No, it's not the Quadratic, algebraic sort that we're talking about here....Instead, consider this:


Imagine a 6 x 6 grid of 36 equal minisquares, or tiles. A "shape" consists of interconnected adjacent tiles on this grid. [Adjacent tiles must share an edge, not merely a vertex.]


**Prove the number of discrete combinations of 4 congruent shapes (9 tiles each) that will complete the grid.** 


One obvious combination is 4 congruent "sub-squares" of 9 tiles each. But there are several other valid shapes which one can find graphically by experiment. Which method(s) of proof is best used to determine the precise number? How would this proof be constructed? Is this problem and its proof methodology generalizable to completing similar grids, matrices etc.?


Look forward to hearing your ideas on this,


Scott



Completing the Square
You can find about 30 examples at [Mike Reid's Rectifiable polyomino page.](https://www.cflmath.com/%7Ereid/Polyomino/rectifiable_data.html) It would appear that for the 6x6 square there are two types of 9 cell tilers:


* Take the [center edge of a 3X6](https://www.cflmath.com/%7Ereid/Polyomino/9omino12_rect.html) 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](http://www.math.cflmath.com/%7Ereid/Polyomino/9omino22_rect.html) 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](http://www.cflmath.com/%7Ereid/Polyomino/7omino4_rect.html) 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.


