Completing the Square

You can find about 30 examples at Mike Reid's Rectifiable polyomino page. It would appear that for the 6x6 square there are two types of 9 cell tilers:

• Take the center edge of a 3X6 rectangle along with a path from one end of that edge to the boundary and the 180- degree rotation starting from the other end

and

• Take a path from the center of a 6x6 square and rotate it by 90,180 and 270 degrees.

So a method to do all this (which I have not done) would have these steps: Find all tiles of these two types. (One needs a path which does not intersect its 1 or 3 partner paths, there can only be so many so go about it systematically). Then show that a tiling must decompose into 2 3x6 rectangles or else have a 4-fold central rotational symmetry.

Most work I know starts with a tile and asks what rectangles it tiles. I would consider it pretty interesting to prove or disprove that any partition of a square into 4 congruent pieces is of one of these two types. Maybe that is already known one way or the other. I am not aware that anyone has looked systematically at dividing a checkerboard into 4 congruent pieces. Here is a decomposition of a 14x14 square into 16 7-cell tiles. It does have a 4-fold rotational symmetry.

further comments This is a fascinating field. Search the literature and you will find some contributions by well known names but you will also find that some of the best results come from "amateurs". I use the term only in the sense of someone who does it exclusively for the love of the subject. An easier problem than yours is: How many ways to split a $k$ by $2k$ rectangle into two congruent pieces? You would have to solve that to solve your problem. Already that is a difficult lattice path problem. For particular small $k$ you could do it but in general asymptotics might be the best one could expect.