Solver for n-dimensional variants of the MPMP7 Unique Distancing problem.
Project with tools for the 7th Matt Parker Math Puzzle problem.
- Solution for the problem stated in the MPMP7 youtube video.
- Solutions for smaller and larger grids.
- Solutions in 3 or more dimensional grids
- Can I fit more markers on the grid?
- Can I solve this on an 8x8 grid?
- Do solutions exist for larger 2D grids?
Yay, Matt mentiond my solution in MPMS: Unique Distancing Problem BTW, note the time offset 🤘
The problem
Arrange N counters on an NxN grid, such that all the distances between the counters are different.
First a solution to the problem stated in the MPMP7 youtube video:
python3 mpmp7_unique_distances.py --width 6 --verbose
This will output these two solutions.
*.....
*.....
.....*
.*....
......
...*.*
or
*.....
......
*.....
...*..
....**
*.....
The script will not terminate immediately, since it will keep searching for more solutions, which it will not find.
I also wrote a C++ program to do the same, but much faster:
./mpmp7-unique-distances -p 6
Solutions for smaller and larger grids.
First, as stated by Matt in his video, for the 3x3 case there are 5 solutions:
..* ... .** ... *.*
**. ..* ... *.* *..
... **. *.. *.. ...
Though you could argue that the first two, and also the last two look identical, with respect to translation. I did not look into counting those as duplicates.
Then the 2x2 grid, this one is pretty obvious, these are the two solutions:
*. *.
*. .*
The simplest grid, being the 1x1 grid has 1 solution:
*
Or is that the simplest? what about a 0x0 grid:
Well, I am not sure if that counts as 0 or 1 solution.
Here is a table listing the number of solutions for the remaining grid sizes:
| size | solution count |
|---|---|
| 0 | 0 or 1 |
| 1 | 1 |
| 2 | 2 |
| 3 | 5 |
| 4 | 23 |
| 5 | 35 |
| 6 | 2 |
| 7 | 1 |
| 8 | 0 |
Here is the solution for the 7x7 grid:
*..*...
.......
**.....
.......
.......
.....*.
..*...*
Solutions in 3 or more dimensional grids
Now Why stop at flat grids, you can solve this for 3-D ‘grids’ as well:
| size | 3D solution count |
|---|---|
| 2 | 3 |
| 3 | 50 |
| 4 | 3983 |
| 5 | >=1185 |
| 6 | |
| 7 | >=3446 |
Then in 4-D I was only able to solve this for 2x2x2x2 and 3x3x3x3 grids. The following table list the number of solutions found for the various higher dimensional grids:
| size | 4D | 5D | 6D | 7D |
|---|---|---|---|---|
| 2 | 4 | 5 | 6 | 7 |
| 3 | 261 | >=255 | >= 37 | >=16 |
| 4 | >=766 | >=81 |
in 5-D there are 5 solutions for the 2-sized grid. and more than 800 for the 3-sized grid, I did not wait for my program to complete it’s search.
Can I fit more markers on the grid?
For the 3D and higher dimensional grids you can, here is a table of the most markers you can fit for a given size and dimension:
| size/dim | 2D | 3D | 4D | 5D | 6D | 7D |
|---|---|---|---|---|---|---|
| 2 | 2 | 3 | 3 | 3 | 4 | 4 |
| 3 | 3 | 4 | 5 | >=6 | >=6 | 4 |
| 4 | 4 | 6 | >=7 | >=4 | >=3 | >=3 |
| 5 | 5 | >=7 | ||||
| 6 | 6 | >=8 | ||||
| 7 | 7 | >=8 |
The empty slots in the table were computationally too expensive to determine.
Can I solve this on an 8x8 grid?
Not with 8 markers, but there are 927 ways you can solve this with 7 markers.
Here is one solution:
*......*
*.......
........
*.......
.*......
.......*
..*.....
........
Do solutions exist for larger 2D grids?
With less markers you can solve this ( obviously ). Here is a table listing the maximum number of markers you can fit on 2D grids:
| grid size | max counters |
|---|---|
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 7 |
| 8 | 7 |
| 9 | >=8 |
| 10 | >=8 |
| 11 | >=9 |
| 12 | >=9 |
| 13 | >=9 |
| 14 | >=8 |
| 15 | >=8 |
code
For my code, see github
