKO6 O4-6 RBK4 R5-7 R8-13 JRB5 O5-8 JRB5 O3-8 JRB5 O8-11 O3-7 JRB5
hand B10 K10 B10 K10 B10 K10 B10 K10 B10 K10 O7 B10 K10 O7 O5 K3 K4 K6 K7 K3 K4 K6 K7 O2 K3 K4 K6 K7 O2 K3 K4 K6 K7
nal board state R7-9 R11-13 BKR10 R4-9 R11-13 BKR10 BKO4 R4-9 R11-13 BKR10 O5-7 BKO4 R4-9 R11-13 BKR10 O5-7 BKO4 R4-9 R11-13 BKR10 RBK7 O5-7 BKO4 R4-9 R11-13 BKR10 RBO5 K3-7 RBO5 O6-8 K3-7 O2-4 O6-8 ORB5 K3-7 ORB5 K3-7 O7-11 O2-4
Figure 1: Test cases Test #8 heuristic IC ICOB ICC ICCOB DTL DTLOB Test #9 heuristic IC ICOB ICC ICCOB DTL DTLOB Test #10 heuristic IC ICOB ICC ICCOB DTL DTLOB BFS tiles left BFS tiles left BFS tiles left predator nodes tiles left predator nodes tiles left predator nodes tiles left 4554 5

Twining Village Calendar

SUNDAY

MAY 2011

TUESDAY WEDNESDAY THURSDAY FRIDAY SATURDAY

MONDAY

7:30 St. Bede's Catholic Services (Bus Stop #1, 2, 3, 4, 5, and 10) 9:30 United Methodist Church (Bus Stop #1, 2, 3, 4, 5, and 10) 10:15 Presbyterian Church in Newtown (Bus Stop #1, 2, 3, 4, 5, and 10) 2:30 Ecumenical Worship Service (AUD)
10:00 Grandma's Attic (MB) 11:00 Tai Chi (AUD) 1:00 Balance & Mobility Training (AUD) 1:30 BINGO (HH) 2:00 Open Swim Time (PL) 6:00 Rummy Q (HR) 6:00 Hand & Foot Canasta (HR) 6:30 Pinochle (HH)
8:30 Wii Bowling (AUD) 9:00 First National Bank 10:00 Recreation Committee Meeting (RA) 10:30 Knit or Crochet (VC) 1:00 Group Strength & Flexibility Class (AUD) 1:00 Hand & Foot Canasta (FS) 1:00 Shopping - Richboro 1:15 Rummikub (HR) 2:30 Water Volleyball (PL) 6:00 Hand & Foot Canasta (HR) 6:30 Bridge (HH)

_ _ 130

9:30 Duplicate Bridge (HH) 10:00 Weaving Group (HR) 1:00 Arthritis Foundation Exercise Program (AUD) 1:00 Shopping - Richboro 1:35 Rosary & Prayer Circle (CC) 2:00 Open Swim Time (PL) 7:00 Fran Schwartz (AUD)
8:30 Open Swim Time (PL) 9:00 First National Bank 10:00 Craft Class (HR) 1:00 Group Strength & Flexibility Class (AUD) 1:15 Special Shopping Warminster Walmart 2:30 Rolling Hills Elementary School - 5th Grade Chorus (AUD) 6:00 Rummy Q (HR) 6:00 Hand & Foot Canasta (HR) 6:30 Pinochle (HH)

_ _ 113

9:00 Annual Resident Mini-Bazaar (MB) 10:00 Oxford Valley Mall & Feasterville Giant 1:00 Arthritis Foundation Exercise Program (AUD) 2:00 Northampton Library 2:00 Wii Bowling (AUD) 7:00 Friday Night at the Movies (HH)

7:10 Classic Movie (HH)

Mothers Day
10:00 Grandma's Attic (MB) 11:00 Tai Chi (AUD) 1:00 Balance & Mobility Training (AUD) 1:30 Dining Room Committee Meeting (RA) 1:30 BINGO (HH) 2:00 Open Swim Time (PL) 6:00 Rummy Q (HR) 6:00 Hand & Foot Canasta (HR) 6:30 Pinochle (HH)
8:30 Wii Bowling (AUD) 9:00 First National Bank 10:30 Knit or Crochet (VC) 1:00 Group Strength & Flexibility Class (AUD) 1:00 Hand & Foot Canasta (FS) 1:00 Shopping - Richboro 1:15 Rummikub (HR) 2:30 Water Volleyball (PL) 6:00 Hand & Foot Canasta (HR) 7:00 Residents' Association Membership Meeting (AUD)
9:30 Duplicate Bridge (HH) 10:00 Weaving Group (HR) 10:00 Transportation for Catholic Mass (BS#2) 10:30 Catholic Mass (MC) 1:00 Arthritis Foundation Exercise Program (AUD) 1:00 Shopping - Richboro 1:35 Rosary & Prayer Circle (CC) 2:00 Open Swim Time (PL) 2:00 History, Culture, & Religion (CC) 7:00 Katie Eagleson & Jazz Ensemble (AUD)

_ _ 125

8:30 Open Swim Time (PL) 9:00 First National Bank 9:15 Free Hearing Screening and Aid Check (CC) 1:00 Group Strength & Flexibility Class (AUD) 2:00 Current Events (RA) 6:00 Rummy Q (HR) 6:00 Hand & Foot Canasta (HR) 6:30 Pinochle (HH) 6:30 Bridge (HH) 7:00 Construction Project Presentation (AUD)
10:00 Neshaminy Mall & Feasterville Giant 12:45 Princeton University Art Museum (BS#2) 1:00 Arthritis Foundation Exercise Program (AUD) 2:00 Wii Bowling (AUD) 7:00 Friday Night at the Movies (HH)

3:30 Happy Hour (VC) 7:10 Classic Movie (HH)
8:30 Wii Bowling (AUD) 9:00 First National Bank 10:00 Voting - Primary Election (BS #2 & #10) 10:30 Knit or Crochet (VC) 11:00 Voting - Primary Election (BS #2 & #10) 1:00 Group Strength & Flexibility Class (AUD) 1:00 Hand & Foot Canasta (FS) 1:00 Voting - Primary Election (BS #2 & #10) 1:00 Shopping - Richboro 1:15 Rummikub (HR) 2:30 Water Volleyball (PL) 6:00 Hand & Foot Canasta (HR) 6:30 Bridge (HH) 7:00 Fred Miller (AUD)
9:30 Duplicate Bridge (HH) 10:00 Weaving Group (HR) 1:00 Arthritis Foundation Exercise Program (AUD) 1:00 Shopping - Richboro 1:35 Rosary & Prayer Circle (CC) 2:00 Open Swim Time (PL) 3:00 Malcolm Brashear Conductor (AUD)
8:30 Open Swim Time (PL) 9:00 First National Bank 11:15 Bristol Riverside Theater Presents - Little Women (Musical) (BS#2) 1:00 Group Strength & Flexibility Class (AUD) 1:15 Trader Joe's - Special Shopping (BS#2) 4:00 Let's Get Together! (CC) 6:00 Rummy Q (HR) 6:00 Hand & Foot Canasta (HR) 6:30 Pinochle (HH)
10:00 Oxford Valley Mall & Feasterville Giant 1:00 Arthritis Foundation Exercise Program (AUD) 2:00 Wii Bowling (AUD) 7:00 Friday Night at the Movies (HH)
1:00 Council Rock South Spring Concert (BS #10) 7:10 Classic Movie (HH)

TUESDAY

WEDNESDAY

THURSDAY

FRIDAY

SATURDAY

10:00 Grandma's Attic (MB) 11:00 Let's Talk with Jeanne (AUD) 11:00 Tai Chi (HH) 1:00 Balance & Mobility Training (AUD) 1:30 BINGO (HH) 2:00 Open Swim Time (PL) 6:00 Rummy Q (HR) 6:00 Hand & Foot Canasta (HR) 6:30 Pinochle (HH)
8:30 Wii Bowling (AUD) 9:00 First National Bank 10:30 Knit or Crochet (VC) 1:00 Group Strength & Flexibility Class (AUD) 1:00 Hand & Foot Canasta (FS) 1:00 Shopping - Richboro 1:15 Rummikub (HR) 2:30 Water Volleyball (PL) 6:00 Hand & Foot Canasta (HR) 6:30 Bridge (HH)

24 Senior Fitness Day

9:30 Fitness Walk (Village Flag Pole) 9:30 Duplicate Bridge (HH) 10:00 Weaving Group (HR) 10:30 Dr. Nicholes DiNubile's "Framework" (AUD) 11:30 Birthday Celebration (VDR) 1:00 Get Acquainted with Fitness Room & Pool (VFR) 1:00 Shopping - Richboro 1:35 Rosary & Prayer Circle (CC) 2:00 Fitness Fun Day (AUD) 2:00 History, Culture, & Religion (CC) 7:00 Richard DiBlassio & RIchard DeLello (AUD)
8:30 Open Swim Time (PL) 9:00 First National Bank 11:30 Special Luncheon (BS#2) 1:00 Group Strength & Flexibility Class (AUD) 2:00 Group Crossword Game (HH) 2:00 Current Events (RA) 6:00 Rummy Q (HR) 6:00 Hand & Foot Canasta (HR) 6:30 Pinochle (HH)
10:00 Neshaminy Mall & Feasterville Giant 1:00 Arthritis Foundation Exercise Program (AUD) 2:00 Wii Bowling (AUD) 7:00 Friday Night at the Movies (HH)

7:30 St. Bede's Catholic Services (Bus Stop #1, 2, 3, 4, 5, and 10) 9:30 United Methodist Church (Bus Stop #1, 2, 3, 4, 5, and 10) 10:15 Presbyterian Church in Newtown (Bus Stop #1, 2, 3, 4, 5, and 10) 2:30 Ecumenical Worship Service (AUD) 6:30 Shuffleboard (SC)

29 Memorial Day

6:00 Rummy Q (HR) 6:00 Hand & Foot Canasta (HR) 6:30 Pinochle (HH)
Please check Channel 24 on your television, Lobby Digital Signage, or weekly activity events on bulletin board for changes, additions, and/or special announcements to this calendar.

Room Key

(HR) = 800 Hobby Room (PL) = Village Pool (RA) = Residents' Association Office (VC) = Villa Caf_ (VDR) = Village Dining Room (VFR) = Village Fitness Room (V Lib.) = Village Library
(AUD) = Kelly Auditorium (BS#2) = Bus Stop #2
(BS #2 & #10) = Bus Stop #2 & #10 (CC) = Casual Corner
(FS) = Fireside Lounge (HH) = Holland Hall