Chessgammon: A Playable Hybrid of Chess and Backgammon
Darse Billings
Copyright (C) Darse Billings, August 1994.
While playing around with idea of combining the nice properties of
backgammon and chess, I hit upon the following highly playable game,
which I call "chessgammon" for obvious reasons.
The existing forms of dice chess, such as Modern Chaturanga, are
rather unsatisfying. By applying the dice rules and doubling cube
from backgammon to the game of chess, an interesting and exciting
hybrid is formed. The result is similar to backgammon in tempo, but
is (arguably) richer in strategy, because the goal is to capture the
enemy King instead of winning a race.
Chessgammon should appeal to anyone who enjoys backgammon and knows
the rules of chess. Chessgammon could also become highly popular
among serious chess players who are always looking for fast and fun
alternatives to their favourite game.
In chessgammon, each value of the dice corresponds to a type of piece,
as follows:
roll piece
---- -----
1 Pawn
2 Knight
3 Bishop
4 Rook
5 Queen
6 King
These values are easy to remember because they correspond to the strengths
of the pieces. The pips on the dice can also serve as a mnemonic: one dot
for a pawn, three diagonal dots for a bishop, a square pattern for the
rook, five dots like the crown of a Queen, and so on.
On each turn, a player rolls two dice and moves two men (if possible) as
indicated by those values. As in backgammon, the player must use both
rolls if possible to do so. For example, on an opening roll of 6-1, the
player must move a pawn and the King, so the only pawns that may be
moved are the d, e, and f pawns, and there are exactly six legal moves
to choose from.
If a player has no piece of the type indicated by a die roll, or if all
pieces of that type have no legal move, then that die roll cannot be
used (but the other die roll must still be used if possible). It is not
uncommon for all or part of a roll to be unusable.
Rolling doubles allows a player to move two pieces of that type, or the
same piece twice (which can be a very powerful roll). At the beginning
of the game, each player rolls one die, and the player rolling the
higher value moves first, using those two die rolls. Ties are re-rolled,
so no game may begin with doubles.
The rules for piece movement are exactly as in chess, except that it is
legal to move into check and to ignore checks. The object is to capture
the enemy King, which ends the game. Pawns may promote to any piece
other than a King. All other rules, including castling and en passant
captures, apply in the obvious way.
As most serious backgammon players know, the use of the doubling cube
adds a lot to the strategy and skill element of the game. Use of the
doubling cube is strongly recommended for chessgammon.
One difference between the two games is that backgammon has the
possibility of winning a double game (gammon). This sometimes
discourages the use of the cube in very strong positions, making for
longer games. Chessgammon has no such deterrent to doubling, and
combined with the sudden-death nature of victory, the average game may
be considerably shorter. For those who enjoy the possibility of
gammons, a double game can be awarded for capturing all sixteen of the
opponents pieces (ie. baring the enemy King and then capturing it).
The following is a sample game which should clarify how the game is
played, and illustrate some of the strategic aspects of the game.
First I will give the raw game score, then I will give an annotated
version.
1. 6 4 --- 2. 5 3 --- 3. 2 1 Nc3 e3 4. 4 1 h5 Rh6
5. 6 1 Ke2 b4 6. 3 3 --- 7. 6 4 Ke1 Rb1 8. 6 5 ---
9. 5 2 Qf3 Nb5 10. 3 2 Na6 11. 4 2 Rb3 Nc7 12. 6 6 ---
13. doubles 14. accepts 15. 3 1 Ba6 h4 16. 3 1 e5 Bd6
17. 6 4 Kf1 Rh3 18. 5 3 Qc7 Be7 19. 5 1 Qf7 d3 20. 4 1 Rf6 ba6
21. 5 3 Qe8# White wins 2 points
White rolls 6
Black rolls 4
White starts the game.
1. 6 4 ---
2. 5 3 ---
3. 2 1 Nc3 e3
White chooses e3 instead of e4 to avoid opening lines against her own
King. King safety is a key strategic element in chessgammon, just as
it is in chess. Indeed, many chess principles carry over to the
probabilistic game.
4. 4 1 h5 Rh6
As in backgammon, Black must use both die rolls if possible to do so,
and this is clearly the best of the six legal moves.
5. 6 1 Ke2 b4
White loses the right to castle and is forced to a more vulnerable
square, both of which are common occurences in chessgammon. White
selects b4 to develop the queenside bishop, again not wanting to expose
the King by moving the d or f pawns.
6. 3 3 ---
7. 6 4 Ke1 Rb1
8. 6 5 ---
9. 5 2 Qf3 Nb5
White has a large lead in development resulting from Black's three
useless rolls, and now makes two serious threats against the King. A
one-on-one match-up between two pieces, each having an 11/36 chance of
capturing the other, gives the first player to act about a 59% chance
of succeeding, assuming no other pieces intervene. Thus, Nxc7 would
give White about a 40% chance of winning immediately, with relatively
little to lose, and is therefore a dangerous threat to Black. In
addition, White threatens to win immediately with rolls of 2 2 or 5 5.
10. 3 2 Na6
Defending the c7 square is clearly the highest priority for the Knight.
11. 4 2 Rb3 Nxc7 ?!
White captures on c7 anyway, attacking the King but leaving the knight
"en prise" to two pieces. Black has a 20/36 chance of capturing, and
White will have an 11/36 chance should Black fail. This gives White
about a 20% chance of winning instantly. If the knight is lost, White
estimates her chances of winning at around 50%, since she still has a
lead in development which compensates for the lost material. Thus her
equity for the game is about 0.60, and it looks like a good move.
However, Nd6 was a clearly superior move, since Black could not then
use fives, and because he already has good uses for fours, such as Rf6.
Note also that this move would not be as good if White did not have
another knight on the board, since then she would be risking the use
of all rolled twos. Keeping at least one piece of each type on the
board is another important strategic principle.
12. 6 6 ---
13. doubles
With Black's miss, White now has about a 44% chance of capturing the
King with the knight, increasing the equity to around 0.72. Equity
values in chessgammon are often highly volatile, since you usually
have many "market losers" in threatening to win the game instantly...
14. accepts
15. 3 1 Bxa6 h4
Not c4 or b5, because White wants to keep the c4 square open for the bishop.
16. 3 1 e5 Bd6
17. 6 4 Kf1 Rh3
18. 5 3 Qxc7 Be7
Probably better was Bxc7 Qf6. Both Queens are defended three times, so
there is about a 90% chance of recapturing, making the risk of losing
the Queen outright around 6%. In return, Black relieves some of the
pressure on f7, and threatens to take the bishop on a6 with check.
19. 5 1 d3 Qxf7 !?
White defends against 5 5 Qc4 Qf1# with d3, and plays the very sharp
move Qxf7. Black has 13 rolls to capture the Queen (including 4 4 and
5 5), and failing will give White an 11/36 chance to win. Ignoring
secondary defences (which are not great, but not insignificant), this
gives White about a 35% chance of winning immediately. If the King
captures the Queen, it will still be exposed on f7, and the h3 rook or
the a6 bishop may finish the job. White estimates she has a 40% chance
of winning without the Queen (perhaps optimistic), giving a cubeless
equity of about 0.61. However, this does not account for Black's
potential double in the event that White fails. This, combined with
the loss of all rolled fives, probably makes a less risky move
preferable.
20. 4 1 Rf6 ba6
21. 5 3 Qe8# White wins 2 points
Rack'em up and play again! Despite this lengthy analysis, this game
would have probably taken less than five minutes to play.
There is an easy method for computing the probabilities commonly
encountered in chessgammon. For example, suppose two Kings are
adjacent to each other, with no other pieces on the board (an unlikely
occurence!). Each King has an 11/36 chance of capturing, so the
probability that the first player to act will win is:
(11/36) + (11/36)(25/36)(25/36) + (11/36)[(25/36)^4] + ...
while the second player's chances are:
(11/36)(25/36) + (11/36)[(25/36)^3] + (11/36)[(25/36)^5] + ...
But notice that each pair of terms in this infinite sequence have the
same ratio (because (25/36)^2 is a common factor). So to compute the
exact probability, we need only compute the ratio of the first two terms:
Player A first move: (11/36) = 0.3056
Player B first move: (11/36)(25/36) = 0.2122
Prob [Player A wins] = 0.3056 / (0.3056 + 0.2122) = 0.5901 = 1296/2196
In general, if Pa and Pb are the probabilities of player A or B
hitting, respectively, then the probability that player A wins is:
Pa / (Pa + Pb(1 - Pa))
Computing actual probabilities during a game is not essential. As
with backgammon, accurate estimates become second nature with
experience, and familiar patterns become recognizable.
Cheers, - Darse.
--
Go is better than Chess. Poker is more lucrative. Sex is more fun.
Darse Billings, 7 kyu; 2065 CFC; meaningless IRC sb/hand ratios:
(rayzor on IRC) Hold'em +0.22 ; HiLo Omaha +0.56
Erik Robert Wilson
proposed would make pawns relatively worthless except as blocking
pieces. So here is a proposed alternative dice roll scheme:
1 any P
2 N or NP (b or g pawn)
3 B or BP (c or f pawn)
4 R or RP (a or h pawn)
5 Q or QP (d pawn)
6 K or KP (e pawn)
I haven't actually played out a game or anything that involved, but it
may make it more possible to get a recognizable "pawn structure".
--
| \ o IO | "If we do not find | Erik Wilson |
|-----| o GANYMEDE |anything pleasant, at | University of Illinois |
| O | o EUROPA |least we will find | (217) 359-7547 |
|----/ o CALLISTO |something new" -Voltaire| eri...@uiuc.edu |
Darse Billings
>As someone once said "pawns are the soul of chess". Chessgamon as
>proposed would make pawns relatively worthless except as blocking
>pieces. So here is a proposed alternative dice roll scheme:
> 1 any P
> 2 N or NP (b or g pawn)
> 3 B or BP (c or f pawn)
> 4 R or RP (a or h pawn)
> 5 Q or QP (d pawn)
> 6 K or KP (e pawn)
>I haven't actually played out a game or anything that involved, but it
>may make it more possible to get a recognizable "pawn structure".
I've had a chance to try this alternate scheme, and I'm not sure I like
it, personally. Pawns already have about the same relative worth in
chessgammon as in chess (whereas Queens, for example, are probably worth
less in chessgammon).
However, I think using this as an option for each player's *first* roll
may improve the game by making it easier to develop pieces.
I welcome more suggestions. Keep in mind that chessgammon is just the
preliminary result of a straight-forward combination of chess and
backgammon rules, and many improvements are undoubtedly possible. I
strongly encourage people to try new ideas (hopefully only minor changes
that do not affect the spirit of the game), and please let us know of
experiments you thought were successful.