Here is a simulation approach:

use strict;
use warnings;

my $n = shift;
my $sum = 0;
my $count = 0;
while( $sum != ($n+1)*$n/2 ) {
    $sum = 1 + int( rand( $n**2 ) );
    $count++;
}
print "Sum of 1 to $n: $sum (found after $count iterations)\n";
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