Problem of the Month (June 2002)

This month we consider tilings of squares with consecutive squares. It is well known that the only non-trivial solution of 12 + 22 + . . . + n2 = k2 is n=24 and k=70. Unfortunately, it has been proved that a square of size 70 cannot be tiled with squares of sizes 1 through 24. (There is a packing without the square of size 7. Can you find it?)

It turns out there ARE tilings of squares by consecutive squares if we allow either 1 or 2 squares of each size. We such a tiling a diverse tiling. The smallest non-trivial diverse tiling of a square is the 20×20 square below:

What other diverse tilings of a square can you find? Are there many diverse tilings of rectangles? What's the largest one you can find?

I'll pay $10 for the first diverse square tiling where repeated squares don't touch. I'll also pay $10 for the first diverse square tiling that contains 1 square each of odd sizes and 2 squares each of even sizes. I'll also pay $10 for the first diverse tiling of a triangle by smaller equilateral triangles.


ANSWERS

Antonio Ianiero found diverse square tilings of sizes 1-10 in a 25×25 square, sizes 1-12 in a 30×30 square, and sizes 1-12 in a 33×33 square.

Patrick Hamlyn wrote a computer program to search for diverse square tilings. He found that all diverse square tilings smaller than 48×48 have two equal squares that touch. His results are below:

Number of Diverse Square Tilings

kNumber of Diverse
Tilings of k×k Square
Author
11Trivial
201Patrick Hamlyn
231Patrick Hamlyn
251Patrick Hamlyn
261Patrick Hamlyn
2993George Sicherman
306George Sicherman
31205George Sicherman
32439George Sicherman
33412George Sicherman
3483George Sicherman
35240George Sicherman
36136Patrick Hamlyn
37359George Sicherman
3864George Sicherman
3964George Sicherman
4054George Sicherman

Here are some pictures of small diverse square tilings:

Diverse Square Tilings

nk
11
920, 23
1025, 26
1129
1229, 30, 31, 32, 33
1332, 33, 34, 35, 36, 37
1435, 36, 37, 38, 40

And here are some pictures of small non-square diverse rectangle tilings:

Diverse Rectangle Tilings

nRectangles
12×1
23×2, 5×2
35×3, 8×3
48×7
59×8
715×13
817×15, 18×15, 28×14
920×17, 21×15, 24×14, 24×18, 25×21, 26×15, 28×17, 30×14
1026×22, 26×24, 27×20, 27×23, 28×17, 28×22, 28×26, 30×15, 30×20, 30×24,32×16, 36×15, 36×19, 42×14
1129×21, 30×25, 30×26, 30×29, 31×27, 31×29, 31×30, 35×25, 37×20, 37×22, 37×26, 39×19, 39×22, 40×20

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 2/12/13.