Problem of the Month (March 2007)

The problem of finding the order of a polyomino, the fewest number of copies of the polyomino that can tile a rectangle, has been well-studied. See Mike Reid's page for example. This month we study using polyominoes to tile rectangles with identical square corners missing. What are the smallest rectangles with 1×1 corners missing that polyominoes tile? What about rectangles with 2×2 or larger corners missing?


ANSWERS

Rectangles

1-5

6
(David Klarner)

(T. W. Marlow)

7
(David Klarner)

(T. W. Marlow)

8
(David Klarner)

(Mike Reid)

(Mike Reid)

9

10

(William Marshall)

(William Marshall)

11
(William Marshall)

(Mike Reid)


Rectangles Without 1×1 Corners

1-4

5

6

7

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Mike Reid)

8

(George Sicherman)

(George Sicherman)

9
(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Mike Reid)

10

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

11
(George
Sicherman)

(George
Sicherman)

(George
Sicherman)

(George
Sicherman)

(George Sicherman)

(George Sicherman)

(Mike Reid)


Rectangles Without 2×2 Corners

1-4
(Patrick Hamlyn)

5

6

(Patrick Hamlyn)

(Mike Reid)

(Mike Reid)

(Mike Reid)

7

(Patrick Hamlyn)

(Patrick Hamlyn)

(Mike Reid)

8

9
(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Mike Reid)

(Mike Reid)

10

(Mike Reid)

(Mike Reid)

11
(George
Sicherman)

(George Sicherman)

(George Sicherman)

(George Sicherman)

(George Sicherman)

(George Sicherman)

(George Sicherman)

(George Sicherman)


Rectangles Without 3×3 Corners

1-3

4

5

6

(Patrick Hamlyn)

(Mike Reid)

7

(Patrick Hamlyn)

(Patrick Hamlyn)

8

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

9

(Patrick Hamlyn)

10
(Joe DeVincentis)

11
(George Sicherman)

(George Sicherman)

(George Sicherman)

(George Sicherman)

(George Sicherman)

(George Sicherman)

(Mike Reid)


Rectangles Without 4×4 Corners

1-3

4

5
(Patrick Hamlyn)

6

(Mike Reid)

7

8

(Patrick Hamlyn)

(Mike Reid)

(Mike Reid)

9

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Mike Reid)

(Mike Reid)

10

(Mike Reid)

11
(George Sicherman)

(George Sicherman)

(George Sicherman)

(George Sicherman)

(George Sicherman)

(George Sicherman)

(George Sicherman)


Rectangles Without 5×5 Corners

1-3

4
(Mike Reid)

5
(Patrick Hamlyn)

6

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Joe DeVincentis)

(Mike Reid)

7
(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Mike Reid)

(Mike Reid)

8
(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

9

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Mike Reid)

10


Rectangles Without 6×6 Corners

3
(Patrick Hamlyn)

(Patrick Hamlyn)

4
(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Mike Reid)

5
(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

6
(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

7
(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Mike Reid)

8
(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Mike Reid)


10
(Mike Reid)


11
(Mike Reid)


Patrick Hamlyn also considered the variant where non-symmetric polyominoes could not be flipped over. Even-sized corners were not very exciting, but here are the best results for odd-sized corners:

Rectangles Without 1×1 Corners


(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)


(Patrick Hamlyn)

(Patrick Hamlyn)


(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)


Rectangles Without 3×3 Corners


(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)


(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)


Rectangles Without 5×5 Corners


(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)


(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)


(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)


(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)
Ed Pegg thought these tilings were worthy of mention, even if they don't exactly meet this month's criteria:

George Sicherman found tilings of rectangles missing only one 1×1 corner...

...two opposite 1×1 corners...

...two adjacent 1×1 corners...

...and three 1×1 corners.

Brian Astle thought that the smallest square missing two opposite 1×1 corners that could be tiled by more than 2 polyominoes other than monominoes was an 18×18 square tiled with 46 copies of a heptomino. Mike Reid found that a 10×10 square could be tiled with 14 copies of a different heptomino.

Brian Astle also conjectured that no square missing opposite corners could be tiled with an odd number of polyominoes, but Mike Reid found a counterexample using 41 hptominoes to tile a 17×17 square.

Claudio Baiocchi noted that if two copies of a polyomino could tile a 4×5 rectangle, then there are trivial tilings of rectangles missing 4×4 or 5×5 corners:

Gavin Theobold noted the same trick works for other size rectangles too.


If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 3/31/07.