According to Bruce Whitehill a game must meet the following two criteria to be considered as a variation: 1) a player's piece are required to finish in the starting space of the opposing player; and 2) piece movement includes the allowance for jumping all pieces, including a players own as well as an opponent's, without any pieces being removed from play
David Parlett (1999) mentions these three variations of Halma:
Colour Halma: The colour of the square determines the rules of movement. If a piece start on a dark square it must remain on dark squares only, moving one step diagonally and is not allowed to jump. A piece who start on a light square must remain on light squares only, and can only move by orthogonal jumps.
Marked Halma: Each player marks his corner piece and must get that piece to the opposite corner exactly
Grasshopper : A chessboard variant where the players start from the ten squares of diagonally opposite corners, or fifteen on the 10x10 draughtboard
Perkis suggest this a standard Super Halma game: The game is played with 19 pieces per player on a 10x10 International Checkers board. The main feature is the altered rule for Jump move: A piece may jump over any other piece any number of empty spaces away, either orthogonally or diagonally, provided it can land the same number of empty spaces beyond it in a straight line. It is also possibly to trap a opponents piece.
Or said with other words: ...Where a stone jumps at any distance, provided that the jumped stone lies at the exact midpoint of the jump.
Super Chinese Checkers
Guess what is it :-). The Chinese Checkers entry in Wikipedia explains the rules (calls it the 'fast-paced variant'). First named by Wayne Schmittberger.
João Neto mentions some Halma variants on his homepage:
Top Secret: Another curious variant of Halma, is Top Secret, where each player drops in turns their 18 stones on the central 6x6 square area of the 16x16 board. Then, after all drops were made, the goal is to be the first to remove all stones moving or jumping over the board.
Le Grec: A simple variation is Le Grec (by Dominique Huguenin and Yves Chédel) played in the next diagram. The rules: (1) Move one friendly stone (orthogonal or diagonal) to an empty cell, or jump over an enemy stone (no jumps over friendly stones) landing on the immediate next cell. Every move/jump must be forwards or sideways never backwards. (2) Wins the player that moves his stones to the last two rows. [Play as Java applet].
There is also Renpaarden played on a 9x9 square board: Both players
start with the first two rows filled with friendly stones (so, 18
stones each). Stones move like chess knights: either to an empty
square, which ends the turn, or to a square with a piece of the
opponent; in the latter case, the piece can make another jump. (there
is no capturing.). Winner is who has all his stones at the two rows at
the opposite of the board.
Le Zug is played on the following 17x17 board. Below is a initial setup (the stones on the second row can be placed at any column where they are diagonally adjacent to the first row stones). Stones move diagonally forward and can jump diagonally over stones of either color (jumps can be multiple and are not mandatory). A piece which performs a jumping move, whether single or multiple, is allowed to make an additional diagonal step. All steps and jumps are forward only. The first four rows of each side is considered the camp of that player. No piece may be moved into an enemy camp while their are still friendly pieces in their own. Wins the player that first occupies the last row, and place the remaining three stones at the penultimate row. According to L. Lynn Smith the title appears to be an amalgamate of French and German. Its meaning "The Course".
Carrera de Caballos, from Héctor Canteros, is played on the next board, where pieces move like non-capture chess Knights, i.e., they jump to one of the nearest cells not in the same column or row.
Hexma: Check also Hexma (by Cameron Browne) a variant of Chinese Checkers with a connection touch: "a player wins by completing a chain of their pieces between their home and away board edges" (it should be use one of the mechanisms explained above to remove drawish strategies). The same game on a typical hex board:
Variant Applets page mentions two Chess-Halma
The idea to create a crossover between Halma and Chess is not new. 'Schachhalma' ('Schach' is German for Chess) was described by Hajo Bücken and Dirk Hanneforth in their book 'Klassische Spiele ganz neu' which was first published in 1990.
Ugolki (russian for 'Corners')
The most popular variant is supposed to be played with 3x3 start standing. Qoute from Wikipedia as of August 6, 2011:
Ugolki is a two-player board game that is typically played on an 8×8 grid board with 16 game pieces per player. It is said to have been invented in Europe in the late 18th century. Variations on the size of the board and the number of game pieces also exist.
Both players start off with square arrangements of 16 pieces in opposing corners of the game board. Each player's goal is to move all of the game pieces from the starting corner to the corner occupied by the opponent at the start of the game.
Players take turn moving one game piece. A game piece may move only away from the starting location into a destination that is empty, provided the following conditions are met:
Main sources: Whitehill (2002) and Parlett (1999) - se sources in History section. Other sources listed.