Problem of the Month (December 2002)

A rectangle has aspect ratio n if the ratio of the larger side to the smaller side is n. This month we consider dissections of a square into rectangles with consecutive integer aspect ratios starting with 1. We call these aspect ratio dissections. For example, there are two essentially different ways to cut a square into rectangles with aspect ratios 1, 2, 3, and 4:

Can you prove there are no other ways to do this? Can you show that we can always cut a square into n rectangles with aspect ratios 1 through n? How many such ways are there? The first square above is dissected into rectangles which are all arranged vertically. Are such aspect ratio dissections where all the rectangles are vertical rare? Can you find a aspect ratio dissection where no rectangle of aspect ratio k touches a rectangle of aspect ratio (k+1)?


Patrick Hamlyn found all the aspect ratio dissections of squares of size 12 or less, and the the dissections of squares of size 28 or less into no more than 8 rectangles. They are sorted by the size of the large square here:

3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28

Patrick Hamlyn considered aspect ratio dissections of squares 2 through n as well, and didn't find many:

8, 9, 13, 15, 16, 18, 20, 21, 22, 23,

Patrick Hamlyn also looked for aspect ratio dissections of squares only using odd aspect ratios, and found these:

7, 8, 13, 15, 18, 19, 20, 21, 22, 26, 28

I found this dissection into even aspect ratios, using only vertical rectangles:

Mike Reid also found aspect ratio dissections by hand corresponding to the sequences 3, 5, 7, 9, ... , and 3, 4, 5, 6, ... , and my personal favorite 1, 4, 9, 16, ... .

Patrick Hamlyn also looked for aspect ratio dissections of squares using only prime aspect ratios, and found these:

15, 23, 31

Patrick Hamlyn also searched for non-touch aspect ratio dissections. He found all of them of size 13 of less, and all of them of size 19 or less with no more than 10 rectangles. Andrew Bayly found one of these of size 12.

10, 11 12, 13 14, 15 16, 17 18, 19

Patrick Hamlyn confirmed that a 18×18 square is the smallest that admits a vertical aspect ratio dissection except for the 4×4 square. Andrew Bayly also found another larger vertical aspect ratio dissection:

Patrick Hamlyn also found the smallest tiling that does not contain a subrectangle, and the smallest vertical (or in this case, horizontal) tiling with the non-touch property:

Mike Reid, Clinton Weaver, Sasha Ravsky, and Andrew Bayly showed that aspect ratio dissections exist for all n ≠ 2 since we can extend such a dissection of n rectangles to one with n+2 rectangles, and such dissections exist for n=1 and n=4.

Robert Reid came up with a similar but more complicated extension that could extend non-touching examples. He asks whether n=5 is the only example where an n×n square cannot be dissected.

Jeremy Galvagni noticed that if we pick and choose which aspect ratios to use to dissect a square, that any set of numbers whose reciprocals sum to 1 (like {2,3,6}) easily forms a dissection with vertical rectangles.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 12/15/02.