#!/usr/bin/perl
use warnings;
use strict;

my @AoA = (
#    [11,29, 10,25],
    [15,35, 11,29],
    [15,15, 11,21],
    [10,25, 11,21],
    [15,35, 21,39],
    [25,40, 21,39],
    [21,11, 25,10],
    [15,15, 21,11],
    [35,35, 29,39],
    [29,39, 25,40],
    [35,15, 29,11],
    [25,10, 29,11],
    [40,25, 39,29],
    [35,35, 39,29],
    [39,21, 40,25],
    [35,15, 39,21],
);

my @line = @{ shift @AoA };
LINE:
while (@AoA) {
    for my $index (0 .. $#AoA) {
        for my $indices ( [0, 1, -2, -1, 0, 0 + @line],
                          [2, 3, -2, -1, 2, 0 + @line],
                          [0, 1,  0,  1, 0, 0],
                          [2, 3,  0,  1, 2, 0],
                        ) {
            if ($AoA[$index][$indices->[0]] == $line[$indices->[2]]
             && $AoA[$index][$indices->[1]] == $line[$indices->[3]]) {

                my $new = splice @AoA, $index, 1;
                splice @$new, $indices->[4], 2;
                splice @line, $indices->[5], 0, @$new;
                next LINE
            }
        }
    }
    die "No solution for @line.\n";
}

print "@line\n";
[download]

Thanks choroba, I'm amazed how quickly you worked that. I will try to understand your code to solve my next problem but in the meantime I'll tell you what it is: when there is more than one line the code seems to go into an infinite loop - how to break out of the loop in that situation? (and sorry for being a pest)

my @AoA = (
    [11,29, 10,25],
    [15,35, 11,29],
    [15,15, 11,21],
    [10,25, 11,21],
    [15,35, 21,39],
    [25,40, 21,39],
    [21,11, 25,10],
    [15,15, 21,11],
    [35,35, 29,39],
    [29,39, 25,40],
    [35,15, 29,11],
    [25,10, 29,11],
    [40,25, 39,29],
    [35,35, 39,29],
    [39,21, 40,25],
    [35,15, 39,21],

    [61,29, 60,25],
    [60,25, 61,21],
);
[download]

My version of the code stops with the following error:

No solution for 39 21 40 25 39 29 35 35 29 39 25 40 21 39 15 35 11 29 
+10 25 11 21 15 15 21 11 25 10 29 11 35 15 39 21.
[download]

لսႽ† ᥲᥒ⚪⟊Ⴙᘓᖇ Ꮅᘓᖇ⎱ Ⴙᥲ𝇋ƙᘓᖇ