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.
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:
k | Number of Diverse Tilings of k×k Square | Author |
---|---|---|
1 | 1 | Trivial |
20 | 1 | Patrick Hamlyn |
23 | 1 | Patrick Hamlyn |
25 | 1 | Patrick Hamlyn |
26 | 1 | Patrick Hamlyn |
29 | 93 | George Sicherman |
30 | 6 | George Sicherman |
31 | 205 | George Sicherman |
32 | 439 | George Sicherman |
33 | 412 | George Sicherman |
34 | 83 | George Sicherman |
35 | 240 | George Sicherman |
36 | 136 | Patrick Hamlyn |
37 | 359 | George Sicherman |
38 | 64 | George Sicherman |
39 | 64 | George Sicherman |
40 | 54 | George Sicherman |
Here are some pictures of small diverse square tilings:
n | k |
---|---|
1 | 1 |
9 | 20, 23 |
10 | 25, 26 |
11 | 29 |
12 | 29, 30, 31, 32, 33 |
13 | 32, 33, 34, 35, 36, 37 |
14 | 35, 36, 37, 38, 40 |
And here are some pictures of small non-square diverse rectangle tilings:
n | Rectangles |
---|---|
1 | 2×1 |
2 | 3×2, 5×2 |
3 | 5×3, 8×3 |
4 | 8×7 |
5 | 9×8 |
7 | 15×13 |
8 | 17×15, 18×15, 28×14 |
9 | 20×17, 21×15, 24×14, 24×18, 25×21, 26×15, 28×17, 30×14 |
10 | 26×22, 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 |
11 | 29×21, 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.