## Monday, July 12, 2010

### [gfhqlpbf] Dodgem results

In 3x3 Dodgem, the unique position with the longest forced win is

> . .
. > .
^ . ^

with ^ to move, win in 22.

In all diagrams, we will use the convention of ^ to move. One move is one ply, a single player moving (so not the chess convention where a "move" consists of a pair of a White and a Black turn). A win in 0 is a terminal position, where the player to move has no moves available. This can happen if a player has no pieces left, or a stalemate position. Thus, a position where the player to move has one remaining piece left at the end of the board is (usually) a win in 2. Wins are always in an even number of moves.

The standard initial position is one of 9 positions that are win in 20.

In 4x4 Dodgem, the standard initial position is a draw. Under optimal play, the game loops. The following 14 positions have the longest forced win, win in 42.

. . . .
. . . .
> > . ^
> ^ ^ .
> . . .
. . . .
. > . ^
> ^ ^ .
> . . .
> . . .
^ . . ^
. > . ^
> . . .
> . . ^
. . . .
^ > . ^
> . . .
> ^ . .
^ . . .
^ . > .
> . ^ .
> . . .
^ . . .
^ . > .
> . . .
> > . .
. . ^ .
^ . . ^
> . . .
> > . ^
. . . .
^ . . ^
> . . .
> ^ > .
^ . . .
^ . . .
> . ^ .
> . > .
^ . . .
^ . . .
> > . .
> . . .
. ^ . .
^ . ^ .
> > . .
. > . .
. . ^ .
^ . . ^
> > . .
. > . ^
. . . .
^ . . ^
> ^ > .
. . . .
> . . .
^ . ^ .

In 5x5 Dodgem, the standard initial position is again a draw. The following 5 positions have the longest forced win, win in 82.

. . . . .
. . . . .
> . ^ . .
. > ^ . .
. > . ^ ^
. . . . .
. . . . .
> . ^ . .
. > . . ^
. > ^ ^ .
. . . . .
. . . . .
> . ^ . .
. > . ^ ^
. > . ^ .
. . . . .
. . . . .
> . . ^ .
. > ^ . ^
. > . ^ .
. . . . .
. . . . .
. . ^ . .
. . ^ ^ .
> > . . ^

That final position above looks funny; I'm not completely convinced my program is correct.

In all of the 5x5 positions above, there are an uneven number of pieces on the board. The positions with the longest forced win which still have all 8 pieces in play are the following 6 positions below which win in 68.

> > > . ^
. . . . .
. . ^ ^ .
^ . . . .
. > . . .
> > > . .
. . . ^ ^
. . ^ . .
^ . . . .
. > . . .
> > > . .
. . . . .
. ^ . ^ ^
. ^ . . .
. > . . .
> > > . .
. . . ^ .
. ^ . . ^
. . ^ . .
. > . . .
> > > . .
. . . ^ .
. ^ . . ^
^ . . . .
. > . . .
> > > . .
. . . ^ .
^ . . . ^
. ^ . . .
. > . . .