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King versus Man + King
If two kings can trap a lone one, as in
then the key questionconcerning the above endgame, is whether it can be converted to a 2x1 kings endgame.
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 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.
The same holds for Turkish Checkers. I don't have any theory on the game, but 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 white moves in both diagrams.
diagram 2
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diagram 3
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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.
in 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.
It thus can allow itself to be driven to a6 after which black interposes the king again: 5 ... ha5, see diagram 4.
diagram 4
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diagram 5
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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.
Here's a different man with a different king. Since the man can still move sideways, promotion cannot be prevented, so
in Armenian the endgame is an unconditional win.
Again a different man, now with a "turkish" king. The man has three directions of movement, but here 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 needs access to a corner to promote.
New are the facts that:
- it cannot reach one, once it has crossed both main diagonals
- it must go forward and remains 'on color' if it moves out of frontal attack
- without cover it must approach via b2 or g2 or it will be trapped by the king in the corner
The importance of the first point would appear evident.
diagram 6
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To illustrate the consequences of the second one, let
white in diagram 6 move 1. de1, blocking black's access to e4.
- If black moves 1 ... d5-c4 he misses the win after 2. ec1, see diagram 7: black cannot cover and 2 ... c4-d3 loses the corner,
so the man is driven onwards to a2, where it meets its doom
- If he moves 1 ... d54 white also moves 2. ec1 and black moves 2 ... d68,
see diagram 8.
Since the black man must proceed to c3, white makes it its homebase, guarding c3 either
from the rank or the column, depending on the black king's actions. So no go for black
- 1 ... da6 keeps him on the winning track, see diagram 9
diagram 7
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diagram 8
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diagram 9
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diagram 10
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If white attacks the black man with ed1, black moves d5c4, 2. dc1 c4b3 3. cb1 b3a2 4. ba1? a68 resulting a black win.
1. e14 on the other hand meets with a64 (diagram 10), threatening capture and also opening the way to a2.
The black strategy does not apply if the black man has passed both main diagonals.
diagram 11
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In diagram 11 white moves 1. dc1 after which d3e2 2. ce1 is a draw. If black moves 1. ... d32, then 2. cd1 d43
3. da1 dh3 4. ah1! also secures the draw. Finally 1. ... dh4 2. cd1! forces hd4 and a draw by 3-fold.
Here's the same man with a different king. Generally speaking, if the man has not crossed both main diagonals,
black can win. Even if it has, white can make some serious mistakes. In diagram 11 - now considered a Dameo position -
if 1. dc1 is followed by d3e2, then 2. ce1 d4e3 3. eh1 secures a draw. On 1. dc1 d4a1 however, white must not answer
2. ch1? d32 and black wins. Instead 1. dc1 d4a1 2. c1b2! secures a draw. Finally white can prevent 1. ... d4a1 altogether
and reach a draw by 1. da1 d3c2 2. ac1 d4c3 3. ca1! (not ch1??) ch3 (to prevent white from attacking on a2 or c1) 4. ah1.
Draughts, Shaski, Brazilian Checkers and HexDame do not even qualify, so in this category they're the most drawish.
In Checkers, Croda and Dameo draws are possible but rare. In Turkish and Armenian Checkers
the endgame is an unconditional win due, ironically, to sideways movement.
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