There was once a Duke who had an awkwardly
shaped duchy, laid out like this:
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 showed up for me as a bonus problem!
I have successfully divided the shape into pieces that are 2 units, 3 units, and 5 units. I believe 4 units and 6 units are not possible.
As I have now started to try pieces of 10 units (the next factor of 60), I feel like I have leaped off a cliff! It is less cut-and-dry since there are so many more shapes, and it is more difficult to demonstrate why it is not possible.
I had no luck with 10 units, so I decided to instead start with 30 and work down. I couldn't get 30 or 20. That just leaves 10, 12, and 15.
Then I had the idea to try solving it for 2-unit pieces, and then put 5 together to get 10-unit pieces, or 6 to get 12-unit pieces. I also tried putting 5 3-unit pieces together to get 15-unit pieces.
Still no luck.
But I'm still working on it!
Posted by Dustin
on 2011-12-27 03:49:17