Count Carefully! Abalone Openings
The board game Abalone is played with black and white spheres on a hexagonal board. The spheres can be moved individually one unit in (up to) 6 directions, but spheres can also be moved in pairs or as straight lines of 3; again, this would be one unit in (up to) 6 directions. When moving a pair or line of three, the line of the pieces must point in the same direction after the move as they were before.
Below we can see the 6 possible moves for A. If A moved to the right it would end up on the circle highlighted in red.
The pair of spheres at B are shown moving to the left, but they could also move down-left, down-right or right. They are unable to move up-left or up-right because the other (in this case white) sphere blocks them. Pieces can only move into empty spaces.
The pair of spheres at C are shown moving "in parallel" to the left, but they could also move up-left or down-left. They are unable to move in the other directions because they are at the edge of the board.
The line of 3 spheres at D are shown moving "in parallel" up-right, but they could also move in all six directions.
The next image shows the position at the beginning of a normal game. How many opening moves are there altogether for Black?
The answer is 44.
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1 solution
There are 14 moves for just one sphere, 20 moves for spheres in pairs and 10 moves for spheres in lines of 3.
This gives a total of 14 + 20 + 10 = 44 possible moves.
A further breakdown
For a single piece, there are 6 up-left moves, 6 up-right moves, 1 left move and 1 right move.
For a pair that is pointing up-left there are 5 up-left moves and 1 up-right move.
For a pair pointing up-right, this is exactly mirrored.
For a pair point left, there is 1 left move, 1 right move, 3 up-left moves and 3 up-right moves.
For a line of three pointing up-left, there are 3 moves up-left.
A line of three pointing up-right mirrors this.
Finally for a line of three pointing left, there is 1 left move, 1 right move, 1 up-left move and 1 up-right move.