choroba's Explore-all-possible-combinations mechanism works okay for smallish sets (max:32 or 64 depending upon your Perl), but will get very slow for anything much larger than 20 or so.

This will very quickly (less than 0.001 of a second) find a solution, if one exists, for sets of 100s or 1000s of elements. :

`#! perl -slw
use strict;
use Time::HiRes qw[ time ];
use List::Util qw[ sum ];
sub partition {
my $sum = sum @_;
return if $sum & 1;
$sum /= 2;
my @s = sort{ $b <=> $a } @_;
my @a;
my( $t, $n ) = ( 0, -1 );
$t + $s[$n] <= $sum and $t+= $s[$n] and push @a, $n while ++$n < @
+s and $t <= $sum;
@a = delete @s[ @a ];
@s = grep defined, @s;
return unless sum( @a ) == sum( @s );
return \@a, \@s;
}
our $N //= 64;
my( $a, $b ) = partition 1,3,5,7;
print "sum( @{ $a } ) == sum( @{ $b } )" if $a;
my @set = map int( rand 100 ), 1 .. $N;
my $start = time;
( $a, $b ) = partition @set;
printf "Took %f seconds\n", time() - $start;
if( $a ) {
printf "(%u) == sum( @{ $a } ) == sum( @{ $b } )\n", sum @$a;
}
else {
print "No solution existed for the $N element set @set";
}
`

A few runs:

`C:\test>1031699 -N=1e1
sum( 7 1 ) == sum( 5 3 )
Took 0.000042 seconds
No solution existed for the 1e1 elemet set 19 41 38 74 95 79 28 90 86
+87
C:\test>1031699 -N=1e1
sum( 7 1 ) == sum( 5 3 )
Took 0.000093 seconds
(184) == sum( 99 53 28 2 2 ) == sum( 91 48 24 11 10 )
C:\test>1031699 -N=1e2
sum( 7 1 ) == sum( 5 3 )
Took 0.000186 seconds
(2418) == sum( 99 98 97 97 96 95 95 93 90 89 88 88 84 84 82 82 82 81 8
+0 79 75 74 73 73 73 72 71 69 59 ) == sum( 67 67 66 65 64 62 61 60 58
+57 57 55 55 51 51 51 49 49 47 47 45 44 42 42 41 41 40 38 38 37 36 35
+34 34 34 33 33 32 31 31 29 28 28 27 27 27 26 26 24 24 23 23 21 20 20
+16 16 15 14 13 13 12 12 11 10 8 7 6 5 5 2 )
C:\test>1031699 -N=1e3
sum( 7 1 ) == sum( 5 3 )
Took 0.001379 seconds
(25172) == sum( 99 99 99 99 ... 71 71 71 70 70 70 33 0 0 0 ) == sum(
+70 70 70 70 70 70 ... 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 )
C:\test>1031699 -N=1e3
sum( 7 1 ) == sum( 5 3 )
Took 0.000053 seconds
No solution existed for the 1e3 elemet set 26 ... 93 10 10
C:\test>1031699 -N=1e4
sum( 7 1 ) == sum( 5 3 )
Took 0.013463 seconds
(247975) == sum( 99 99 ... 0 0 0 0 0 0 0 0 0 ) == sum( 70 70 ... 1 1 1
+ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 )
C:\test>1031699 -N=1e4
sum( 7 1 ) == sum( 5 3 )
Took 0.000238 seconds
No solution existed for the 1e4 elemet set 60 ...72 43 21 61 24 16
`

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