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 20x20 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 seach for diverse square tilings of certain sizes. He found that all diverse square tilings smaller than 48x48 have two equal squares that touch. His results are below:
| n | Number of Diverse Tilings of nxn Square |
|---|---|
| 1 | 1 |
| 20 | 1 |
| 23 | 1 |
| 25 | 1 |
| 26 | 1 |
| 29 | 92 |
| 30 | ? |
| 31 | ? |
| 32 | ? |
| 33 | ? |
| 34 | 66 |
| 35 | 276 |
| 36 | 136 |
| 37 | 422 |
| 38 | 62 |
Here are some small diverse square tilings:
| n | Sizes of Square |
|---|---|
| 1 | 1 |
| 9 | 20, 23 |
| 10 | 25, 26 |
| 11 | 29 |
| 12 | 29, 30, 31, 32, 33 |
| 13 | 34, 35, 36 |
| 14 | 35, 36, 37, 38 |
And here are some small non-square diverse rectangle tilings:
| n | Rectangles |
|---|---|
| 1 | 2x1 |
| 2 | 3x2, 5x2 |
| 3 | 5x3, 8x3 |
| 4 | 8x7 |
| 5 | 9x8 |
| 8 | 17x15, 18x15, 28x14 |
| 9 | 21x15, 24x14, 24x18, 26x15, 30x14 |
| 10 | 26x22, 27x23, 28x22, 30x20, 32x16, 36x15, 36x19, 42x14 |
| 11 | 29x21, 30x25, 30x26, 30x29, 31x27, 31x30, 37x22, 39x22 |
If you can extend any of these results, please
e-mail me.
Click here to go back to Math Magic. Last updated 7/28/02.
