I wrote a proof of theory script which uses this fixed strategy technique. I don't know if it satisfies all of the requirements of the challenge, but it does appear to work.
Each of the three fixed weighs results in either 0, 1, or -1. These weigh results are joined together to create a unique key for each ball/weight combination which are then used to generate a lookup table that can then be used to quickly find the oddball and its weight in any given set.
use strict;
use warnings;
my $ballfinder = My::Oddball->new(
[5,6,8,10], [7,9,11,12], # L/R set 1
[2,3,4,7], [5,6,11,12], # L/R set 2
[1,4,10,11], [2,5,7,8], # L/R set 3
) or die "Bogus ball arrangement";
# Now we have a valid table, let's use it to find some random oddballs
for (1..5){
my $weight = (0.9, 1.1)[ int(rand(2)) ];
my $oddball = (1..12)[ int(rand(12)) ];
my $balls = [ qw/ 0 1 1 1 1 1 1 1 1 1 1 1 1 / ];
$balls->[$oddball] = $weight;
print "Randomly chosen ball: $oddball, ".($weight < 1 ? 'light' : 'he
+avy')."\n";
print $ballfinder->detect($balls)."\n\n";
}
package My::Oddball;
sub new {
my ( $class, $L1, $R1, $L2, $R2, $L3, $R3 ) = @_;
my $table = { };
# Iterate through all possible combinations: balls 1-12, light & heavy
foreach my $weight qw/ 0.9 1.1 / {
for my $oddball (1..12){
my $balls = [ qw/ 0 1 1 1 1 1 1 1 1 1 1 1 1 / ];
$balls->[$oddball] = $weight;
my $key = canon_key(
balance( [ @$balls[@$L1] ], [ @$balls[@$R1] ] ),
balance( [ @$balls[@$L2] ], [ @$balls[@$R2] ] ),
balance( [ @$balls[@$L3] ], [ @$balls[@$R3] ] ),
);
# If the table is valid, we should get a unique key
# for each ball/weight combination
return if exists $table->{$key};
$table->{$key} = "Ball $oddball is ".($weight < 1 ? "light" : "he
+avy");
}
}
return bless { table => $table, sets => [ $L1, $R1, $L2, $R2, $L3, $R
+3 ] }, $class;
}
sub detect {
my ($self, $balls) = @_;
my ( $L1, $R1, $L2, $R2, $L3, $R3 ) = @{$self->{sets}};
my $key = canon_key (
balance( [ @$balls[@$L1] ], [ @$balls[@$R1] ] ),
balance( [ @$balls[@$L2] ], [ @$balls[@$R2] ] ),
balance( [ @$balls[@$L3] ], [ @$balls[@$R3] ] ),
);
return $self->{table}{$key};
}
sub canon_key { join( '_' => @_ ) }
sub balance {
my ( $setA, $setB ) = @_;
return weigh($setA) cmp weigh($setB);
}
sub weigh {
my ($set) = @_;
my $sum = 0;
$sum += $_ for @$set;
return $sum;
}
