 onnection
games have always attracted me. I liked the simplicity and the mathematical perfection of Hex. Later, I rediscovered
the Game of Y without knowing that it had been invented already. Since that time, I became convinced that a connection
game on a hexagonal board should exist. Of course, Christian Freeling invented Havannah, which has very simple rules,
but it has not got the mathematical perfection I desired.
On 26th December, 2001, only five days before
the deadline of the Unequal Forces Game Design Competition, I was playing a game of Lur, a territorial game inspired by
Anchor and Hexgo. Suddenly I became aware of a property of hexagonal boards with hexagonal cells: If the board is
completely filled with pieces of two colors, one player must have created a line or a Y. (This property has now been
mathematically proved by Taral Guldahl Seierstad.) This would give me the mathematical perfection, and I just needed
some simple rules.
Initially I tried the rule that the first
player who creates a line or a Y wins the game. Unfortunately, this does not produce a balanced game. An attacking
player has too strong an advantage since he has five different ways to win—three lines and two Y's.
Suddenly I remembered the Unequal Forces
Game Design Competition, and that Kerry had suggested a connection game with different objectives. There it was!
One player attempts to win by creating a line, and the other by creating a Y. However something remained, as creating
lines is easier than creating Y's, and the first move advantage was not enough to balance the game. On the other hand,
draws are possible, as we can see in the diagram below.
White cannot achieve a line; Black cannot achieve a Y
The first problem was solved by adopting the rule that I call the contract. At the beginning of the game both
players place black pieces on the board, until one considers that Black has a better position and decides to play Black.
The other problem was how to avoid draws. A
game cannot be a good game if it is often drawn, and when one first invents a game one does not know how drawish it
is going to be. I decided to eliminate possibility of draws. I realized that the only way for a draw to happen is
when one player creates the opponent's objective, so I decided that if a player creates the opponent's objective he
loses. This was a lucky choice. I later realized that this rule makes the game very different strategically from Hex,
and it affords scope for new tactics. It also allows for new concepts, such as "topologically won positions," that we
will discuss below.
After Unlur won the competition, Juan Zubieta
decided to include it on the Ludoteka website so that it could be played in real time. We subsequently were able
to test the game deeply.
The first thing we observed is that Unlur over a
board with eight cells per side becomes very complex and difficult to understand, so now we prefer to play on a board
with six cells per side. We do not really know what is the best size yet, but this article will be based on the
smaller board.
The other refinement of the rules is related to
the contract. I have adopted the rule that forbids play on the border cells of the board during the contract stage.
Why? We have discovered that some Black moves on the border of the board are a liability for Black. So, if during
the contract one player felt that playing inside the border would lead to a good position for Black, he would make a
weak move by playing on a border cell, thereby extending the contract. This situation could be repeated several times,
with the contract continuing indefinitely.
Some preliminary discussions of the tactics and
strategy of this unusual game follow these rule clarifications.
- Jorge Gomez Arrausi
Other Winning Conditions
In addition to the normal winning conditions, there are a few other configurations that guarantee a win for one of the
players. In other words, when the game reaches such a position, one of the players cannot help but win, even if the
players collaborate not to let that happen. I know of three such winning configurations.
I will say that a player controls a side, when
he has a chain of stones connecting the two sides adjacent to this side. For example, White controls the right if
there is a white chain connecting the top-right side to the bottom-right side.
The three winning configurations are:
1. If White controls two opposite sides, then White wins.
2. If Black controls three non-adjacent sides, then Black wins.
3. If White and Black control one side each opposite to each other, and the two controlling groups are adjacent to
each other, then White wins.
In most cases it is probably easier to win by using
the ordinary winning method (create a line or a Y), but I think these other methods can have an impact on the game in a
given situation. In fact I have seen each of them used in games I have played or in variations thereof.
- J- Taral Guldahl Seierstad
Tactics and strategy
As a hexagonal connection game, Unlur inherits much of the theory about edge templates from Hex.
For the basics about templates we therefore refer readers to Cameron Browne's articles on Hex in
earlier issues. There are, however, certain important cases where Unlur differs from Hex. One of
the most notable ones is that a 0-connected path does not guarantee a win. If Black has a 0-connected
path creating a Y, it may be possible for White to force Black to give up a secure connection in order
to avoid creating a line. (See Diagram 1.) For White, a 0-connected path only rarely does not imply
a win, since it is very hard for Black to force White to make a Y before he makes a line.
To avoid this, Black should
try to make a 0-connected path that does not touch sides that he does not plan to use in his final Y.
White, on the other hand, should try to force Black to touch such sides; typically, it means that
White should force the black stones to touch sides belonging to both sets of three sides that make
up a Y. Clearly corners are dangerous for Black, since they belong to two adjacent sides. If a black
stone is securely connected to an edge by an edge template that covers a corner, it means that White
can force the connection to go to the corner. (See Diagram 2.)
Another important difference
between Unlur and Hex is that in Unlur all sides are friendly. This means that we have another family
of edge-templates, namely those that guarantee that a stone can be connected to at least one out of two
different sides, or to two different sides independently. (See Diagram 3.) There are too many
possibilities for a useful catalogue.
A final unique difference is
that there is a set of positions that topologically guarantees a win for one player, before
the game is completed. This is completely unlike most other connection games, where a player
can usually complete his goal if the opponent has not already achieved his. (See Diagram 4.)
The Unlur Contract
In Unlur, an ingenious variant of the pie rule is used. The players have different objectives, and
it is quite clear from the rules that White has a distinct advantage on an empty board. So, the
players even out the game by alternately placing black stones on the board. We will refer to the
positioning of black stones when a player passes as the contract. Thus the goal for the players in
the opening should be to create an even contract. Because the opening generally consists of several
moves, it is possible to make the game very evenly balanced. Also, even on the smaller Unlur board
with six hexagons on each side, there are 11 opening moves, and around 50 possibilities for each move
thereafter. So, for a four-stone opening, there are already 2 million possibilities. The number of
openings that are actually played may be smaller, since players may repeat certain patterns, but in
general this makes opening analysis very difficult. We will look at two types of contract.
The block contract
When the position gets so close that both players may consider swapping, you do not want your next
black play to have too large an effect on the position. The moves that are least influential to the
position are usually those that are adjacent to one, two or three stones already placed. Thus, when
the initial plays are near the sides, it is quite reasonable to expect a contract with one or two
blocks of stones grouped together. Playing like this makes sense for both players because it will
never force them to play a move that will greatly affect the position.
A contract of this type can
thus be played close to one or two (often opposite) sides, effectively leaving only two possibilities
for White to make his line. (See Diagram 5.)
In this type of contract,
it is usually reasonable to expect that White will play the first move of the game on the center
point, so this is what the potential Black player has to defend against.
The center contract
The first or second move for Black is played in the center. Because the center point lies on diagonals
linking up all the sides, it is a very powerful point. So, a typical contract can be expected to last
for only two or three moves. Similarly, if the first moves are near the center, again the contract can
be expected to be short, perhaps three or four moves. (See Diagram 1.)
Generally, looking at reasonable
center and side openings, we can interpolate and get an impression of what is fair and what is not.
This gives a good idea of how long a contract should be, but the particulars are more difficult,
because of the specific relations between stones. Generally, fewer stones are needed if they are
spread out, and fewer stones are needed if they are close to the center. As for tactics, you should
always expect White to start on the center point if it is not already occupied. Also, you should be
ready to answer an attachment by White on a point adjacent to the center point if you take Black with
a stone at the center point.
Also, the number of possible
openings is quite astonishing. Only three randomly distributed stones allows for around 200,000
possible openings. However, it is only near the center that three stones is sufficient, so the
total number of reasonable contracts is likely to be much larger.
Diagram 1 — The "arrow" opening
The diagram shows the "obvious continuation" from the opening with the unmarked black stones.
It is a natural choice for both Black and White from a center-point opening. If Black's only goal
were to connect three alternating sides, he could simply play A, creating a 0-connected path between
the left, top-right, and bottom-right sides. However, as noted above, a 0-connected path cannot
guarantee a win. Black would not be allowed to connect to the bottom-right side until he has connected
the top-right side, because A is also on the top-left side. In this particular case, it appears hopeless
for Black, since White in effect gets five free moves at the bottom while Black connects to the top right.
The continuation from the
diagram would be Black A, then White B!. White C may also work here, but in general B is a good
example of Unlur technique. If Black has more material on the right side, a move such as D could
be an effective replacement for Black A.
Diagram 2
The connection template above left guarantees a connection to the bottom-left side. In Hex, this
would always be sufficient information, but in Unlur it also makes a difference that there is a corner
in the template at C. After the sequence 1-2-3, shown on the right, Black needs to play in the corner
to complete the template. As seen in Diagram 1, this may be a liability for Black.
Diagram 3
In the template on the left, White is guaranteed to connect to either the bottom-left or bottom-right
side. In the template on the right (including A, B, and C), White is guaranteed to connect to both the
bottom-left and bottom-right sides. Without one of the points A, B, and C, the connection templates to
the two sides are overlapping, thus White is not guaranteed to connect to both. (Black plays at D, with
a follow-up at either E or F.)
Diagram 4
The above configuration of white stones guarantees a win for White, no matter where the Black stones
are placed. Notice first that Black can never form a Y, since a white group separates the top-left
side from the right side and the bottom-left side, and another white group separates the bottom-right
side from the left side and top-right side. Furthermore, White can never be forced to form a Y before
he forms a line, since whenever the two white groups are connected he also achieves his winning condition.
So, the game will end either with White forming a line, and winning, or Black forming a line in the center
area, which is also a win for White.
Diagram 5 — A fair contact?
If Black gets all the stones at A, B, C, and D, he has almost eliminated the possibility of a White top-left
to bottom-right win, so, in effect, his goal reduced to connecting his two large groups. Surely, this goal seems
much easier than White's goal. However, with only the black stones as played so far, it is hard for Black after
White G. The answer is probably that Black needs one or two of the points marked A-F, in addition to the six
stones on the board.
- Carl Johan Ragnarsson and Taral Guldahl Seierstad
Sample Game
The following game was played online between Elvire Scheibling and Carl Johan Ragnarsson. Carl Johan passed on
the 10th move, taking Black.
The contact
Considering the general ideas about a "block" contract discussed above, the contract looks favorable for Black.
However, any opening where Black has no stones near the center point will be difficult to analyze. On a small
board, the 7, 8, and 9 stones have nice potential toward the center.
Position up to White 20
The beginning of the game has nothing special. All the moves here are quite natural. Even more than in Hex, it is
important, especially for White, to be connected in the center, to avoid Black breaking through here. The center is
the focal point of the game, and there is not as much room for keeping your options open.
In this type of game Black has to rely on
sneaking around on the sides, trying his best to avoid the corners. White was already thinking that she was in
a bad position, giving Black control of so many sides. She thought it was now urgent to take control of the
lower side. So White made the unexpected move 20. Maybe it is a mistake and she should have pushed Black to
the corner, playing first the A-B exchange. However, the situation is not easy since Black is not forced to
answer at B immediately. In any case, Black was happy to avoid the corner and played A himself.
Position after Black 53
Here, White is happy she did not force Black in the corner A in the beginning of the game. With a Black stone at
A, Black could play B and win, achieving both the Black and White aims in a move.
But, Black can win anyway! Even though Black
needs an extra tempo to play at one of the points marked C, Black has enough time to connect to the left,
by playing at D or E. But, if Black had only one more disconnected bridge on the right side, White could have
sneaked by on the bottom left with both moves E and B. This shows clearly how important the forcing moves can
be for the end result of the game, not only because of their placement, but also because of the sides a player
connects to, and the tempos it takes him to connect.
Final position
Black wins, achieving the aims of both Black and White.
- Carl Johan Ragnarsson
There is now a mailing list for discussion of Unlur
at http://groups.yahoo.com/group/unlur, and one may play
online at the Ludoteka website at http://www.ludoteka.com/. Jorge has a
webpage for Unlur at
http://www.abstractboardgames.com/unlur, and he hopes to be
running the first Unlur tournament by the
beginning of 2003. If you wish to get some practice in the meantime, an Unlur board is given on the back
cover of this magazine. It is the right size to play with small coins such as pennies and dimes, but I
expect that most game players would be able to find other suitable counters. - Ed.
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