Cutting a rectangle into an odd number of congruent pieces - MathOverflow

Cutting a rectangle into an odd number of congruent pieces

2010-01-14T14:24:22Z

We are interested in tiling a rectangle with copies of single tile (rotations and reflexions are allowed). This is very easy to do, by cutting the rectangle into smaller rectangles.

What happens when we ask the pieces not to be rectangular?

For an even number of pieces, this is easy again (cut it into rectangles, and then cut every rectangle in two through its diagonal. Other tilings are also easy to find).

The interesting (and difficult) case is tiling with an odd number of non-rectangular pieces.

Some questions:

Can you give examples of such tilings?
What is the smallest (odd) number of pieces for which it is possible?
Is it possible for every number of pieces? (e.g., with five)

There are two main versions of the problem: the polyomino case (when the tiles are made of unit squares), and the general case (when the tiles can have any shape). The answers to the above questions might be different in each case.

It seems that it is impossible to do with three pieces (I have some kind of proof), and the smallest number of pieces I could get is $15$, as shown above:

This problem is very useful for spending time when attending some boring talk, etc.

Answer by Nurdin Takenov for Cutting a rectangle into an odd number of congruent pieces

2010-01-14T14:55:35Z

Ok. It's easy to see that rectangle cannot be cut into odd number of equal triangles: Monsky proved [here] that a square cannot be cut into an odd number of triangles with equal areas. But if we have rectangle divided into odd number of equal triangles, we could shrink it in one direction and get a square divided into an odd number of triangles with equal areas - contradiction.

Update: If the tile could be cut into odd number of triangles with the same area, then obvious that rectangle cannot be cut into odd number of such tiles. Example:

Answer by Michael Lugo for Cutting a rectangle into an odd number of congruent pieces

2010-01-14T17:35:54Z

Golomb's book Polyominoes has a section on this. Call the smallest odd number of copies of a polyomino that can tile a rectangle its "odd-order". Then Golomb says there are polyominoes of odd order 1, 11, and 15+6t for all $t \ge 0$. The polyomino of odd order 11 is due to Klarner [1], and is illustrated here by Michael Reid.

Reid has lots of pictures of tilings of rectangles with polyominoes. In particular the 15+6t family can be seen: here are polyominoes with odd-order 15, odd-order 21, odd-order 27, and so on. Reid has shown [3] that other odd orders exist, including 35, 49, and 221, but I don't know if there's a general pattern.

Finally, Stewart and Wormstein [2] proved that polyominoes of order 3 do not exist. (Stewart's book Another Fine Math You've got Me Into suggests that Wormstein is a fictional character.)

[1] David A. Klarner, Packing a rectangle with congruent N-ominoes, J. Combin. Theory 7 (1969) 107-115, [2] Stewart, Ian N. and Wormstein, Albert. Polyominoes of order 3 do not exist. J. Combin. Theory Series A 61 (1992) 130-136.
[3] Michael Reid. Tiling Rectangles and Half Strips with Congruent Polyominoes. J. Combin. Theory Series A 80 (1997) 106-123.

Answer by subshift for Cutting a rectangle into an odd number of congruent pieces

2010-01-18T21:50:06Z

I am posting the 11 pieces solution shown in the article cited by Michael (it is not freely available online).

This is the smallest known number of pieces. Some remarks:

The question is open for 5, 7, or 9 pieces. Get your pencils!
Everything so far is with polyominoes. Any suggestion with more complicated shapes?
Unlike the other solution I posted, this one cannot be resized along the $x$ or $y$ axis.