The New Sudoku Players' Forum

[lonogod]

Hello all,

I've got a weird one for you. I'm working on a variant and have a conundrum.

It's an 8x8 grid puzzle. The numbers 1 to 8 MUST appear exactly once in each row and column. There are no regions though. It is strictly just rows and columns. All digits of one number (as in all the 3s) MUST be connected to each other by a knights move. I've been working on it for hours and can't find a solution that works for even one digit. Does anyone know if it's possible?

Thank you,

Lono

[HATMAN]

I think not, but check with Bill as chess interactions are his specialist area.

[999_Springs]

yes, there is a solution

found one by hand in about 20 minutes of shuffling knights around with my finger on lichess's board setup editor on my phone. it's a fun little exercise

step 1: look for patterns with 1 digit

i started by looking for patterns with 1 digit where all the digits are a knight's move apart and there is only 1 knight in each row and column. i did this manually on lichess's board setup editor

i found 5 of them. there may well be more, but i stopped here. note that all these patterns have various rotations and reflections, which doesn't change the pattern

Code: Select all

pattern A:
1.......
..1.....
.1......
...1....
.....1..
.......1
....1...
......1.

pattern B:
1.......
..1.....
....1...
......1.
.......1
.....1..
...1....
.1......

pattern C:
.1......
1.......
..1.....
....1...
...1....
.....1..
.......1
......1.

pattern D:
.1......
1.......
..1.....
......1.
...1....
.....1..
.......1
....1...

pattern E:
..1.....
.1......
...1....
1.......
....1...
.......1
.....1..
......1.

step 2: combine 2 of them in the grid trying to avoid symmetric digits

when trying to find 1-digit patterns, it is noticeable how restrictive the given conditions are, so it seems likely that to find a solution for a full grid it's worth exploiting some of the symmetries of the grid, instead of trying to fit 8 of these patterns in one at a time.

so what this means is it's worth trying to fit 2 of these patterns into the grid so that no pair of horizontally or vertically symmetric cells contain a pair of digits. (example, using chess notation: b1 and g1 are row-symmetric; a4 and a5 are column-symmetric) this is because we can then fit 4 copies of this pattern using reflections into a full solution grid without any clashes.

trying to find 2 of these to fit together was done pretty much by guessing until i found one that worked, but there are some combinations that can be instantly ruled out, such as any two patterns that contain a corner cell can't be used here because then you'd have 2 corner cells, and corners are invariant under rotation/reflection so you'd end up having to fit 8 corner cells into the grid after you fit 4 copies of this pattern in, but there are only 4 corners. the same logic goes for the centre cells, and the diagonal cells (b2,b7,g2,g7) and (c3,c6,f3,f6)

after a while i found that diagonally-reflected pattern B combines with pattern E to get the combination i want:

Code: Select all

pattern E in 1s + pattern B diagonally-reflected in 2s:
2.1.....
.1.....2
.2.1....
1.....2.
..2.1...
.....2.1
...2.1..
....2.1.

this works because no row or column contains a 1 and 2 that are symmetric with respect to the centre line of the grid, and the diagonal cells are only represented once each for each equivalence class

step 3: apply reflections

this is the easy part. apply all combinations of horizontal and vertical reflections to the above partially completed grid and you get a solution

Code: Select all

25187364
41578632
52813746
14756823
68231475
37465281
86324157
73642518

there may be other solutions but i stop here

[lonogod]

I appreciate the thorough examination, 999_Springs! Unfortunately, none of those solutions satisfy the requirement that all digits be a knights move apart. In every one of the solutions you posted, there are 1s touching at a corner.

I came up with many instances of that as well...to my dismay. I played with it more and just can't find a solution. If someone finds one, let me know. I will sing your praises from on high!

[HATMAN]

Ionogod

That is how I interpreted it so did not get a solution; however, if you check the exact wording of your requirement, Springs is right as you do not exclude king move, or indeed long-knight (ogeima)

[lonogod]

Very good point! I didn't think about that.

It feels like it's easier to make a puzzle with restrictions rather than something like this. With restrictions/constraints it's clear what is, and isn't, allowed. Not with this though. Lesson learned.

Thank you both for your help!

Lono