Solving Mahjong Solitaire Positions

Bachelor Thesis Presentation

Tijmen Stam

December 6, 2007

info@iivq.net

Overview

● Introduction● The Game● Research Question● Heuristics and Strategies● Results● Conclusion

Introduction

Mahjong is the name of two games:– Mahjong, a 4-player table game invented by

Confucius (or in the 19th century)

– Mahjong Solitaire, a 1-player computer game first implemented by Brodie Lockard in 1981

The subject of our research is Mahjong Solitaire.

The Game (1)

Mahjong Solitaire is a tile removal game.

Stones: 144, in 36 quadruplets

Goal: Remove all stones

The Game (2)

Rules:– Stones have to be removed in matching pairs

– A stone has to be free when it is removed

– A stone is free when● there is no stone to the left and to the right of it, and● there is no stone on top of it

In this research, we have imperfect information: we don't know the stones we don't see.

Research Question

How well do various solving strategies handle the solving of Mahjong Solitaire positions?– In the imperfect-information game

– Using the Turtle layout

Solutions

Two solving methods:● Heuristics (1 + random researched)● Search strategies (4 researched)

Strategies & Heuristics are combined: 8 test configurations

Random Strategy

“Hey, there is a free pair. Let's pick it!”

Randomly pick one pair from all available free pairs.

MultipleFirst Strategy

“I can freely pick quadruplets – Let's pick them first.”

If there are quadruplets, pick them first.

Else, pick two stones from all triplets first.Else, if there are only pairs, pick those.

Greedy Strategy

“Free stones gives possibilities. I want maximum possibilities.”

Pick a pair such that the increase in free stones is maximal.

Obstruction-Tree Strategy (1)

“Stone X is blocking many other stones. I want to remove stone X first.”

Process:● Find stone a that blocks other stones

● Find matching stone(s) b to a (findMatches)

● If b is free, pick pair (a,b). Else, find all stones c blocking b (expandObstructors)

● Find matching stone(s) d to stones c (findMatches)

Obstruction-Tree Strategy (2)

Example: remove stone with ID 123

Heuristics

If in one of the strategies there are multiple equal choices, a heuristic is used.

2 heuristics: Random and MaxBlock

MaxBlock ≈ “# stones a stone n is blocking”● For each layer of stones, the “distance” to the

center of the row. ● MaxBlock = sum of row values for stones below

Results

Percentage of solved positions

Conclusion

To solve positions of the imperfect-information game of Mahjong, using a heuristic without using a complex search strategy is a simple yet very effective way to achieve that goal.

Questions?

Literature:

● Allis, L. Victor (1994). Searching for Solutions in Games and Artificial Intelligence. Ponsen & Looijen, Wageningen.

● Butler, Jonathan (2001). The tiles of mah jong. http://www.atdesk.com/jon/mahjong.html.

● Elonen, Jarno (2003 - 2004). There is no deterministic way to solve a mahjongg solitaire game. http://elonen.iki.fi/code/ misc-notes/no-alg-mahj-solit/.

● Emmett, Christopher James (2007). Mahjong solitaire. Technical report, The University of Manchester.

● Fregger, Brad (1999). Lucky That Way. Sunstar Publishing,U.S.

● Gimeno Fortea, Pedro (1998-2005). Mahjongg solitaire solver. http://www.formauri.es/ personal/pgimeno/mj/mjsol.html.

● Marsaglia, George (1968). Random numbers fall mainly in the planes. Proc Natl Acad Sci U S A., Vol. 61(1), pp. 25-28.

● Matsumoto, Makoto and Nishimura, Takuji (1998). Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Transactions on Modeling and Computer Simulation, Vol. 8(1), pp. 3-30.

● Petersen, Vegard Krog (2003-2005). Solitaire mahjongg - a guide to the world of the computer tile-matching, solitaire mahjongg games - history. http://home.halden.net/vkp/vkp/.

● Schaeffer, Jonathan, Burch, Neil, Bjornsson, Yngvi, Kishimoto, Akihiro, Muller, Martin, Lake, Rob, Lu, Paul, and Sutphen, Steve (2007). Checkers is solved. Science, Vol. 317.

● Shannon, Claude E. (1950). Programming a computer for playing chess. Philosophical Magazine, Vol. 41, No. 314.

Images based upon the "postmodern" tileset released with Gnome Mahjongg under the GNU General Public Licence