Routes

[cinelli]

.---.---.---A---.---.---.---.
|   |   |   |   |   |   |   |
.---D---.---.---.---B---.---.
|   |   |   |   |   |   |   |
.---.---.---.---.---.---.---.
|   |   |   |   |   |   |   |
.---.---.---.---.---.---.---.
|   |   |   |   |   |   |   |
.---.---.---.---C---.---.---D
|   |   |   |   |   |   |   |
.---.---E---.---.---.---.---E
|   |   |   |   |   |   |   |
.---.---.---.---.---B---.---.
|   |   |   |   |   |   |   |
C---.---.---A---.---.---.---.

In this puzzle you are asked to connect A to A, B to B, etc so that
. the routes follow the lines of the grid
. they do not cross
. no route goes along the line of another route
. no route passes through the start or end of another route

Cinelli

[UncleEbenezer]

The easy solution:

a a a A d d d d
a D d d d B b d
a a a a a a b d
c c c c c a b d
c a a a C a b D
c a E a a a b E
c a e e e B b e
C a a A e e e e

(approach that made it easy: note that A-A spans the entire grid, and therefore must take the path it does around E,C and D from bottom to top, then fit the others around it)


The more interesting issue: generalise. Is there a solution for every placement of five pairs of endpoints on an 8x8 grid (answer is clearly No, but are there rules/criteria)? Likewise, what are the rules/criteria for a solution to be unique, or require every grid point to be used?

[UncleEbenezer]

The more interesting issue: generalise. Is there a solution for every placement of five pairs of endpoints on an 8x8 grid (answer is clearly No, but are there rules/criteria)? Likewise, what are the rules/criteria for a solution to be unique, or require every grid point to be used?

Scrub that. I don't think there's anything really interesting in generalising. It's not exactly Königsberg.

[SteelCamel]

note that A-A spans the entire grid, and therefore must take the path it does around E,C and D from bottom to top, then fit the others around it

Yes, I realised the same thing in a slightly different way - A cuts the board completely in two, so each of the other lines must have both ends in the same one of the halves created by the A line. C, D, E all have one end on the perimeter, so the A line must go around the other end of each. The only way to avoid the top D is to use the top left corner, and while there are two lines left of E, using the left one will block the bottom C. Once you factor in the need to go right of the other C, the A-A line is almost fixed - and it's quite obvious which routes will block other lines, leaving only one option.
C is now obvious, as it's completely surrounded by A. E only has two options left, one of which obviously blocks B. And once that's in, B takes the remaining lines.

[cinelli]

Yes, UncleEbenezer has a solution within an hour and a half. I like how the routes are unique and that every grid point is covered.

Cinelli