Distancing uniquely

The latest Maths Puzzle from Matt Parker concerns itself with distancing, not socially but uniquely. Can you put three markers in a 3 by 3 square such that no two marker-to-marker distances are the same? You can:

Here A and B are 1 unit apart, B and C are √5 units apart, and A and C are 2√2 units apart. Not counting reflections and rotations, there are 5 distinct ways to put three markers in a size 3 square at unique distances:

Parker asks for a solution for six markers in a 6 by 6 square. I very quickly convinced myself that if there’s an intelligent, elegant way to get such a solution by pencil and paper, I wasn’t going to find it. Instead I wrote a Python script to find all solutions for any size square. Any size to a point, anyway.

How many ways are there to put n markers in an n by n square? If you number the positions 1 through n^2 then selecting n positions to receive markers amounts to choosing n out of n^2; the number of ways of doing so is \binom{n^2}{n} = n^2!/(n!(n^2-n)!). For 3 markers in a 3 by 3 square there are 84 ways. For 6 markers in a 6 by 6 square there are 1,947,792 ways. My Python script gets very slow very fast.

The way it works is this: There’s a subroutine that takes a string pattern like “*.*……” which is the concatenation of “*.*”, “…”, and “…” corresponding to:

The subroutine does nothing if it finds non-unique distances within the pattern. If the distances are unique and the number of markers is equal to the largest number previously seen with unique distances it adds the pattern to a list of solutions; if the number of markers exceeds the largest number previously seen, it erases the list of solutions and starts a new list with this pattern. Then it calls itself recursively to investigate patterns made by taking the original one and adding a marker at each of the rightmost empty positions. (In this case it checks “*.**…..”, “*.*.*….”, “*.*..*…”, “*.*…*..”, “*.*….*.”, and “*.*…..*”)

Actually it does the above only after looking at the eight possible rotations and reflections of the pattern and seeing whether it’s the lexicographically smallest of the eight.

In a 2 by 2 square you can put 2 markers, 2 different ways. (Adjacent to one another orthogonally or diagonally. Of course there is only one distance either way, so it’s unique.) In a 3 by 3 square you can put 3 markers 5 ways, as seen above. For 4 by 4: 4 markers, 23 ways. For 5 by 5: 5 markers, 35 ways.

So it looks as though the number of ways to put n markers in an n by n square just keeps increasing. Until you try it with 6. You can put 6 markers in a 6 by 6 square… only 2 ways. 7 markers in a 7 by 7 square — only 1 way. When you get to 8 markers in an 8 by 8 square, it can’t be done. At best you can put 7 markers, but there are 3350 different ways!

Having done that for squares, I modified the script to do other dimensions: Rows of n positions or cubes of n^3 positions. Rows are easy — the number of combinations grows slowly. Cubes grow very fast. Worse yet, checking all 48 possible rotations and reflections in three dimensions gets complicated, and once I finally got that code working, I discovered that even though checking for rotations and reflections cut way down on the number of positions it needed to examine, it was slower than examining all positions, because the reflection and rotation checking took so much time! Then I realized it’d make more sense to check for duplicate distances first, then check for rotations and reflections before doing the recursive calls, and that did speed things up somewhat, but not enough to make a lot of difference. There are probably smarter ways to do it in Python, and probably even smarter ways in C, but I’ll leave those to someone else to work out.

For rows I looked at sizes up to an arbitrary limit of 20. For squares and cubes I stopped when I got to a size that took more than about three hours. Successive size squares took about ten times as long, and successive size cubes about 200 times as long! The results:

dimensions ->123
size vmaxsolutionsmaxsolutionsmaxsolutions
2212231
32235420
431423636
53353574223
63762
74171
84573350
94138547
10430952
11455
1252
13511
14534
15578
165160
175292
1864
19612
20640

For instance, in a row of 5 positions, there are 3 ways to put 3 markers at unique distances. In a 5 by 5 square, there are, as said above, 35 ways to put 5 markers. In a 5 by 5 by 5 cube, there are 4223 ways to put 7 markers, and so forth.

The two solutions for 6 markers in 6 by 6 are:

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