I like that sort of problems. If only to construct recursive algorithms for them.

Here is a recursive procedure to construct and print out all paths of given length from a last visited node, provided a head path is already given and we know what nodes are left for a potential visit:

use strict;
use warnings;

my %adj = (
    1 => [2,5,6],
    2 => [1,3,5,6,7],
    3 => [2,4,6,7,8],
    4 => [3,7,8],
    5 => [1,2,6,9,10],
    6 => [1,2,3,5,7,9,10,11],
    7 => [2,3,4,6,8,10,11,12],
    8 => [3,4,7,11,12],
    9 => [5,6,10,13,14],
    10 => [5,6,7,9,11,13,14,15],
    11 => [6,7,8,10,12,14,15,16],
    12 => [7,8,11,15,16],
    13 => [9,10,14],
    14 => [9,10,11,13,15],
    15 => [10,11,12,14,16],
    16 => [11,12,15]
);

sub boggle {
    my ($sofar, $last, $remains, $lentogo) = @_;
    return print "@$sofar\n" unless $lentogo;
    return () unless @$remains;
    my (@right, @left) = @$remains;
    while (my $next = shift @right) {
        boggle([@$sofar, $next], $next, [@left, @right], $lentogo-1) i
+f grep {$next==$_} @{$adj{$last}};
        push(@left, $next);
    }
}

for my $len (3 .. 6) {    # len 3 .. 6
    for my $start (1 .. 16) {    # all starting points
        boggle([$start],$start,[1..$start-1,$start+1..16], $len);
    }
};
[download]

the last lines show the application to print out all paths of lengths 3 .. 6 starting anywhere on the 16 grid points.

Edit: fixed a bug in the while loop.