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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.

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

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.

 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.