```Newsgroups:     rec.games.chess,rec.games.go,rec.games.abstract
From:           gasser@inf.ethz.ch (Ralph Gasser)
Subject:        Nine Men's Morris is a DRAW
Organization:   Dept. Informatik, Swiss Federal Institute of Technology (ETH), Zurich, CH
Date:           Tue, 23 Nov 1993 12:42:10 GMT
```

```The game of Nine Men's Morris (NMM) has been solved. The game is a draw.

As there are different variations of NMM being played, I will first specify
the rules I used, then give a short summary of how the solution was computed.

RULES
-----

Nine Men's Morris is played on a board with 24 points where stones may be
placed.

+--------+--------+
|        |        |
|  +-----+-----+  |
|  |     |     |  |
|  |  +--+--+  |  |
|  |  |     |  |  |
+--+--+     +--+--+
|  |  |     |  |  |
|  |  +--+--+  |  |
|  |     |     |  |
|  +-----+-----+  |
|        |        |
+--------+--------+

Both players start with nine stones each.

The game goes through three phases
-  opening phase
Players alternately place stones on an empty point.
-  midgame phase
After all stones are placed, players slide stones to
any adjacent vacant point.
-  endgame phase
When a player has only three stones left, she may
jump a stone to any vacant point.

When closing a mill (three-in-a-row), any opponent's stone
which is not part of a mill may be removed. If all the
opponent's stones are part of mills, any stone may be removed.

Closing two mills simultaneously (opening phase) only allows
one of the opponent's stones to be removed.

-  The first player who has less than three stones loses.
-  The first player who cannot make a legal move loses.

HOW IT WAS SOLVED
-----------------

Retrograde Analysis (well known from Chess) was used to compute
databases for all mid- and endgame positions (about 10 billion
different positions). These positions were split into 28
separate databases characterized by the number of stones on the
board i.e. all the 3 White stone against 3 Black stone
positions, the 4-3, 4-4 ... up to 9-9 positions.

An 18 ply alpha-beta search for the opening phase then found
the value of the initial position (the empty board). Only the
9-9, 9-8 and 8-8 databases were needed to establish that the
game is a draw.

Ralph Gasser (gasser@inf.ethz.ch)
Informatik, ETH
CH-8092 Zurich
Switzerland
```

```Newsgroups:     rec.games.abstract
From:           gasser@inf.ethz.ch (Ralph Gasser)
Subject:        Nine Men's Morris: More about the draw
Organization:   Dept. Informatik, Swiss Federal Institute of Technology (ETH), Zurich, CH
Date:           Wed, 24 Nov 1993 10:31:38 GMT
```

```I would like to answer some questions that came up regarding
the solution of Nine Men's Morris.

- How can a draw be reached?

A draw is reached through cyclical play, whereby neither
player can (or will) close a mill. Note that since the proof
was obtained using only the 9-9, 9-8 and 8-8 databases, neither
player need close more than one mill to reach a draw ,thereby
avoiding a loss with less than three stones.

- What is an optimal opening strategy (for humans)?

I must admit that I have not examined the opening moves
for any easily describable patterns.
However, the examination of mid- and endgame databases
has repeatedly shown optimal play to be beyond human
comprehension. Therefore, I very much doubt that any such
simple-to-describe strategy exists.

- Is a paper describing these results available?

I am now working on such a paper. I will send a copy to those
of you who have requested one as soon as it is completed.

There is however a paper describing some initial results
of the retrograde analysis phase of the project:

Gasser, R., "Applying Retrograde Analysis to Nine Men's
Morris", Heuristic Programming in Artificial
Intelligence 2: the Second Computer Olympiad, Levy, D.N.L.
and Beal, D.F. (Eds.), pp. 161-173, Ellis Horwood, London.

- Is the program available?

Making the program available is something of a problem
because of the databases it uses. In total the databases
amount to a few hundred MBytes.

There is of course a program available which plays without
any database (or with a few small db's). While this program
cannot claim to be invincible, it plays quite strongly, having
won a match against the British Champion.

This program was developed under the Smart Game Board and
therefore only runs on Macintosh computers. The program
is shareware and depending on the number of requests, I will
either make it available via ftp or snail mail.

Ralph Gasser (gasser@inf.ethz.ch)
Informatik, ETH
CH-8092 Zurich
Switzerland
```

```Newsgroups:     rec.games.abstract
From:           colley@qucis.queensu.ca (Paul Colley)
Subject:        Re: Nine Men's Morris: More about the draw
Organization:   Computing & Information Science, Queen's University at Kingston
Date:           Wed, 24 Nov 1993 15:41:54 GMT
```

```In article <1993Nov24.103138.21488@neptune.inf.ethz.ch>
gasser@inf.ethz.ch (Ralph Gasser) writes:

>A draw is reached through cyclical play, whereby neither
>player can (or will) close a mill. Note that since the proof
>was obtained using only the 9-9, 9-8 and 8-8 databases, neither
>player need close more than one mill to reach a draw ,thereby
>avoiding a loss with less than three stones.

This appears to be a mistake.  Er, let me rephrase that.  "The evidence
supplied does not prove your conclusion."

In building the 9-9 and 9-8 databases, you would have used the smaller
data bases.  Retrograde analysis starts from the smallest, and works
up.

Say some position in the 8-7 database is a draw.  Then in the 8-8
database, a position where the player may close a mill and reduce to
the drawn 8-7 position gets marked as a draw (assuming there's no
winning move).

You don't need the 8-7 database in doing the opening search, because
you don't need to know why the particular 8-8 position is drawn.  This
is NOT the same thing as

>neither player need close more than one mill to reach a draw

The fact that you didn't need the smaller databases in the opening
search just means that no player need close more than one mill *DURING
THE OPENING* to reach a draw.  It doesn't mean the player doesn't need
to close a mill during the middle or end game.

The correct statement, based on the evidence you give, is "neither
player need close more than one mill, and only during the opening, to
reach a book draw."  Well, now that the initial position of NMM is
solved, we could even say "neither player needs to make a move to reach
a book draw."  Of course, if your opponent doesn't agree, you still
have to play out the book draw, and that (may) require the smaller
databases.  Or may not require the smaller databases.  I have nothing
to contradict your statement.  I'm just asking for more support.

- Paul Colley
University:  colley@qucis.queensu.ca
Home:        pacolley@ember.uucp   watmath!ember!pacolley  +1 613 545 3807
"Anyone who attempts to generate random numbers by deterministic means is,
of course, living in a state of sin."  - John von Neumann
```