Cancellation rules can be added to any chess variant in which pieces can be assigned numerical values.
In cancellation rules, captures work according to the following rules, instead of the normal rule of attacking piece replaces attacked piece.
In standard chess, pieces are usually valued at P=1, B=3, N=3, R=5, Q=9. Suppose that a white Q is on a1, and a black R on d4. If the Q moves to attack the R:
The black R is removed from the board. Its value (5) is subtracted from that of the Q (9), leaving 4. Since there is no piece in standard chess with a value of 4, a piece of value 3 (either B or N, white's choice) is placed on d4. Since there is additional value (1) remaining, white also places a P on any vacant square adjacent to d4.
Suppose a white P is on d4, and a black Q on e5. If the P moves to attack the Q:
The white P is removed from the board. Its value (1) is subtracted from that of the Q (9), leaving 8. Since there is no piece in standard chess with a value of 8, 7, or 6, a piece of value 5 (a R) is placed on e5. Since there is additional value (3) remaining, black also places either a B or N on any vacant square adjacent to e5.
If the Q were valued at 10 instead of 9, in both examples, the player owning the Q would have placed an additional P on a vacant square adjacent to d4 (in the first example) or e5 (in the second example).
A variant on this variant would be to allow placement of pieces which have the necessary value, but which do not normally appear in the game (for example, placing a wazir (of value 2) in a game of standard chess).
In Cancellation Shogi, the lower-valued piece goes into the opponent's reserve. Pieces placed on the board as a result of a cancellation are considered unpromoted, and may promote normally. Optionally, the reserve may be designated to contain _any_ piece of the necessary value, with the exact piece to be determined upon re-entry.