There was once a Duke who had an awkwardly shaped duchy, laid out like this:

       XX
 XX  XXXX
 XXXXXXX
XXXXXXXXX
XXXXXXXXX
 XX  XXXXX
 XX XXXXXX
    XXXXXX
     XXXX
     XX

Each X represents a 1000-hectare square.

The Duke was out of favor with the King, and when it came time for the Duke to pass down his land, the King hatched a plan to break up the Duke’s holdings (and thus the political power of his heirs). The King proclaimed that the Duke must split all of the land among his heirs such that:

1) The land is split into at least two pieces, and
2) Every piece has the same shape and size, allowing for rotations and/or reflections of the shape.

What are the largest pieces the Duke can break his duchy into, in accordance with the King’s edict?

This type of puzzle isn't really my forte, but I figured I'd give it a shot.  Have we confirmed whether or not the pieces need to be connected?

Also, I think the way the problem is drawn is throwing everyone off.  There doesn't seem to be any restriction on splitting the 1000-hectare squares apart.  Marking them with X's makes them appear to be distinct units, when in actuality this is just one large irregular shape with an area of 60,000 hectares.  I'm guessing that perhaps the divisions include angles less than 90 degrees...

Maybe someone who is better at this type of puzzle can take that idea and run with it. ;)

Posted by tomarken on 2006-03-02 10:37:29