This is still a little raw. But I don't have time to do more with it, and you can see the thinking process. Note in particular the "is_guaran" field - that tracks the heart of why this spooky math works.

Give it a shot. Read this through to see how it works, run it, and follow through the explanation I gave earlier. This weekend I will try to catch up on any remaining confusion.

#! /usr/bin/perl -w
use strict;
use Getopt::Std;
use vars qw(@nums $opt_n $opt_N);
getopts("n:N:");
use Data::Dumper;
$Data::Dumper::Indent = 1;

if ($opt_n) {
  @nums = split /:/, $opt_n;
  unless (2 == @nums) {
    die "Need -n option to have an argument [number:number]\n";
  }
}
else {
  @nums = (ask_num(), ask_num());
}

unless ($opt_N) {
  print "How many times should I run this: ";
  $opt_N = <STDIN>;
  chomp($opt_N);
}

for (1..$opt_N) {
  my %info = run_experiment(@nums);
  print Dumper(\%info);
  <STDIN>;
}

sub ask_num {
  print "Please enter a number: ";
  my $num = <STDIN>;
  chomp($num);
  return $num;
}

sub flip_coin {
  0.5 > rand() ? 1 : 0;
}

sub guess_other_num {
  return (log(rand) * (flip_coin() ? 10 : -10 ));
}

sub run_experiment {
  my %status;
  @status{'low', 'high'} = sort {$a <=> $b} @_;
  if (flip_coin()) {
    $status{"gave_high"} = 1;
    @status{"guess_high", "aux"} = think_high($status{high});
  }
  else {
    $status{"gave_high"} = 0;
    @status{"guess_high", "aux"} = think_high($status{low});
  }
  $status{is_right} = ($status{guess_high} == $status{gave_high});
  if ($status{low} < $status{aux} and $status{aux} < $status{high}) {
    $status{is_guaran} = 1;
  }
  else {
    $status{is_guaran} = 0;
  }
  if (wantarray) {
    return %status;
  }
  else {
    return $status{is_right};
  }
}

sub think_high {
  my $num = shift;
  my $other_num = guess_other_num();
  my $guess =  $num > $other_num;
  return wantarray() ? ($guess, $other_num) : $guess;
}
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