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draughts dissected

At the age of six I had mastered Fox & Geese. My geese were invincible and I was far from reluctant to prove it. In sheer desperation my father taught me to play Draughts. It was my entrance to the world of abstract games.
Now I must betray it and explain why a good game has become a bad sport.

I'm sorry to say that this is not, as many would have it, a matter of opinion. For those who would like to dismiss these findings as ridiculous: if by some twist of fate Dameo had been the dominant draughts form of the 20th century and onwards, you would have naturally accepted that as a child. If someone had come up with 10x10 Draughts as it is played now, you might well have ridiculed the very game you're defending now - and this time maybe rightly so. So don't kid yourself.

Draughts is so globally recognized that there's hardly any culture that hasn't accommodated it in one form or another. There are dozens of variants. We'll look at the traditionals Checkers, Shaski, Draughts, Turkish Draughts, Armenian Draughts and Spanish Draughts and compare them with two modern games, Hexdame and Dameo. HexDame differs in only one respect from the international game: it is a literal translation of Draughts to the hexgrid. Dameo is a different story altogether.
We'll compare the games' basics because these have a deep impact on their behaviour.



Straight versus oblique
An initial position suggests forward movement so it implicitly leads to the concept of promotion. A straight move has one direction forward, two sideways and one backward, while an oblique move has two forward and two backward.

straightobliqueI will call this "the straight plane" and the "oblique plane".
Before you think there's any fundamental difference, please consider the 'transformation' depicted below:

schuinrecht

So the "oblique plane" is the same straight plane, only rotated by 450.

Since men may not move backward, the oblique move would appear to be more in the game's spirit - progress being forced rather than optional - and most variants indeed use only the light or the dark squares.
That choice sacrifices three advantages of the straight plane:

  1. The board aligns with the direction of capture, giving combinations more scope
  2. Only two instead of four long-range kings are needed to trap a lone one
  3. Algebraic notation

To play devil's advocate: the third one isn't quite true.
But the first two are so basic that it's hard to imagine any serious player would choose to ignore them. 9x9 is an unusual but average board size. For the sake of argument let capture be in four directions.

schuinrecht
  • On the left the playing area is 81 squares, 40 of which allow a capture by the white man
  • On the right the playing area is 41 squares, 16 of which allow a capture by the white man

Here's the first ratio difference:
straight scores 50%, oblique 40%.

During a multiple capture in Draughts, a man may visit a square more than once, but not jump a man more than once, and the move must be completed before the captured men are taken off the board. Under these conditions, on the left the man can capure 33 of the 40 men, the man on the right 10 of 16.
Here's the second ratio difference: straight scores 82%, oblique 62%.

The differences are the consequence of alignment: in the straight plane a man can capture along a side and through a corner, in the oblique plane a man bounces off the side and gets stuck in a corner.

Multiplying each plane's percentages gives an indication of the overall difference in terms of scope for combinations.

  • Straight scores 50x82=4100
  • Oblique scores 40x62=2480

Apart from the above, there's no fundamental difference in the nature of capture in either plane. The straight plane is represented by Turkish and Armenian Draughts and the modern variants Croda and Dameo, the oblique plane by the Checkers, Shaski and Draughts.

Conclusion
The straight plane provides about 65% more scope for combinations and halves the number of kings needed to capture a lone one.

Since tactics provide the spice and beauty of draughts games, one might reasonably argue that more is better. Of course alignment is not the only factor, or there would be little difference between Checkers and Draughts. The tactical scope of the latter is based on backward capture and a long-range king and gives no reason for complaints. Checker's tactical scope pales in comparison.

The consequences for the 'kings versus a lone king' endgame however are very dramatic. Draughts needs four kings to trap a lone one and is currently dying from the consequences. The difference in material needed to win is far too high for the current top level players. The same holds for Shaski and Spanish Draughts.
Checkers ironically doesn't suffer from it's oblique orientation because short range kings are not affected. This game died of exhaustion, not of an intrinsic flaw.
All 'square' games in this comparative investigation need only two kings to capture a lone one - as uncompromising as it should be.



Mobility
It would appear that in draughts type games, greater freedom of movement grants a wider choice of action. This in itself does not necessarily make a game any more interesting, or we might be playing Draughts on a 20x20 board with 90 men per side. Since this looks just as advertising as playing on a 3x3 board with 3 men per side, we may safely conclude to the existence of an optimum boardsize for every draughts variant.
How large exactly a board should be is arguable: Spanish, International and Canadian all use the same rules, but on three different sizes. This does affect mobility because the ratio of all squares to squares on the edge increases (32/14 - 50/18 - 72/22). But creating greater freedom of choice by enlarging the playing area doesn't necessarily make the choices any more interesting.
In my opinion the games presented here all have a size that allows about all they have to offer to be made explicit with the minimum material required to do so. The size of the playing areas therefore will only be a criterion for mobility in relation to other criteria such as the number of men, the size of the promotion area and the direction and scope of movement of pieces. By the same argument Shaski will be represented by Draughts - its slightly higher mobility, due to absence of compulsory majority capture and a 'flying king', is counteracted to about the same degree by its smaller boardsize.

Checkers
Initial moves Checkers
number of vacant choice of directions in which a man may:
squares men per side area opening moves move capture
32 12 25% 7 2 2
forced promotion average number of range directions in which a king may:
progress squares moves to promotion king move capture
yes 4 6 short 4 4

This obviously doesn't look too impressive in terms of mobility. Marion Tinsley, the greatest Checkers player ever, described playing the game as 'looking into a bottomless pit'. It certainly may have looked that way, but among the top players a draw has become a very common result. A game is exhausted if nothing new can be found anymore and on top of that Chinook has the whole game wrapped up.


Draughts
Initial moves Draughts
number of vacant choice of directions in which a man may:
squares men per side area opening moves move capture
50 20 20% 9 2 4
forced promotion average number of range directions in which a king may:
progress squares moves to promotion king move capture
yes 5 7.5 long 4 4

Here piece mobility is higher due to backward capture and a long range king, but with only 20% of the board vacant things are fairly crammed in the opening. Promotion looks bad: only 5 squares and the men at an average distance of 7.5 moves.


Turkish
Initial moves Turkish
number of vacant choice of directions in which a man may:
squares men per side area opening moves move capture
64 16 50% 8 (4) 3 3
forced promotion average number of range directions in which a king may:
progress squares moves to promotion king move capture
no 8 5.5 long 4 4

The number of opening moves from here on is deceiving in that all next games have left-right symmetry in their opening array which effectively halves the number. However, symmetry is soon ends: in the square games, black on his first move already faces a non-symmetric situation.
Turkish Checkers' vacancy percentage is a bit flattered as an indicator of initial mobility: a player's own back rank hardly counts as part of his playing area.
Initial mobility looks somewhat meager because all sideways movement is blocked.
The game attracts some attention in the Draughts world in their quest to prove that Draughts isn't dying under their very eyes. They envy the two kings versus one endgame.


Armenian
Initial moves Armenian Draughts
number of vacant choice of directions in which a man may:
squares men per side area opening moves move capture
64 16 50% 22 (11) 5 3
forced promotion average number of range directions in which a king may:
progress squares moves to promotion king move capture
no 8 5.5 long 8 4

Five directions of movement, three of capture and a supermobile king. This game offers flexibility and variety. The ratio in strength of a piece and a king may seem about the same as in Turkish draughts, but the king despite its increased mobility still can't capture a lone men.


Dameo
Initial moves Dameo
number of vacant choice of directions in which a man may:
cells men per side area opening moves move capture
64 18 44% 52 (26) 3 4
forced promotion average number of range directions in which a king may:
progress cells moves to promotion king move capture
yes 8 6.2 long 8 4

All rules and conventions concerning capture are the same as in Draughts, but applied to the straight plane. Movement however is linear: alingned pieces support rather than block one another. Though white's initial options are halved by left-right symmetry, the ones shown, black on his first move has 52 opening choices.
Piece mobility in general is extremely high: none of the other games offer such a degree of flexibility in tempo and pace. At the same time it is effectively restricted by forced progress and an abundance of tactical dangers.


Hexdame
Initial moves Hexdame
number of vacant choice of directions in which a man may:
cells men per side area opening moves move capture
61 16 48% 15 (8) 3 6
forced promotion average number of range directions in which a king may:
progress cells moves to promotion king move capture
yes 9 5.9 long 6 6

The 3-6 and 6-6 piece mobility is very high and so is the vacancy ratio, but here too mobility is effectively constrained by forced progress and tactical dangers.
A promotion area of 9 cells, pieces less than six moves away on average, no 1-on-1 opposition and an exeptionally strong king - this is very much a king's game.

Conclusion

Checkers
May we excuse Checkers? The game isn't designed for great mobility. It's the essential draughts type game and has great strategic depth, but its tactics and their scope cannot compete with any of the others. It is very much a game of a time gone by. Chinook has the game wrapped up and as a sport it's as dead as the dodo.

Draughts
Draughts obviously used to do better than Checkers, but things have changed. The game is fairly crammed in the opening and not very advertising in terms of room for promotion. It suffers most under its oblique plane: because it restricts its scope for tactics and increases the number of kings needed to capture a lone one to four.

Turkish and Armenian
Armenian tops the classix in terms of mobility: it superimposes the moves of men in the staight and the oblique plane. If the game is designed too improve on Turkish however, the intention may be better than the implementation. Despite its mobility it doesn't appear to operate more effectively. The need for such an archaic rule as declaring king versus man a win doesn't speak in favor of these games' favor.

Dameo
Dameo's linear movement is new to the game, yet the underlying logic is simple. It obviously accellerates the mechanism and offers a wider choice of pace and tempo. This might be an ideal game to replace Draughts as a sport weapon, but most draughts players shudder at the suggestion that anything might be wrong with their beloved game. Even after a century of widely discussed drawishness at top level, and countless suggested remedies.

Hexdame
In full recognition of the fact that a hexgrid may inspire some players to throw up, we nevertheless find a better scoresheet here than in the square game.



One-on-one opposition
We're considering basics, so let's put two opposing men on the boards:

1-on-1 Turkish1-on-1 Draughts1-on-1 Hexdame

  • The straight plane has no forced progress and has one-sided opposition: the player to move cannot get out of it, while his opponent can either keep opposition or move forward.
  • The oblique plane has forced progress and absolute opposition: the player to move must sacrifice the man.
  • The hexagonal plane has forced progress too, but has no opposition whatsoever: the player to move cannot be stopped.

The diagrams above represent the situation in Turkish Checkers (left), Anglo-American Checkers, Shaski and Draughts (middle) and HexDame respectively.
To enlarge the field, we'll show one-on-one in games that employ both planes: Dameo and Armenian.

1-on-1 Dameo1-on-1 Armenian
  • Dameo has forced progress, but no opposition: the player to move cannot be stopped because all capture is straight only.
  • Armenian Draughts has no forced progress but no opposition either: the player to move cannot be stopped because all capture is straight only.

Conclusion
The existence or non-existence of 1-on-1 opposition is not a criterion for quality but an indication of character. Games that have it, naturally offer less freedom of movement and more emphasis on blockade as a means to win. Games that don't, are not devoid of opposition in a wider sense but they tend to rely more on breakthrough and promotion and usually require weak men and strong kings to prevent drawishness.
Here we can label a quality aspect: a lone king should at least be able to trap a lone man.



Man versus king
Obviously the outcome of this endgame depends - if not on opposition - on position. Its clear that conditions for a man on his own back rank differ from those of a man one step away from promotion. Here the focus should not only be on a man and a king in direct opposition, but more generally on the conditions - if any - under which the king can win. Lets start with the simplest structured games.

Checkers and Turkish Checkers
opposition Checkersopposition Turkish

  • Checkers
    is the essential game in the oblique plane: it takes the simplest form of forward movement and capture and rewards promotion with the same options backward. As a result having 'the move' is everything. Having the move means being able to take opposition: in the diagram, the player to move does not have the move!
    The only escape is a tric-trac corner. To put it another way: if in the above diagram the pieces were in a similar position on the light colored squares, the king would even lose (!) if he were to move. As it is he can retreat to the back rank and come up the side if black takes opposition. So a lone king in Checkers isn't very strong, and if unable to reach a tric-trac corner, it may even lose against a single man. Its strength lies in teamwork.
     
  • Turkish Checkers
    is almost the essential game in the straight plane: it takes the simplest form of non-backward movement and capture and rewards promotion with the same options backward ... and a bonus. A very understandable bonus because a promotion from 3 to 4 directions of movement and capture is a rather meager reward. So the king became long-range, moving as the rook in Chess. Good move!
    Despite this it cannot win against a single man. This was perceived as a problem. A wise man was consulted and he came up with a brilliant solution: let's declare it a win! Bad move.


Draughts and Hexdame
opposition Draughtsopposition hexdame

  • Draughts
    In direct opposition a man loses if he must move. Having the move as in Checkers plays less of a role, because in the majority of cases, a king can turn the tables with a tempo move. The tric-trac corner is more than a doghole, it's an underdog hole and the graveyard of many a man. That's in fact where the above situation would end if it were white's turn.
  • Hexdame
    In HexDame the king is stronger: not only can it trap a man in the above manner, it can do so till the very last moment because unlike Draughts it can cover the back rank while standing on it. In the bottom corner a white king covers both sides of the back rank up to (but excluding) the corners. If a man comes down the side, the king can trap it by taking the corner. In a race for promotion, corners are often the best option.


Dameo and Armenian
opposition ad1opposition ad2

  • Dameo
    Dameo has forced forward progress, so obviously a man cannot move onto a rank covered by a king other than along the side. This eventually boils down to the situations depicted above.
    Note that if it's white's turn, the situation on the left is a draw. White therefore should not take the corner while black is still on the third rank.
  • Armenian
    In Armenian the situation on the left, white to move, is also a draw, but with black to move both situations are a draw rather than a win. Black can only promote in a corner and if white prevents it in the above manner, black simply moves sideways. This man may be too strong for the game's good.




King versus man+king

Checkers
If two kings can trap a lone one, as in Checkers, in Turkish, in Armenian and in Dameo, then the key question concerning the above endgame, is whether it can be converted to a 2x1 kings endgame.

King v. man+king checkers
diagram 1
It will be clear from diagram 1 that black - to move - has the choice between losing and reaching a draw, so saying that the endgame is unconditionally won is stretching it a bit.
On the other hand, you might have trouble working your way into such a predicament, having the benefit of two pieces for 'keeping the move'. Usually, it would seem, the conversion and thus the win can be secured.
If in doubt, consult your local Checkers site - I'd hardly be breaking new ground here.


Turkish
The same holds for Turkish Checkers. A man can move sideways, and can move onto a rank covered by the opponent's king along the side columns. The king therefore must oppose the man to have a chance. It would seem that - black to move - the situation in diagram 2 is one white might want to avoid, and the situation in diagram 3 one black might want to avoid.
So lets assume for argument's sake that it's white to move in both diagrams.

King v. man+king turkish
diagram 2
King v. man+king turkish
diagram 3


Diagram 2
White moves say dc1 and black can progress with 1... d54: white cannot move back to d1 but he can block the man with 2.c13.
Black moves 2... d68. Now white must allow the man access to either d3 or c4. This is the general idea of progress towards the side column and down, with the black king either attacking its opponent or covering its man. Clearly this is a win.

Diagram 3
White moves say 1.da1 and the black king must vacate the d-column or 2.a16 would trap the man and force a draw.
After 1 ... dh5 however black has his own threat: white cannot move a15 or a16, and any move on the back rank except 2. ad1 will result in the black king moving to h8 to cover the man from behind. Since this allows promotion, white is forced to keep attacking the black man.
The black man thus can allow itself to be driven to a6 after which black interposes the king again: 5... ha5, see diagram 4.

King v. man+king turkish
diagram 4
King v. man+king turkish
diagram 5

White moves anywhere on the back rank, say ah1, and black takes the corner, diagram 5. Now nothing can prevent the black man from promoting on a1 or b1. Usually, it would seem, the conversion and thus the win can be secured.

Armenian
Armenian is Turkish with diagonal movement: capture remains straight only. In the above example, it's black rather than white who may profit from this additional freedom, so in Armenian the conversion is even less problematic.

Dameo
The 'Armenian' king, but with a different man, one for which the option to move sideways has been replaced by the option to move diagonally forward. In terms of initial appreciation of the positions in diagram 2 and 3, nothing much changes. The black man usually needs access to a corner to promote. New are the facts that:

  • it cannot reach one, once it has crossed both main diagonals.
  • unless covered, it must approach via b2 or g2 or it will be trapped by the king in the corner.

King v. man+king dameo
diagram 6

To illustrate the some of the consequences, consider diagram 6, where the black man is beyond reaching a corner, white to move.

  • 1.da1?-d32 (threatening d4a1) and now 2.ad1-d48, while other moves are followed by either 2... d4a1 or indeed promotion, either way securing a black win.
  • 1.dh1?-d32, 2.ha1-d4h8! and black wins.

So allowing black access to the second rank isn't such a great idea. White can keep a draw by:

  • 1.d1c2-d4a1, 2.cb2-a12, 3.b2c1 and the black man cannot move to d2(e2) because of 4. cd1(e1).


Conclusion
Draughts, Shaski, Spanish and Hexdame do not even qualify, so in this category they're the most drawish. In Checkers and Dameo draws are possible but rare. In Turkish and Armenian the endgame is an unconditional win, which is ironically due to sideways movement.



King versus kings

Checkers
The number of kings needed to capture a lone one is a rough but pretty reliable indicator of a game's margin of draws. Here are some key positions.

King v. kings checkersKing v. kings checkers

In Checkers it's all about 'the move' and the tric-trac corner. 'Having the move' means being able to take opposition - in the left diagram, the player to move does not have the move!
On an odd-sided board without tric-trac corners, a king versus king position will always be a win for the player with the move. On 8x8 it can go either way: on the left black to move loses, white to move is a draw because white can retreat into the tric-trac corner.
To ensure a win you need two kings. On the right white to move loses: black invades the corner. If it's black to play he can reverse the move with g3f2 and f4e3.

Turkish
King v. kings turkishWhite can start from the position shown.
Black cannot occupy the 6 center files, so he can only place his king in a1 (on h1 white moves a71). The sequence is:

  • 1.ag7-ah1 (in Dameo g7h6 already wins here) 2.gb7!-ha1, 3.b72 (the symmetric key-position) a13 (or anywhere up to a7) 4.ha8+


Armenian and Dameo
King v. kings dameoArmenian and Dameo have an identical endgame, that differs from Turkish because of the games' diagonal movement. Yet the same premiss regarding the above position holds. Other than in Turkish however, the position below - black to move - is an immediate win.
If it is white's move, he can reverse the tempo:

  • 1.ah8-ha1, 2.bg7+


Hexdame
King v. kings hexdameKing v. kings hexdame

In HexDame 3 kings will always capture a lone one in three moves or less. A key formation is given on the left. On the right white, to move, will obviously lose a man, yet he wins the game. Barring rotations and reflections,the problem gave rise to 16 unique (but quite regular) 2x1 positions, that are a win.
This shouldn't shroud the fact that generally speaking three kings are needed.

Draughts
King v. kings draughtsKing v. kings draughts

In Draughts, barring some 10.000 of over 2.500.000 three against one positions, one needs four kings to capture a lone king. With four kings the capture is so easy that it suggests overkill. On the left a position where three kings do the job, but it is the exception, not the rule. On the right one of many positions to indeed capture the reluctant sucker.
In comparison with the other games, the situation Draughts is embarrassing.
One way to remedy the problem is to make king's subject to the demotion rule as in 'KillerDraughts': if the king is obliged to stop on the square immediately behind the last piece captured, then two kings in a tric-trac corner are sufficient to capture a lone king!
However small this change may be, and however good, it will affect a large part of the existing theory and that may prove too much to swallow for the draughts community at large. Few may be waiting for a slightly different rerun of basically the same game, despite the smaller margin of draws it provides.

Conclusion
The games in the straight plane are the clear winners here. Draughts players, please hide your hidden agendas, open your minds and listen carefully: Draughts is the most drawish of the long-range games due to an intrinsic flaw.

It can't be remedied, it won't go away - you're playing the wrong game.