Markers can move forward one square, or can jump over a marker of either colour if there is an empty square to land on.
Markers are not removed from the board if jumped.
You DO NOT have to alternate moving black and white markers.
The x's move E or S and o's move N or W and the goal is to make the x's and o's change places in these two board positions and in the general case.

Easy:

+-+-+-+    N
|x|x|x|  W + E
+-+-+-+    S
|x| |o|
+-+-+-+
|o|o|o|
+-+-+-+


Medium:

+-+-+-+-+-+
|x|x|x|x|x|
+-+-+-+-+-+
|x|x|x|x|x|
+-+-+-+-+-+
|x|x| |o|o|
+-+-+-+-+-+
|o|o|o|o|o|
+-+-+-+-+-+
|o|o|o|o|o|
+-+-+-+-+-+

I will try to see if we can be more efficient by using 
two dimensions. Consider the pieces on the spaces 
A,B,C,D,E,F,G,H as indicated. We really just want to 
rotate these 180 degrees. Let us move H into the center 
and have A-G chase eachother's tails until they are in 
thier opposite positions. The free square will now be 
H's destination.
Before:
+-+-+-+-+-+
|A|x|B|x|C|
+-+-+-+-+-+
|x|x|x|x|x|
+-+-+-+-+-+
|H|x| |o|D|
+-+-+-+-+-+
|o|o|o|o|o|
+-+-+-+-+-+
|G|o|F|o|E|
+-+-+-+-+-+
After 30 moves:
+-+-+-+-+-+
|E|x|F|x|G|
+-+-+-+-+-+
|x|x|x|x|x|
+-+-+-+-+-+
|D|x| |o|H|
+-+-+-+-+-+
|o|o|o|o|o|
+-+-+-+-+-+
|C|o|B|o|A|
+-+-+-+-+-+
Now rotate I,J,K,L,M,N similarly in 17 moves:
+-+-+-+-+-+
|o|x|o|x|o|
+-+-+-+-+-+
|I|x|J|x|K|
+-+-+-+-+-+
|o|x| |o|x|
+-+-+-+-+-+
|N|o|M|o|L|
+-+-+-+-+-+
|x|o|x|o|x|
+-+-+-+-+-+
Likewise for O,P,Q,R,S,T except before closing off the
loop on the 17th move (64th overall) leave a space open 
to rotate the last four:
+-+-+-+-+-+
|o|O|o|P|o|
+-+-+-+-+-+
|o|x|o|x|o|
+-+-+-+-+-+
|o|T| |Q|x|
+-+-+-+-+-+
|x|o|x|o|x|
+-+-+-+-+-+
|x|S|x|R|x|
+-+-+-+-+-+
After 63 moves:
+-+-+-+-+-+
|o|o|o|o|o|
+-+-+-+-+-+
|o|U|o|V|o|
+-+-+-+-+-+
|o|o|T| |x|
+-+-+-+-+-+
|x|W|x|X|x|
+-+-+-+-+-+
|x|x|x|x|x|
+-+-+-+-+-+
Now swap U,V,X,W in nine moves and finish off with T
for a total of 73 moves.

I guess this isn't very efficient as it isn't hard to 
do in 48 moves using the direct approach.

Oh well.

Edited on June 8, 2006, 12:51 am

Edited on June 8, 2006, 12:57 am

Posted by Eric on 2006-06-08 00:49:06