Do the (Forum) Dice Really Hate Me?

[Oladon]

Introduction
I've heard it from others, and I've said it myself... but do the dice really hate us? I haven't seen any analysis of the Paizo forum dice, so I've decided to do my own. Let's find out if the dice really are biased against us.

Method
The plan is to roll 10000d20, plug the results into a script (or a couple of scripts), and put the results below (in the same post). The script (which I'm writing now) should calculate the mean, mode, and standard deviation of the set, as well as the frequency of each roll. A fancy ASCII graph of the last item would be a big plus (hurray for gnuplot).

I'm composing the bulk of this post before previewing, in order to preserve the dice rolls (in case I can't actually post until tomorrow or later) as well as to avoid posting a thread that (for some period of time) consists of nothing but 10000 rolls.

dicestats.pl:

Quote:

#!/usr/bin/perl

use strict;
use warnings;
use Statistics::Basic qw(:all);

my @allrolls;
my @nums;
my %dice;

while (<STDIN>) {
my @rolls = ($_ =~ /\(([0-9 ,]+)\)/g);

foreach (@rolls) {
for my $roll (split(", ")) {
push(@allrolls,$roll);
$dice{$roll}++;
}
}
}

@nums = @{[sort {$a<=>$b} keys %dice]};

foreach (@nums) {
print "$_\t$dice{$_}\n";
}

print "\n";
print "Mean:\t" . mean(@allrolls) . "\n";
print "Median:\t" . median(@allrolls) . "\n";
print "StdDev:\t" . stddev(@nums) . "\n";

(I'd rather have used Common Lisp for this, but it's late and it was faster to do it in Perl.)

dice.plt:

Quote:

set term png

set font "Arial"
set output "dicechart.png"
set xrange [0:21]
set yrange [*:*]
set xlabel "Value"
set ylabel "Count"
plot "dice.dat" using 1:2 title "" with boxes

Loading up gnuplot and running:

Quote:

load "dice.plt"

... will produce the graph we want.

Execution
At this point I'll preview the post, copy the dice rolls into a file, and run the scripts. I'll then copy the output, add it as the "Results" section, submit the post, and let the numbers speak for themselves (though I'll be happy to answer any questions that come up).

Results
Value: Count
1: 508
2: 513
3: 541
4: 469
5: 495
6: 485
7: 501
8: 502
9: 474
10: 519
11: 507
12: 514
13: 476
14: 493
15: 468
16: 479
17: 495
18: 520
19: 511
20: 530

Mean: 10.5
Median: 10
StdDev: 5.77

Sorry everyone, no ASCII chart -- I'd forgotten that the forums don't allow multiple consecutive spaces. Anyway, here's the frequency chart in png form.

Lastly, I performed a Chi-squared test on the data, with the following results.

graphpad.com/quickcalcs/chisquared1 wrote:

P value and statistical significance:

Chi squared equals 15.896 with 19 degrees of freedom.
The two-tailed P value equals 0.6642
By conventional criteria, this difference is considered to be not statistically significant.

Lots of dice:

1000:
100d20 ⇒ (7, 18, 20, 4, 17, 11, 16, 20, 4, 10, 11, 20, 8, 3, 4, 10, 11, 5, 19, 7, 4, 1, 15, 17, 13, 18, 15, 9, 14, 13, 20, 15, 14, 17, 3, 11, 15, 17, 1, 8, 12, 7, 3, 18, 10, 17, 15, 16, 9, 3, 2, 13, 16, 4, 6, 18, 16, 16, 12, 13, 6, 7, 7, 20, 20, 5, 7, 16, 3, 3, 19, 14, 14, 15, 6, 17, 3, 20, 14, 14, 15, 19, 17, 16, 16, 6, 17, 6, 3, 19, 2, 12, 13, 8, 14, 3, 3, 19, 16, 10) = 1155
100d20 ⇒ (5, 5, 8, 2, 15, 14, 7, 13, 20, 8, 11, 15, 8, 3, 6, 9, 18, 1, 16, 19, 11, 6, 12, 17, 13, 20, 18, 15, 8, 14, 19, 15, 14, 11, 6, 20, 12, 18, 15, 1, 8, 13, 7, 6, 14, 15, 11, 15, 11, 14, 14, 12, 13, 13, 18, 1, 10, 3, 19, 4, 7, 3, 11, 20, 6, 13, 2, 4, 19, 9, 1, 18, 12, 2, 10, 1, 16, 17, 11, 7, 14, 17, 8, 13, 3, 13, 20, 4, 1, 17, 1, 14, 4, 13, 20, 12, 20, 18, 3, 15) = 1108
100d20 ⇒ (5, 15, 5, 3, 19, 17, 16, 8, 10, 15, 6, 9, 9, 10, 2, 2, 3, 5, 15, 9, 6, 13, 16, 13, 16, 16, 6, 16, 4, 20, 3, 8, 19, 11, 15, 1, 18, 5, 20, 1, 19, 9, 1, 8, 4, 13, 4, 19, 15, 6, 14, 17, 20, 1, 15, 2, 17, 13, 6, 4, 12, 1, 20, 15, 15, 1, 8, 4, 9, 17, 4, 10, 9, 14, 14, 18, 20, 10, 6, 5, 16, 15, 1, 16, 12, 15, 3, 3, 12, 11, 3, 3, 5, 10, 19, 20, 19, 14, 15, 15) = 1056
100d20 ⇒ (10, 6, 18, 13, 19, 19, 16, 20, 12, 1, 7, 6, 7, 16, 1, 5, 9, 17, 4, 10, 6, 3, 17, 3, 17, 4, 7, 15, 17, 9, 4, 11, 8, 9, 10, 12, 9, 10, 12, 13, 17, 5, 4, 16, 15, 2, 13, 5, 4, 17, 2, 2, 9, 4, 9, 17, 4, 9, 6, 20, 19, 10, 16, 16, 5, 16, 14, 18, 3, 18, 8, 7, 15, 12, 1, 14, 19, 12, 13, 20, 8, 16, 8, 11, 7, 18, 10, 12, 9, 14, 10, 8, 8, 10, 17, 6, 1, 4, 8, 7) = 1040
100d20 ⇒ (9, 20, 19, 2, 1, 9, 14, 12, 5, 11, 9, 13, 10, 2, 11, 9, 14, 6, 16, 2, 5, 11, 8, 5, 19, 12, 10, 20, 9, 17, 13, 8, 7, 10, 13, 4, 7, 19, 2, 2, 15, 6, 6, 5, 9, 15, 19, 5, 4, 19, 7, 11, 3, 16, 16, 17, 14, 1, 20, 6, 5, 3, 13, 15, 1, 12, 15, 3, 2, 3, 6, 8, 15, 13, 3, 7, 6, 15, 4, 1, 3, 6, 8, 1, 8, 16, 2, 3, 15, 4, 4, 14, 8, 6, 20, 6, 11, 5, 7, 19) = 925
100d20 ⇒ (6, 9, 6, 19, 16, 1, 4, 16, 20, 10, 18, 3, 13, 2, 14, 10, 4, 17, 13, 15, 20, 2, 15, 6, 3, 18, 10, 11, 13, 13, 11, 14, 2, 17, 10, 2, 1, 13, 14, 4, 8, 12, 16, 18, 5, 19, 5, 10, 14, 5, 10, 17, 16, 2, 5, 7, 17, 14, 2, 8, 15, 8, 15, 2, 19, 15, 18, 5, 4, 10, 16, 7, 14, 8, 3, 10, 14, 18, 10, 11, 6, 6, 17, 18, 11, 16, 16, 8, 11, 4, 13, 12, 10, 4, 2, 19, 12, 7, 13, 16) = 1068
100d20 ⇒ (8, 3, 10, 4, 20, 20, 3, 3, 10, 11, 19, 7, 1, 10, 14, 8, 13, 13, 6, 2, 16, 2, 19, 4, 4, 6, 19, 19, 10, 1, 20, 19, 17, 12, 16, 8, 1, 6, 8, 9, 16, 4, 6, 18, 18, 7, 17, 17, 9, 16, 10, 1, 1, 14, 11, 6, 18, 16, 9, 12, 20, 17, 7, 10, 7, 6, 7, 18, 11, 19, 5, 9, 3, 3, 15, 17, 18, 7, 9, 12, 17, 14, 12, 4, 5, 2, 19, 20, 4, 17, 9, 6, 7, 10, 14, 13, 6, 5, 18, 3) = 1052
100d20 ⇒ (3, 12, 8, 17, 17, 4, 20, 8, 5, 7, 3, 13, 8, 6, 17, 12, 9, 3, 9, 8, 18, 14, 17, 19, 2, 2, 20, 13, 14, 15, 17, 15, 4, 11, 15, 3, 17, 6, 19, 11, 14, 7, 11, 12, 14, 8, 4, 20, 8, 1, 18, 20, 9, 18, 4, 8, 18, 11, 19, 13, 2, 6, 9, 18, 2, 13, 19, 18, 11, 13, 4, 9, 10, 3, 20, 2, 15, 17, 17, 10, 7, 12, 2, 7, 11, 2, 14, 12, 2, 11, 13, 5, 4, 18, 13, 3, 10, 14, 3, 14) = 1073
100d20 ⇒ (7, 14, 1, 3, 4, 6, 16, 5, 11, 8, 20, 16, 1, 8, 4, 17, 10, 4, 16, 17, 15, 12, 19, 5, 11, 18, 13, 11, 20, 6, 13, 3, 16, 20, 4, 9, 1, 14, 17, 7, 10, 13, 18, 2, 19, 9, 13, 7, 20, 3, 17, 16, 9, 15, 4, 17, 4, 2, 1, 13, 7, 20, 19, 2, 8, 15, 12, 14, 20, 10, 20, 15, 12, 9, 10, 10, 17, 15, 5, 11, 18, 12, 1, 9, 11, 20, 14, 10, 1, 3, 2, 6, 8, 18, 6, 9, 18, 16, 13, 11) = 1091
100d20 ⇒ (13, 13, 13, 1, 1, 8, 4, 5, 14, 13, 4, 3, 17, 18, 5, 7, 13, 7, 11, 6, 11, 8, 7, 14, 18, 3, 17, 14, 2, 11, 4, 7, 18, 9, 4, 17, 17, 17, 17, 14, 18, 17, 18, 6, 11, 14, 7, 15, 20, 12, 20, 11, 10, 9, 20, 8, 2, 10, 10, 12, 12, 14, 20, 8, 17, 12, 12, 7, 17, 8, 12, 11, 2, 5, 2, 16, 6, 2, 14, 16, 8, 11, 20, 4, 2, 19, 13, 1, 7, 16, 15, 19, 8, 4, 17, 7, 14, 7, 14, 11) = 1085
2000:
100d20 ⇒ (7, 6, 17, 13, 10, 18, 10, 9, 4, 1, 7, 7, 12, 12, 1, 5, 7, 14, 19, 9, 2, 10, 9, 15, 19, 18, 16, 15, 10, 2, 14, 6, 8, 2, 16, 12, 5, 9, 8, 1, 11, 12, 3, 20, 12, 6, 9, 6, 5, 12, 5, 19, 19, 6, 8, 5, 3, 2, 19, 18, 8, 13, 4, 7, 20, 4, 14, 7, 16, 8, 12, 16, 14, 6, 7, 19, 2, 7, 9, 11, 5, 11, 15, 4, 1, 13, 16, 7, 17, 3, 20, 13, 5, 16, 11, 10, 19, 5, 10, 11) = 1001
100d20 ⇒ (14, 12, 16, 11, 16, 8, 15, 11, 7, 9, 17, 1, 17, 3, 6, 16, 16, 3, 12, 7, 13, 14, 19, 11, 13, 19, 8, 11, 14, 8, 20, 7, 12, 13, 2, 17, 19, 1, 17, 5, 6, 13, 5, 10, 6, 19, 1, 14, 5, 4, 18, 16, 7, 2, 18, 1, 15, 14, 15, 18, 11, 14, 9, 8, 9, 15, 19, 14, 18, 17, 7, 20, 4, 20, 4, 12, 5, 7, 8, 11, 12, 20, 8, 10, 19, 20, 3, 19, 7, 5, 12, 7, 3, 6, 20, 18, 18, 2, 19, 10) = 1137
100d20 ⇒ (5, 9, 20, 9, 20, 4, 15, 10, 12, 20, 8, 7, 4, 2, 18, 1, 14, 3, 5, 1, 10, 7, 11, 19, 3, 16, 9, 17, 18, 17, 10, 9, 19, 4, 4, 12, 3, 6, 6, 12, 8, 17, 11, 5, 5, 20, 2, 1, 18, 19, 9, 14, 16, 18, 7, 17, 1, 9, 16, 20, 14, 18, 3, 7, 18, 3, 16, 20, 20, 14, 17, 7, 10, 11, 15, 14, 16, 17, 18, 2, 4, 9, 9, 18, 20, 8, 1, 1, 7, 12, 2, 16, 5, 7, 15, 9, 1, 9, 10, 14) = 1069
100d20 ⇒ (9, 15, 20, 5, 4, 1, 20, 14, 5, 16, 7, 1, 1, 12, 11, 19, 7, 2, 9, 3, 3, 10, 10, 12, 12, 18, 14, 1, 12, 14, 5, 10, 10, 3, 14, 12, 8, 15, 8, 8, 11, 2, 1, 17, 7, 13, 19, 11, 4, 5, 3, 7, 11, 18, 20, 5, 14, 9, 12, 5, 19, 15, 2, 20, 20, 1, 4, 10, 20, 10, 5, 9, 15, 12, 16, 2, 7, 19, 15, 15, 7, 19, 14, 17, 5, 5, 12, 7, 4, 13, 1, 16, 18, 7, 11, 19, 14, 12, 18, 13) = 1042
100d20 ⇒ (14, 13, 14, 17, 20, 8, 7, 4, 18, 19, 1, 14, 18, 17, 20, 14, 3, 10, 11, 1, 14, 10, 16, 6, 13, 11, 15, 15, 9, 14, 16, 16, 9, 1, 8, 3, 11, 11, 12, 11, 14, 7, 16, 9, 2, 3, 11, 20, 16, 5, 20, 20, 7, 20, 10, 20, 4, 6, 10, 10, 1, 9, 17, 12, 16, 9, 17, 12, 7, 9, 9, 11, 17, 14, 13, 4, 6, 8, 20, 17, 15, 19, 4, 16, 12, 3, 3, 20, 3, 12, 17, 10, 17, 19, 7, 11, 17, 13, 15, 11) = 1166
100d20 ⇒ (9, 14, 7, 5, 2, 1, 6, 6, 4, 6, 2, 16, 18, 11, 9, 4, 4, 11, 19, 16, 6, 2, 14, 3, 2, 18, 8, 18, 11, 6, 18, 14, 8, 2, 17, 12, 4, 6, 5, 8, 12, 4, 7, 1, 16, 17, 9, 9, 11, 8, 13, 7, 5, 20, 15, 15, 15, 11, 12, 10, 17, 11, 16, 14, 17, 8, 20, 15, 13, 17, 14, 10, 7, 1, 9, 16, 12, 2, 3, 15, 10, 5, 11, 17, 16, 20, 8, 8, 4, 6, 9, 2, 5, 1, 11, 2, 20, 18, 9, 12) = 1000
100d20 ⇒ (15, 17, 15, 3, 8, 7, 17, 14, 20, 18, 20, 13, 1, 16, 3, 18, 11, 16, 8, 6, 19, 7, 1, 18, 18, 3, 18, 12, 4, 2, 2, 11, 12, 6, 2, 4, 16, 20, 7, 5, 7, 19, 7, 12, 19, 7, 6, 2, 18, 12, 2, 7, 1, 2, 1, 2, 20, 13, 14, 1, 13, 14, 11, 18, 11, 13, 14, 17, 11, 9, 10, 7, 5, 20, 16, 12, 9, 8, 11, 13, 18, 7, 5, 20, 9, 14, 9, 14, 11, 20, 5, 20, 8, 4, 6, 2, 19, 9, 7, 1) = 1055
100d20 ⇒ (8, 3, 19, 3, 10, 2, 7, 4, 18, 17, 4, 12, 1, 13, 18, 16, 16, 1, 17, 7, 11, 4, 1, 11, 1, 13, 13, 16, 2, 2, 18, 1, 7, 16, 7, 9, 2, 11, 1, 9, 15, 3, 15, 6, 4, 11, 11, 5, 17, 13, 17, 12, 11, 7, 5, 11, 19, 1, 9, 15, 15, 5, 14, 20, 5, 1, 7, 17, 13, 2, 20, 15, 4, 6, 5, 6, 3, 8, 15, 8, 6, 12, 12, 8, 5, 15, 14, 19, 2, 20, 18, 19, 6, 17, 7, 16, 14, 18, 18, 20) = 1013
100d20 ⇒ (11, 8, 3, 9, 15, 2, 1, 14, 11, 16, 18, 17, 7, 12, 15, 17, 20, 16, 19, 17, 3, 6, 4, 20, 11, 15, 18, 11, 13, 18, 7, 16, 12, 20, 7, 16, 11, 3, 1, 18, 14, 20, 13, 19, 17, 6, 6, 18, 4, 17, 4, 2, 14, 19, 14, 18, 2, 14, 1, 9, 11, 7, 17, 7, 8, 11, 6, 20, 14, 1, 19, 12, 2, 8, 19, 3, 3, 19, 8, 8, 1, 2, 5, 9, 10, 11, 5, 6, 19, 11, 18, 11, 7, 20, 5, 4, 19, 12, 19, 16) = 1122
100d20 ⇒ (12, 12, 4, 1, 13, 2, 4, 5, 10, 17, 7, 13, 20, 8, 19, 15, 8, 2, 1, 6, 2, 13, 13, 1, 18, 20, 19, 3, 1, 7, 6, 5, 18, 11, 16, 2, 13, 12, 9, 16, 6, 5, 1, 3, 19, 18, 1, 16, 7, 7, 18, 12, 3, 12, 20, 7, 11, 17, 6, 16, 2, 13, 4, 19, 12, 4, 3, 2, 20, 14, 4, 9, 2, 6, 15, 8, 11, 20, 11, 7, 16, 8, 11, 7, 2, 1, 17, 8, 3, 5, 7, 19, 6, 5, 12, 9, 17, 17, 4, 17) = 966
3000:
100d20 ⇒ (8, 11, 16, 19, 14, 10, 1, 18, 20, 15, 2, 13, 14, 19, 6, 3, 7, 11, 20, 17, 9, 17, 3, 12, 1, 12, 14, 20, 5, 12, 8, 11, 15, 3, 14, 3, 12, 12, 10, 4, 18, 19, 4, 18, 14, 5, 9, 4, 14, 6, 3, 13, 10, 13, 4, 4, 19, 1, 10, 19, 18, 3, 3, 4, 7, 11, 3, 13, 17, 4, 7, 10, 9, 14, 15, 13, 8, 3, 17, 2, 19, 19, 4, 9, 2, 1, 6, 10, 9, 5, 13, 3, 13, 7, 9, 9, 14, 14, 16, 1) = 1012
100d20 ⇒ (3, 13, 20, 1, 14, 18, 7, 8, 11, 15, 17, 6, 18, 14, 18, 5, 11, 9, 19, 18, 1, 2, 2, 8, 16, 16, 10, 8, 18, 13, 5, 18, 6, 16, 17, 19, 7, 11, 10, 12, 3, 14, 19, 6, 19, 10, 1, 1, 4, 13, 17, 12, 12, 15, 5, 14, 15, 14, 1, 11, 2, 3, 8, 12, 5, 18, 6, 2, 13, 4, 14, 8, 17, 4, 5, 8, 6, 11, 6, 12, 11, 5, 9, 16, 7, 1, 3, 4, 13, 17, 11, 9, 9, 20, 7, 17, 5, 5, 2, 17) = 1018
100d20 ⇒ (9, 1, 18, 2, 1, 20, 12, 19, 20, 13, 12, 3, 16, 1, 11, 6, 20, 6, 10, 3, 11, 9, 4, 18, 17, 5, 1, 7, 5, 14, 15, 8, 13, 3, 19, 6, 2, 8, 12, 15, 19, 17, 11, 13, 12, 12, 2, 11, 8, 14, 16, 6, 12, 17, 4, 10, 1, 17, 18, 19, 14, 13, 15, 15, 6, 5, 8, 17, 20, 7, 12, 5, 16, 7, 19, 3, 19, 8, 12, 10, 16, 4, 5, 18, 5, 7, 10, 18, 5, 5, 15, 4, 4, 6, 2, 6, 13, 8, 1, 15) = 1032
100d20 ⇒ (20, 20, 2, 7, 4, 18, 9, 1, 20, 19, 11, 7, 2, 12, 11, 11, 10, 5, 11, 16, 13, 4, 10, 19, 5, 17, 10, 6, 12, 11, 14, 12, 6, 17, 4, 15, 9, 6, 16, 10, 12, 10, 8, 18, 4, 15, 4, 17, 18, 1, 3, 14, 17, 2, 18, 16, 20, 2, 17, 13, 13, 6, 8, 19, 5, 12, 11, 19, 4, 11, 20, 3, 10, 9, 7, 19, 8, 4, 20, 3, 1, 9, 18, 3, 8, 20, 20, 10, 15, 20, 4, 5, 14, 18, 13, 9, 6, 10, 6, 1) = 1082
100d20 ⇒ (6, 6, 10, 4, 15, 13, 19, 6, 8, 11, 13, 13, 16, 2, 8, 7, 20, 9, 14, 18, 18, 9, 3, 9, 1, 18, 8, 6, 8, 4, 11, 20, 13, 7, 9, 12, 2, 8, 13, 12, 10, 9, 9, 13, 11, 16, 20, 18, 8, 14, 7, 4, 10, 11, 6, 13, 17, 11, 2, 9, 16, 20, 16, 9, 3, 5, 13, 8, 16, 7, 9, 15, 8, 14, 7, 17, 13, 3, 7, 4, 4, 18, 7, 3, 3, 8, 20, 1, 17, 8, 9, 19, 13, 19, 10, 6, 18, 16, 14, 20) = 1070
100d20 ⇒ (4, 5, 7, 9, 9, 16, 11, 18, 6, 7, 17, 17, 20, 9, 19, 12, 20, 18, 15, 5, 19, 5, 16, 11, 16, 12, 17, 5, 4, 17, 13, 7, 7, 12, 19, 13, 12, 9, 2, 11, 19, 4, 10, 14, 17, 20, 17, 7, 18, 10, 17, 10, 18, 7, 3, 15, 4, 9, 16, 12, 19, 11, 9, 14, 16, 15, 16, 19, 15, 5, 15, 13, 8, 2, 9, 10, 17, 6, 3, 7, 8, 12, 2, 11, 1, 9, 18, 17, 14, 5, 13, 13, 1, 2, 10, 13, 16, 5, 12, 3) = 1132
100d20 ⇒ (17, 15, 10, 4, 4, 17, 4, 9, 17, 20, 17, 5, 2, 8, 12, 19, 18, 12, 2, 1, 2, 12, 14, 15, 9, 11, 9, 15, 7, 16, 13, 18, 18, 3, 12, 15, 12, 20, 17, 19, 17, 17, 14, 5, 18, 14, 5, 12, 12, 15, 6, 20, 3, 1, 4, 10, 18, 17, 20, 17, 4, 10, 19, 2, 17, 10, 3, 15, 11, 17, 16, 16, 1, 18, 5, 14, 17, 10, 11, 2, 15, 14, 3, 18, 5, 5, 19, 19, 10, 5, 17, 7, 11, 4, 8, 17, 4, 19, 17, 4) = 1154
100d20 ⇒ (18, 6, 15, 4, 13, 5, 13, 1, 13, 16, 3, 3, 3, 2, 4, 3, 9, 17, 14, 20, 16, 10, 12, 8, 5, 15, 18, 14, 2, 3, 2, 17, 20, 16, 16, 16, 6, 9, 15, 18, 14, 19, 20, 12, 3, 19, 16, 10, 3, 11, 18, 19, 15, 8, 15, 9, 14, 5, 2, 9, 11, 17, 20, 11, 18, 15, 17, 19, 14, 3, 5, 16, 20, 5, 13, 14, 1, 11, 6, 3, 6, 16, 7, 15, 13, 12, 16, 10, 9, 11, 4, 6, 3, 17, 13, 14, 15, 8, 8, 5) = 1108
100d20 ⇒ (14, 18, 14, 4, 5, 4, 9, 4, 16, 12, 19, 4, 2, 12, 5, 7, 9, 14, 19, 18, 10, 16, 4, 14, 6, 20, 8, 13, 16, 13, 13, 2, 3, 3, 18, 18, 16, 10, 17, 2, 20, 14, 9, 7, 4, 15, 20, 18, 15, 12, 2, 10, 18, 14, 17, 19, 4, 13, 14, 9, 8, 18, 2, 18, 3, 16, 6, 5, 11, 9, 5, 8, 17, 4, 3, 9, 16, 9, 11, 5, 19, 16, 15, 13, 17, 8, 12, 7, 5, 9, 10, 5, 12, 9, 17, 1, 7, 12, 1, 14) = 1077
100d20 ⇒ (14, 2, 16, 10, 3, 10, 14, 1, 9, 14, 8, 1, 5, 18, 9, 9, 14, 6, 13, 9, 17, 15, 8, 11, 16, 13, 19, 12, 12, 8, 19, 8, 14, 9, 16, 12, 10, 5, 8, 12, 20, 1, 13, 15, 9, 17, 8, 7, 6, 11, 20, 4, 8, 15, 8, 17, 11, 18, 12, 1, 16, 11, 17, 1, 19, 6, 15, 15, 15, 4, 11, 18, 20, 20, 2, 10, 15, 17, 13, 2, 8, 20, 13, 19, 6, 2, 10, 19, 9, 11, 4, 10, 1, 2, 6, 18, 20, 13, 10, 6) = 1104
4000:
100d20 ⇒ (16, 11, 6, 14, 14, 1, 1, 1, 7, 7, 15, 8, 17, 10, 3, 15, 7, 19, 15, 2, 7, 5, 14, 18, 5, 4, 14, 10, 13, 10, 13, 9, 2, 3, 18, 2, 14, 10, 20, 18, 1, 6, 10, 18, 5, 11, 7, 13, 12, 11, 14, 18, 13, 13, 19, 17, 2, 9, 9, 3, 19, 20, 15, 20, 1, 16, 10, 19, 2, 17, 16, 1, 17, 3, 19, 20, 18, 4, 1, 2, 3, 9, 12, 15, 10, 1, 11, 15, 10, 10, 15, 10, 18, 1, 2, 11, 5, 16, 18, 16) = 1057
100d20 ⇒ (5, 15, 13, 4, 11, 5, 12, 17, 7, 17, 1, 13, 1, 17, 4, 12, 19, 6, 2, 17, 19, 18, 1, 18, 1, 11, 8, 6, 19, 9, 14, 3, 9, 12, 15, 5, 3, 12, 3, 10, 2, 14, 18, 18, 6, 8, 5, 10, 7, 16, 8, 9, 4, 14, 19, 13, 9, 20, 5, 6, 1, 3, 19, 4, 6, 8, 9, 18, 17, 1, 12, 10, 15, 3, 14, 1, 3, 1, 2, 12, 3, 7, 12, 17, 8, 12, 19, 16, 13, 10, 11, 1, 20, 9, 19, 1, 5, 7, 4, 9) = 967
100d20 ⇒ (10, 15, 13, 10, 11, 16, 3, 6, 14, 9, 20, 16, 2, 3, 11, 12, 17, 19, 15, 18, 14, 10, 17, 5, 13, 19, 13, 12, 18, 17, 12, 1, 16, 15, 4, 14, 18, 8, 13, 1, 20, 11, 13, 19, 1, 20, 9, 2, 14, 1, 10, 17, 19, 13, 14, 13, 18, 2, 14, 17, 4, 10, 1, 14, 8, 4, 8, 13, 15, 17, 6, 7, 18, 3, 4, 12, 11, 14, 18, 19, 15, 18, 20, 10, 17, 13, 11, 13, 16, 1, 8, 10, 11, 5, 7, 16, 2, 18, 14, 12) = 1170
100d20 ⇒ (10, 6, 12, 19, 6, 14, 6, 9, 17, 12, 9, 4, 16, 18, 18, 12, 19, 10, 16, 14, 8, 5, 18, 5, 16, 4, 5, 20, 7, 2, 4, 19, 13, 18, 9, 20, 2, 20, 9, 11, 2, 8, 3, 7, 16, 2, 2, 12, 2, 6, 12, 2, 16, 13, 8, 2, 7, 5, 4, 16, 5, 3, 4, 10, 13, 18, 19, 5, 8, 11, 11, 1, 3, 16, 4, 18, 9, 14, 7, 6, 18, 16, 20, 3, 6, 5, 7, 2, 15, 10, 16, 2, 17, 6, 18, 16, 16, 10, 14, 19) = 1028
100d20 ⇒ (2, 13, 19, 4, 12, 13, 16, 20, 10, 5, 10, 5, 14, 20, 17, 3, 8, 2, 2, 13, 5, 13, 9, 5, 16, 14, 16, 17, 14, 19, 20, 11, 10, 14, 2, 4, 4, 11, 3, 17, 11, 3, 1, 6, 10, 13, 16, 20, 20, 3, 13, 19, 12, 19, 18, 9, 4, 19, 16, 7, 14, 3, 3, 17, 14, 13, 3, 8, 3, 15, 16, 1, 8, 1, 19, 13, 7, 17, 20, 1, 17, 9, 16, 15, 13, 8, 8, 18, 15, 6, 9, 8, 17, 6, 12, 6, 16, 6, 14, 9) = 1095
100d20 ⇒ (14, 7, 20, 16, 13, 18, 20, 5, 5, 2, 20, 11, 18, 15, 8, 10, 19, 20, 3, 13, 19, 6, 12, 8, 6, 18, 3, 17, 18, 10, 11, 7, 9, 8, 11, 7, 18, 10, 17, 7, 3, 5, 19, 12, 1, 1, 16, 12, 15, 3, 15, 16, 14, 4, 11, 1, 10, 6, 20, 6, 7, 16, 15, 6, 3, 5, 5, 16, 19, 10, 16, 7, 5, 11, 2, 5, 13, 8, 20, 20, 12, 9, 16, 9, 10, 3, 12, 5, 5, 2, 10, 13, 7, 16, 13, 9, 20, 12, 9, 1) = 1071
100d20 ⇒ (12, 13, 11, 12, 16, 18, 9, 3, 17, 7, 9, 11, 15, 10, 1, 15, 9, 19, 19, 8, 9, 5, 6, 14, 17, 19, 13, 12, 18, 12, 16, 5, 5, 20, 10, 11, 20, 11, 7, 17, 3, 3, 2, 16, 8, 9, 3, 18, 10, 17, 14, 16, 5, 9, 16, 10, 15, 5, 19, 9, 6, 20, 11, 20, 19, 2, 16, 12, 8, 14, 6, 11, 18, 15, 3, 12, 18, 12, 2, 6, 20, 12, 7, 19, 11, 16, 8, 7, 12, 12, 4, 2, 2, 18, 7, 18, 20, 15, 5, 14) = 1148
100d20 ⇒ (13, 10, 20, 5, 18, 12, 4, 11, 13, 9, 2, 3, 1, 17, 17, 17, 7, 15, 1, 11, 4, 3, 11, 15, 6, 16, 11, 12, 17, 12, 9, 6, 6, 17, 2, 11, 3, 8, 10, 5, 18, 13, 7, 10, 20, 15, 4, 20, 6, 14, 1, 9, 3, 3, 8, 2, 6, 3, 1, 20, 18, 5, 4, 20, 12, 10, 15, 6, 8, 6, 8, 5, 1, 5, 8, 13, 13, 19, 5, 1, 20, 10, 6, 13, 1, 14, 16, 13, 12, 4, 6, 16, 16, 20, 15, 3, 18, 12, 1, 8) = 979
100d20 ⇒ (15, 3, 6, 4, 1, 11, 14, 14, 19, 12, 3, 15, 20, 16, 5, 7, 9, 20, 10, 4, 7, 5, 6, 11, 13, 3, 9, 4, 13, 1, 10, 15, 15, 9, 9, 9, 14, 12, 13, 16, 4, 14, 18, 16, 19, 3, 1, 7, 5, 11, 6, 9, 9, 7, 1, 17, 10, 10, 14, 19, 9, 3, 15, 10, 8, 4, 3, 5, 6, 20, 16, 11, 11, 6, 3, 9, 16, 18, 4, 14, 12, 2, 1, 5, 10, 14, 15, 14, 12, 20, 3, 3, 9, 7, 16, 1, 6, 7, 18, 20) = 988
100d20 ⇒ (14, 3, 9, 16, 12, 7, 8, 14, 15, 17, 4, 15, 6, 6, 17, 11, 3, 1, 19, 2, 5, 9, 7, 1, 1, 13, 17, 9, 13, 10, 2, 17, 20, 4, 2, 1, 3, 5, 15, 10, 14, 3, 14, 5, 16, 9, 8, 1, 7, 6, 10, 2, 10, 6, 14, 16, 10, 17, 9, 20, 5, 6, 13, 13, 20, 17, 16, 16, 12, 1, 20, 18, 18, 10, 2, 10, 20, 19, 18, 4, 15, 4, 16, 9, 16, 2, 17, 17, 4, 3, 1, 4, 11, 18, 9, 8, 17, 1, 7, 3) = 1000
5000:
100d20 ⇒ (11, 7, 1, 13, 10, 16, 7, 18, 19, 10, 5, 12, 11, 18, 1, 14, 5, 6, 11, 4, 2, 11, 13, 19, 9, 18, 18, 18, 4, 7, 18, 12, 8, 9, 16, 13, 8, 3, 15, 19, 17, 13, 11, 12, 14, 6, 16, 3, 15, 16, 4, 5, 4, 9, 9, 7, 4, 11, 2, 6, 1, 20, 14, 18, 19, 13, 13, 3, 16, 1, 20, 10, 6, 9, 19, 13, 5, 18, 6, 15, 13, 19, 16, 19, 19, 5, 12, 3, 4, 17, 3, 20, 2, 13, 16, 13, 12, 15, 6, 10) = 1099
100d20 ⇒ (11, 11, 7, 8, 2, 12, 10, 10, 7, 7, 3, 15, 18, 5, 19, 11, 11, 14, 4, 11, 1, 2, 4, 6, 2, 20, 5, 10, 11, 14, 5, 12, 8, 13, 7, 8, 10, 2, 18, 5, 4, 1, 11, 17, 20, 2, 2, 16, 19, 6, 3, 8, 16, 5, 13, 6, 11, 14, 12, 7, 20, 6, 5, 6, 8, 11, 6, 10, 16, 13, 5, 15, 14, 12, 5, 19, 5, 10, 13, 19, 8, 7, 18, 4, 19, 3, 3, 12, 13, 15, 12, 6, 4, 20, 2, 11, 3, 13, 18, 2) = 963
100d20 ⇒ (17, 20, 18, 19, 6, 16, 11, 3, 11, 18, 14, 4, 5, 19, 12, 5, 19, 6, 13, 18, 17, 19, 8, 18, 5, 3, 13, 8, 12, 11, 8, 12, 5, 12, 18, 19, 11, 1, 15, 7, 9, 14, 19, 15, 3, 16, 19, 2, 17, 5, 3, 6, 19, 10, 5, 3, 16, 8, 13, 1, 19, 4, 8, 20, 9, 19, 16, 16, 10, 3, 14, 10, 15, 12, 7, 7, 1, 11, 1, 15, 15, 9, 1, 16, 18, 15, 7, 13, 17, 14, 10, 6, 1, 2, 18, 3, 7, 1, 8, 9) = 1086
100d20 ⇒ (14, 6, 3, 9, 12, 7, 10, 8, 12, 8, 7, 4, 14, 1, 8, 2, 10, 8, 9, 16, 5, 9, 20, 18, 2, 19, 5, 16, 13, 13, 6, 8, 20, 9, 4, 10, 10, 16, 13, 16, 19, 15, 5, 5, 19, 14, 15, 14, 20, 20, 16, 9, 14, 6, 20, 1, 1, 5, 6, 2, 9, 3, 15, 2, 15, 14, 19, 2, 20, 1, 10, 17, 15, 20, 10, 2, 16, 3, 6, 16, 12, 14, 1, 14, 9, 16, 13, 1, 10, 10, 17, 15, 9, 8, 18, 14, 17, 3, 9, 18) = 1069
100d20 ⇒ (18, 2, 8, 16, 15, 2, 8, 16, 7, 8, 6, 5, 13, 11, 17, 13, 15, 4, 11, 16, 11, 19, 18, 7, 11, 7, 3, 14, 14, 1, 14, 1, 12, 18, 4, 20, 8, 3, 17, 14, 6, 10, 18, 17, 12, 8, 15, 5, 10, 13, 7, 11, 13, 8, 12, 20, 1, 4, 19, 11, 11, 13, 5, 15, 14, 1, 20, 6, 18, 2, 14, 12, 15, 16, 2, 16, 12, 7, 10, 10, 20, 3, 8, 13, 15, 1, 9, 15, 1, 20, 4, 16, 8, 9, 20, 12, 20, 7, 11, 14) = 1092
100d20 ⇒ (4, 6, 6, 18, 4, 19, 6, 13, 14, 9, 1, 12, 7, 15, 18, 7, 2, 2, 11, 9, 13, 19, 13, 6, 8, 18, 13, 16, 10, 13, 5, 19, 14, 1, 11, 20, 3, 10, 19, 9, 5, 2, 5, 4, 5, 12, 17, 9, 13, 3, 19, 17, 12, 3, 3, 13, 7, 8, 5, 20, 20, 15, 11, 6, 6, 5, 3, 2, 15, 1, 16, 12, 2, 10, 2, 8, 12, 15, 14, 6, 6, 12, 4, 15, 13, 6, 12, 16, 6, 3, 12, 7, 10, 4, 11, 19, 5, 19, 15, 5) = 986
100d20 ⇒ (12, 13, 13, 5, 4, 13, 11, 18, 14, 15, 12, 11, 5, 9, 16, 7, 14, 8, 18, 20, 5, 6, 5, 20, 16, 20, 11, 5, 13, 19, 19, 12, 1, 13, 15, 1, 4, 2, 6, 17, 9, 17, 7, 16, 19, 1, 9, 13, 14, 4, 4, 9, 3, 19, 5, 9, 6, 4, 11, 13, 14, 17, 2, 14, 12, 2, 1, 15, 10, 7, 5, 9, 11, 15, 12, 12, 3, 6, 20, 10, 9, 10, 7, 14, 5, 8, 16, 9, 12, 7, 10, 15, 11, 13, 6, 1, 20, 11, 7, 8) = 1036
100d20 ⇒ (1, 18, 9, 6, 11, 4, 9, 4, 14, 2, 9, 8, 10, 15, 1, 20, 10, 19, 14, 5, 18, 9, 8, 10, 9, 7, 1, 9, 5, 13, 5, 8, 5, 17, 17, 12, 12, 1, 8, 4, 20, 14, 19, 13, 7, 18, 13, 3, 15, 14, 7, 19, 13, 19, 17, 19, 19, 15, 12, 12, 10, 6, 16, 7, 7, 10, 7, 20, 16, 5, 5, 11, 4, 14, 19, 8, 7, 20, 15, 18, 11, 15, 14, 1, 7, 4, 9, 2, 2, 16, 18, 11, 13, 9, 11, 16, 15, 10, 1, 20) = 1085
100d20 ⇒ (10, 9, 3, 18, 2, 6, 17, 13, 2, 20, 8, 6, 15, 20, 4, 9, 5, 9, 17, 10, 13, 16, 5, 19, 1, 19, 18, 8, 8, 5, 20, 18, 14, 3, 1, 20, 3, 14, 9, 16, 12, 14, 15, 11, 6, 10, 17, 1, 16, 1, 12, 20, 17, 13, 19, 3, 10, 2, 10, 10, 10, 3, 8, 13, 3, 18, 14, 11, 7, 18, 7, 5, 9, 7, 20, 4, 9, 13, 9, 20, 2, 5, 11, 11, 9, 12, 13, 19, 8, 14, 18, 7, 11, 20, 13, 17, 15, 19, 7, 9) = 1100
100d20 ⇒ (13, 10, 2, 1, 5, 18, 11, 10, 8, 1, 16, 6, 4, 4, 8, 1, 18, 18, 19, 3, 11, 2, 6, 7, 4, 12, 2, 14, 1, 5, 10, 18, 10, 10, 12, 15, 4, 16, 5, 13, 17, 14, 1, 4, 19, 10, 1, 8, 4, 9, 13, 15, 15, 18, 18, 13, 8, 19, 16, 17, 17, 17, 2, 11, 9, 1, 7, 6, 14, 12, 17, 12, 3, 19, 9, 4, 9, 8, 13, 20, 2, 18, 18, 19, 20, 3, 3, 15, 4, 1, 11, 20, 3, 10, 19, 12, 7, 13, 7, 16) = 1023
6000:
100d20 ⇒ (3, 10, 7, 12, 4, 6, 16, 16, 16, 5, 17, 18, 20, 1, 11, 4, 2, 12, 8, 6, 3, 1, 15, 5, 3, 16, 2, 9, 15, 16, 18, 13, 15, 7, 4, 12, 12, 13, 9, 7, 18, 1, 12, 2, 19, 18, 20, 5, 6, 17, 1, 9, 6, 17, 8, 11, 17, 16, 2, 17, 2, 20, 18, 11, 2, 14, 11, 2, 13, 2, 14, 5, 9, 9, 2, 4, 11, 17, 1, 6, 6, 11, 10, 6, 8, 7, 5, 6, 9, 13, 15, 1, 3, 15, 17, 15, 1, 4, 8, 11) = 955
100d20 ⇒ (8, 15, 17, 10, 11, 5, 15, 19, 19, 1, 6, 15, 3, 10, 16, 5, 19, 4, 11, 6, 15, 15, 10, 8, 7, 2, 8, 2, 18, 8, 15, 6, 15, 17, 3, 10, 2, 3, 20, 12, 16, 16, 10, 16, 1, 16, 2, 20, 19, 13, 18, 19, 11, 7, 13, 15, 1, 9, 14, 7, 2, 10, 17, 5, 2, 1, 8, 1, 8, 14, 19, 18, 9, 8, 2, 13, 12, 4, 4, 1, 1, 4, 9, 18, 2, 1, 2, 14, 6, 8, 18, 10, 1, 11, 9, 5, 4, 1, 19, 16) = 971
100d20 ⇒ (11, 7, 3, 14, 3, 7, 14, 3, 5, 7, 11, 12, 12, 2, 14, 2, 6, 19, 1, 5, 14, 11, 3, 11, 4, 10, 11, 2, 2, 14, 2, 2, 14, 8, 17, 16, 5, 15, 9, 4, 2, 11, 18, 19, 13, 19, 3, 6, 12, 4, 20, 19, 8, 6, 14, 2, 1, 8, 20, 15, 11, 3, 3, 3, 17, 4, 3, 11, 16, 17, 1, 10, 20, 11, 9, 8, 4, 14, 17, 8, 4, 9, 9, 17, 20, 20, 7, 19, 1, 12, 7, 1, 20, 2, 3, 11, 11, 20, 12, 5) = 952
100d20 ⇒ (7, 17, 20, 8, 4, 4, 13, 11, 20, 5, 12, 4, 10, 4, 7, 6, 19, 3, 9, 19, 1, 18, 4, 11, 8, 2, 5, 2, 18, 15, 12, 19, 3, 15, 11, 2, 12, 1, 4, 6, 9, 3, 15, 12, 7, 1, 15, 11, 1, 9, 10, 17, 3, 11, 15, 2, 20, 2, 19, 12, 5, 10, 16, 4, 7, 7, 18, 4, 2, 15, 8, 15, 13, 3, 12, 9, 13, 10, 3, 12, 13, 4, 8, 12, 1, 11, 17, 14, 15, 20, 3, 2, 12, 12, 7, 6, 4, 13, 16, 7) = 948
100d20 ⇒ (12, 9, 9, 2, 5, 11, 9, 20, 9, 18, 13, 20, 20, 14, 8, 10, 13, 16, 17, 10, 10, 18, 18, 6, 7, 17, 17, 10, 5, 9, 11, 10, 8, 3, 13, 13, 7, 13, 7, 12, 16, 10, 17, 8, 9, 4, 10, 11, 5, 4, 15, 15, 1, 18, 12, 14, 14, 14, 18, 7, 15, 3, 1, 12, 4, 2, 10, 6, 11, 18, 17, 13, 16, 8, 17, 8, 17, 14, 13, 13, 7, 15, 11, 13, 7, 2, 8, 5, 5, 19, 19, 2, 2, 6, 19, 7, 18, 17, 11, 1) = 1093
100d20 ⇒ (4, 11, 1, 7, 15, 12, 3, 15, 11, 16, 18, 1, 13, 12, 6, 20, 16, 3, 12, 19, 19, 2, 15, 11, 9, 20, 8, 14, 12, 5, 17, 19, 3, 6, 9, 10, 3, 20, 14, 18, 4, 14, 9, 20, 1, 15, 11, 11, 17, 1, 2, 13, 2, 5, 18, 17, 16, 19, 14, 5, 16, 1, 18, 20, 11, 16, 2, 18, 6, 7, 3, 5, 15, 20, 6, 1, 5, 9, 7, 14, 2, 16, 11, 13, 6, 4, 1, 5, 8, 20, 11, 1, 1, 1, 16, 4, 1, 16, 4, 8) = 1012
100d20 ⇒ (6, 6, 4, 7, 17, 6, 17, 4, 16, 13, 4, 19, 2, 12, 14, 13, 5, 1, 2, 7, 9, 11, 20, 20, 14, 2, 11, 2, 9, 8, 15, 18, 5, 19, 19, 5, 5, 15, 17, 13, 5, 20, 8, 12, 2, 12, 5, 8, 8, 19, 16, 2, 6, 9, 17, 4, 5, 3, 17, 3, 1, 19, 18, 2, 10, 18, 4, 19, 16, 17, 11, 12, 10, 8, 14, 10, 9, 18, 2, 17, 3, 11, 16, 3, 4, 4, 9, 16, 9, 5, 20, 6, 12, 13, 14, 19, 12, 10, 1, 16) = 1031
100d20 ⇒ (14, 6, 19, 6, 14, 8, 15, 20, 7, 1, 18, 19, 2, 19, 4, 18, 10, 20, 10, 12, 18, 7, 3, 1, 20, 13, 11, 2, 1, 15, 16, 13, 2, 12, 1, 9, 17, 6, 18, 11, 2, 20, 9, 3, 1, 11, 7, 10, 5, 19, 10, 16, 4, 10, 15, 17, 13, 4, 1, 9, 8, 6, 8, 14, 20, 9, 17, 3, 3, 2, 7, 11, 7, 12, 19, 14, 20, 18, 1, 16, 8, 12, 14, 12, 6, 16, 14, 9, 4, 19, 7, 13, 12, 14, 9, 10, 5, 10, 2, 15) = 1050
100d20 ⇒ (11, 20, 18, 15, 5, 6, 15, 1, 20, 20, 18, 12, 8, 19, 18, 1, 6, 12, 14, 7, 17, 4, 15, 19, 3, 19, 16, 2, 5, 7, 17, 5, 15, 5, 4, 20, 5, 16, 18, 14, 19, 14, 7, 5, 8, 7, 5, 1, 7, 19, 15, 8, 11, 20, 19, 10, 20, 10, 8, 6, 11, 8, 14, 9, 2, 5, 19, 5, 18, 7, 8, 3, 10, 11, 15, 20, 1, 18, 4, 14, 17, 19, 15, 19, 7, 7, 7, 18, 3, 3, 1, 13, 7, 1, 13, 19, 8, 18, 2, 4) = 1094
100d20 ⇒ (2, 7, 1, 15, 10, 7, 7, 4, 2, 2, 19, 15, 8, 12, 4, 17, 16, 1, 17, 3, 16, 8, 8, 20, 10, 14, 13, 20, 17, 19, 14, 3, 13, 4, 4, 3, 1, 7, 1, 14, 8, 16, 10, 16, 8, 17, 17, 18, 6, 6, 8, 9, 14, 15, 19, 2, 8, 15, 11, 11, 1, 9, 1, 14, 12, 18, 15, 13, 12, 11, 19, 2, 15, 20, 17, 14, 9, 18, 10, 14, 3, 10, 7, 16, 7, 16, 17, 1, 20, 11, 15, 7, 20, 10, 6, 15, 3, 4, 20, 18) = 1082
7000:
100d20 ⇒ (12, 13, 12, 8, 6, 10, 3, 7, 11, 8, 20, 16, 20, 12, 19, 13, 18, 10, 11, 15, 11, 12, 13, 10, 6, 18, 17, 7, 5, 8, 12, 11, 7, 8, 13, 19, 17, 2, 13, 12, 6, 14, 14, 4, 20, 10, 7, 19, 11, 5, 10, 4, 20, 8, 17, 20, 20, 15, 15, 5, 2, 13, 18, 7, 7, 13, 9, 8, 5, 20, 15, 18, 19, 12, 8, 16, 20, 2, 1, 10, 10, 6, 3, 1, 16, 9, 6, 3, 16, 8, 15, 3, 15, 2, 20, 18, 2, 18, 8, 18) = 1129
100d20 ⇒ (10, 8, 9, 8, 16, 3, 11, 14, 3, 18, 15, 14, 15, 16, 14, 6, 7, 17, 5, 20, 3, 6, 9, 6, 12, 20, 20, 12, 14, 10, 6, 7, 7, 10, 7, 18, 14, 3, 17, 7, 3, 4, 18, 3, 6, 8, 13, 15, 15, 13, 10, 12, 13, 7, 1, 2, 8, 12, 7, 17, 14, 10, 16, 20, 17, 1, 12, 4, 13, 2, 16, 6, 4, 1, 10, 9, 20, 5, 2, 3, 15, 1, 5, 14, 10, 7, 8, 6, 11, 12, 1, 18, 5, 17, 12, 12, 19, 19, 5, 11) = 1017
100d20 ⇒ (5, 11, 4, 16, 7, 13, 7, 5, 19, 15, 20, 11, 13, 9, 17, 14, 7, 5, 5, 3, 11, 13, 13, 1, 5, 8, 1, 13, 1, 6, 14, 5, 15, 19, 19, 13, 7, 16, 8, 5, 4, 15, 7, 6, 9, 1, 3, 16, 10, 1, 12, 12, 9, 19, 1, 8, 13, 18, 8, 16, 5, 3, 3, 8, 5, 20, 19, 12, 13, 14, 18, 3, 3, 8, 16, 1, 12, 1, 20, 12, 12, 3, 2, 18, 12, 10, 3, 6, 8, 7, 11, 16, 12, 19, 19, 15, 4, 17, 7, 10) = 994
100d20 ⇒ (9, 15, 9, 2, 16, 18, 2, 16, 18, 12, 13, 10, 14, 1, 5, 5, 5, 17, 12, 15, 8, 12, 1, 14, 1, 4, 3, 14, 5, 7, 4, 19, 18, 11, 17, 10, 16, 6, 13, 2, 14, 4, 1, 15, 10, 16, 3, 9, 7, 10, 19, 15, 12, 16, 14, 2, 9, 11, 19, 5, 18, 5, 9, 2, 12, 13, 14, 4, 1, 8, 1, 8, 20, 11, 6, 6, 6, 10, 4, 17, 12, 16, 6, 14, 8, 16, 3, 8, 11, 12, 19, 20, 17, 4, 2, 16, 13, 15, 3, 2) = 1002
100d20 ⇒ (16, 3, 3, 2, 13, 11, 17, 6, 12, 6, 7, 10, 20, 6, 7, 9, 15, 7, 11, 11, 1, 7, 10, 3, 20, 11, 15, 8, 14, 11, 1, 18, 14, 19, 18, 20, 17, 10, 6, 17, 10, 5, 19, 15, 12, 5, 3, 15, 19, 5, 6, 7, 19, 11, 18, 15, 1, 18, 13, 20, 13, 7, 16, 14, 13, 16, 18, 17, 3, 1, 9, 7, 6, 2, 7, 1, 7, 18, 2, 15, 15, 3, 20, 6, 2, 6, 1, 15, 10, 6, 18, 10, 10, 5, 12, 16, 2, 16, 16, 13) = 1062
100d20 ⇒ (4, 4, 18, 18, 14, 9, 8, 4, 6, 15, 7, 15, 11, 16, 18, 14, 2, 11, 4, 7, 13, 10, 5, 18, 20, 10, 8, 15, 2, 4, 15, 10, 7, 11, 16, 15, 16, 11, 8, 19, 1, 5, 8, 19, 3, 17, 15, 4, 6, 1, 6, 7, 12, 3, 7, 13, 20, 6, 14, 8, 3, 18, 11, 6, 9, 2, 4, 11, 6, 16, 18, 11, 3, 4, 7, 9, 19, 5, 19, 15, 18, 13, 9, 15, 17, 20, 2, 19, 1, 13, 16, 9, 19, 9, 20, 17, 5, 9, 6, 1) = 1047
100d20 ⇒ (14, 18, 6, 20, 20, 12, 17, 10, 3, 12, 7, 12, 15, 4, 19, 3, 19, 13, 13, 12, 4, 15, 6, 12, 12, 10, 7, 9, 16, 20, 9, 13, 2, 4, 17, 9, 5, 13, 13, 7, 7, 11, 13, 3, 5, 12, 1, 12, 5, 15, 20, 3, 17, 4, 18, 11, 4, 12, 15, 12, 10, 6, 15, 15, 13, 3, 6, 7, 18, 3, 13, 7, 3, 2, 12, 19, 3, 2, 7, 1, 9, 17, 12, 9, 15, 20, 14, 16, 14, 11, 16, 2, 5, 9, 6, 19, 15, 13, 1, 2) = 1042
100d20 ⇒ (6, 9, 3, 6, 4, 3, 5, 5, 6, 10, 13, 17, 2, 20, 7, 5, 2, 18, 20, 2, 17, 20, 11, 17, 16, 15, 11, 14, 15, 5, 11, 5, 10, 1, 12, 16, 20, 16, 10, 1, 13, 8, 10, 18, 16, 7, 3, 5, 11, 4, 17, 1, 11, 2, 10, 3, 17, 14, 6, 2, 15, 4, 10, 13, 13, 5, 4, 5, 12, 9, 6, 10, 1, 4, 3, 8, 7, 15, 20, 10, 9, 20, 4, 2, 15, 6, 7, 12, 8, 4, 6, 1, 10, 9, 4, 2, 8, 7, 7, 5) = 904
100d20 ⇒ (12, 13, 9, 17, 17, 9, 20, 13, 6, 10, 13, 15, 9, 11, 8, 19, 4, 3, 2, 18, 18, 10, 1, 6, 20, 1, 17, 15, 10, 5, 12, 13, 8, 4, 3, 17, 18, 9, 17, 10, 7, 3, 10, 6, 15, 2, 8, 6, 12, 18, 14, 11, 15, 20, 1, 1, 9, 3, 7, 1, 8, 20, 4, 16, 5, 7, 1, 4, 19, 10, 1, 20, 13, 15, 3, 20, 18, 3, 5, 17, 14, 18, 2, 7, 18, 14, 19, 18, 5, 11, 8, 8, 13, 8, 7, 10, 1, 5, 14, 12) = 1032
100d20 ⇒ (17, 2, 13, 18, 20, 5, 20, 4, 19, 11, 7, 10, 18, 19, 17, 19, 16, 16, 10, 5, 20, 14, 15, 19, 2, 5, 1, 7, 17, 6, 7, 9, 7, 15, 10, 8, 8, 6, 18, 7, 9, 6, 17, 8, 16, 2, 8, 3, 12, 6, 12, 11, 2, 19, 5, 3, 15, 3, 5, 11, 18, 19, 14, 4, 18, 3, 17, 6, 5, 12, 20, 5, 8, 12, 5, 8, 16, 10, 5, 19, 6, 8, 3, 15, 3, 7, 13, 20, 11, 6, 5, 14, 6, 19, 1, 5, 17, 4, 17, 12) = 1056
8000:
100d20 ⇒ (3, 14, 7, 14, 2, 16, 5, 4, 10, 3, 19, 18, 8, 1, 12, 12, 19, 17, 19, 1, 18, 2, 7, 17, 15, 19, 7, 8, 9, 3, 4, 2, 18, 13, 5, 9, 12, 12, 10, 2, 14, 10, 6, 1, 7, 19, 7, 4, 10, 1, 3, 19, 13, 14, 16, 14, 11, 2, 17, 12, 14, 20, 16, 5, 2, 20, 16, 6, 9, 7, 20, 20, 16, 18, 3, 3, 2, 20, 3, 7, 12, 8, 17, 6, 16, 4, 3, 19, 12, 1, 14, 14, 3, 11, 11, 16, 12, 4, 15, 15) = 1036
100d20 ⇒ (18, 16, 14, 4, 12, 12, 1, 14, 14, 5, 6, 7, 5, 5, 10, 1, 18, 10, 10, 11, 20, 7, 18, 5, 9, 3, 13, 7, 2, 19, 19, 14, 6, 12, 6, 5, 10, 3, 12, 15, 14, 3, 18, 9, 1, 9, 9, 9, 19, 9, 3, 10, 1, 16, 17, 14, 19, 18, 20, 6, 5, 20, 15, 8, 11, 8, 18, 12, 18, 20, 17, 17, 20, 12, 8, 10, 9, 11, 19, 13, 4, 14, 15, 20, 5, 9, 8, 5, 1, 7, 18, 17, 5, 17, 19, 4, 9, 6, 3, 11) = 1090
100d20 ⇒ (4, 2, 10, 6, 20, 13, 19, 9, 15, 11, 11, 18, 17, 11, 2, 6, 14, 9, 17, 18, 9, 17, 20, 16, 6, 2, 7, 7, 17, 3, 16, 19, 20, 17, 9, 19, 7, 18, 9, 10, 20, 10, 5, 20, 9, 13, 15, 18, 15, 17, 7, 16, 20, 10, 6, 12, 5, 3, 12, 5, 10, 3, 11, 1, 10, 14, 18, 19, 12, 3, 8, 5, 20, 11, 1, 3, 17, 8, 15, 10, 11, 12, 7, 8, 12, 8, 8, 15, 4, 13, 13, 1, 5, 14, 6, 4, 18, 4, 7, 15) = 1102
100d20 ⇒ (20, 15, 17, 6, 20, 7, 11, 2, 4, 4, 7, 5, 16, 10, 7, 9, 9, 12, 9, 19, 12, 8, 10, 8, 10, 4, 6, 8, 9, 8, 18, 3, 3, 4, 16, 13, 2, 9, 16, 18, 15, 4, 4, 11, 5, 8, 18, 4, 2, 2, 10, 15, 10, 14, 11, 10, 8, 8, 11, 5, 13, 5, 14, 11, 8, 13, 13, 7, 4, 2, 3, 18, 1, 14, 5, 16, 1, 13, 16, 17, 4, 10, 7, 11, 16, 13, 13, 10, 12, 16, 16, 7, 19, 15, 16, 18, 8, 1, 11, 3) = 989
100d20 ⇒ (20, 4, 17, 19, 7, 9, 9, 17, 2, 2, 2, 15, 15, 8, 4, 19, 8, 9, 15, 13, 17, 7, 3, 3, 19, 7, 14, 1, 6, 10, 3, 20, 19, 18, 20, 9, 1, 4, 3, 20, 3, 18, 11, 12, 10, 12, 1, 4, 16, 17, 9, 15, 7, 4, 14, 15, 18, 16, 10, 19, 2, 2, 1, 16, 6, 9, 10, 8, 13, 5, 20, 11, 20, 13, 7, 17, 9, 14, 5, 12, 14, 1, 12, 15, 15, 4, 2, 13, 18, 11, 4, 5, 6, 10, 9, 15, 20, 8, 2, 19) = 1052
100d20 ⇒ (14, 14, 12, 5, 1, 20, 10, 9, 1, 17, 1, 1, 16, 10, 11, 18, 17, 12, 14, 4, 1, 15, 14, 3, 15, 6, 18, 15, 4, 4, 9, 6, 2, 16, 18, 10, 6, 14, 13, 2, 1, 3, 1, 6, 8, 11, 6, 11, 20, 18, 4, 4, 17, 8, 6, 12, 8, 14, 6, 1, 12, 7, 18, 3, 17, 19, 12, 14, 7, 20, 15, 4, 10, 14, 1, 19, 20, 17, 13, 8, 17, 3, 14, 1, 8, 7, 7, 17, 10, 14, 7, 18, 10, 20, 5, 8, 20, 11, 16, 1) = 1027
100d20 ⇒ (12, 9, 4, 17, 4, 20, 8, 4, 14, 19, 9, 14, 7, 19, 9, 12, 4, 3, 6, 18, 16, 14, 1, 1, 20, 20, 2, 2, 19, 8, 3, 4, 9, 17, 9, 7, 18, 15, 15, 18, 5, 1, 17, 20, 7, 14, 9, 11, 3, 13, 17, 6, 19, 2, 14, 10, 9, 18, 6, 20, 13, 10, 4, 1, 2, 1, 8, 14, 1, 16, 18, 19, 16, 16, 1, 8, 9, 4, 9, 20, 15, 12, 6, 18, 10, 5, 13, 9, 12, 3, 20, 15, 15, 9, 5, 18, 19, 20, 4, 12) = 1081
100d20 ⇒ (8, 15, 9, 1, 1, 8, 14, 7, 12, 16, 3, 18, 14, 4, 9, 2, 15, 17, 13, 2, 3, 10, 12, 15, 12, 8, 16, 6, 4, 19, 18, 3, 11, 16, 20, 15, 7, 19, 15, 9, 16, 8, 9, 9, 7, 18, 20, 9, 10, 8, 7, 6, 5, 19, 2, 11, 13, 13, 18, 17, 17, 17, 20, 3, 16, 19, 2, 1, 6, 15, 16, 13, 14, 18, 8, 17, 9, 6, 5, 8, 4, 16, 10, 15, 14, 11, 19, 4, 9, 18, 7, 11, 15, 15, 13, 6, 20, 16, 20, 15) = 1139
100d20 ⇒ (17, 8, 5, 12, 13, 12, 20, 18, 2, 6, 3, 15, 9, 7, 20, 7, 2, 9, 10, 6, 9, 1, 10, 9, 8, 15, 1, 14, 18, 15, 2, 5, 10, 5, 15, 17, 14, 4, 12, 1, 13, 8, 6, 5, 4, 18, 3, 13, 20, 13, 8, 20, 12, 8, 5, 13, 9, 12, 4, 15, 11, 11, 4, 8, 20, 2, 9, 1, 13, 7, 1, 16, 17, 20, 19, 19, 1, 1, 12, 13, 6, 20, 3, 15, 7, 16, 6, 17, 6, 19, 19, 2, 14, 14, 13, 6, 15, 11, 3, 12) = 1024
100d20 ⇒ (5, 17, 3, 5, 8, 1, 17, 4, 13, 20, 3, 11, 19, 2, 5, 15, 9, 5, 18, 4, 16, 15, 9, 7, 5, 14, 19, 17, 19, 10, 6, 1, 11, 18, 14, 19, 13, 11, 8, 18, 16, 7, 4, 19, 15, 13, 17, 11, 15, 15, 9, 10, 9, 1, 3, 14, 20, 17, 16, 20, 12, 12, 12, 10, 6, 14, 14, 6, 4, 11, 5, 11, 13, 6, 2, 10, 2, 8, 8, 4, 20, 14, 14, 15, 2, 19, 19, 10, 4, 5, 1, 3, 10, 19, 15, 4, 3, 9, 6, 14) = 1061
9000:
100d20 ⇒ (1, 20, 2, 7, 17, 5, 11, 7, 16, 12, 20, 1, 13, 14, 11, 3, 8, 4, 11, 16, 20, 3, 1, 14, 17, 14, 15, 10, 5, 15, 7, 16, 12, 2, 10, 7, 11, 11, 17, 9, 17, 11, 15, 8, 10, 15, 10, 3, 1, 17, 18, 4, 5, 11, 6, 6, 13, 18, 14, 10, 16, 8, 1, 6, 9, 2, 10, 11, 20, 18, 20, 18, 1, 17, 13, 15, 14, 12, 3, 9, 3, 13, 9, 17, 19, 1, 14, 5, 10, 14, 6, 5, 18, 19, 16, 10, 16, 12, 15, 7) = 1079
100d20 ⇒ (11, 16, 6, 14, 20, 16, 16, 18, 9, 15, 20, 19, 4, 13, 3, 8, 1, 17, 2, 9, 12, 19, 7, 14, 1, 14, 20, 11, 5, 14, 10, 10, 5, 2, 9, 19, 3, 3, 6, 2, 16, 14, 3, 12, 6, 4, 12, 8, 19, 7, 1, 9, 8, 11, 11, 14, 1, 11, 4, 17, 20, 7, 3, 5, 9, 1, 10, 10, 16, 19, 6, 3, 3, 3, 6, 12, 3, 10, 20, 10, 15, 18, 5, 5, 8, 13, 16, 13, 3, 15, 13, 20, 9, 4, 4, 15, 4, 19, 17, 8) = 1011
100d20 ⇒ (4, 6, 9, 1, 1, 8, 19, 8, 9, 4, 12, 10, 6, 20, 12, 5, 17, 19, 13, 18, 18, 15, 19, 9, 16, 3, 3, 17, 13, 9, 2, 8, 19, 20, 1, 11, 7, 14, 14, 18, 19, 1, 10, 16, 6, 4, 3, 10, 15, 5, 10, 14, 19, 9, 14, 2, 9, 15, 19, 11, 13, 3, 16, 10, 10, 20, 4, 2, 8, 5, 13, 4, 16, 5, 1, 19, 3, 13, 2, 10, 7, 19, 8, 9, 17, 18, 17, 13, 4, 5, 4, 5, 8, 5, 17, 14, 7, 6, 6, 17) = 1031
100d20 ⇒ (1, 9, 18, 16, 18, 5, 14, 20, 8, 8, 17, 19, 11, 14, 18, 14, 5, 18, 5, 9, 2, 8, 11, 18, 13, 6, 11, 8, 19, 15, 17, 18, 1, 18, 14, 16, 12, 13, 15, 6, 2, 9, 15, 12, 16, 2, 4, 10, 15, 8, 2, 6, 5, 13, 19, 11, 4, 4, 19, 8, 19, 2, 15, 2, 9, 4, 5, 3, 1, 8, 3, 6, 20, 5, 9, 13, 3, 9, 14, 13, 10, 2, 4, 20, 14, 15, 20, 19, 6, 18, 16, 11, 17, 15, 19, 8, 20, 6, 3, 6) = 1084
100d20 ⇒ (11, 2, 6, 16, 3, 10, 3, 10, 18, 13, 1, 14, 3, 7, 19, 11, 17, 10, 2, 15, 3, 15, 20, 20, 10, 7, 9, 2, 19, 18, 11, 15, 13, 19, 13, 15, 6, 17, 9, 7, 3, 10, 8, 2, 13, 2, 6, 15, 11, 7, 2, 7, 14, 4, 9, 14, 7, 17, 2, 5, 1, 14, 7, 15, 15, 16, 15, 20, 14, 13, 19, 20, 20, 7, 15, 5, 20, 9, 2, 9, 2, 13, 19, 14, 16, 2, 20, 3, 14, 2, 17, 8, 19, 16, 1, 7, 13, 17, 8, 15) = 1079
100d20 ⇒ (3, 16, 16, 2, 15, 4, 3, 10, 6, 10, 3, 20, 19, 13, 13, 17, 4, 7, 4, 15, 5, 13, 2, 12, 16, 6, 17, 10, 19, 9, 12, 7, 16, 13, 13, 20, 14, 10, 11, 6, 20, 10, 17, 12, 15, 2, 5, 16, 7, 12, 11, 16, 19, 19, 18, 13, 17, 13, 1, 20, 13, 4, 8, 10, 13, 11, 2, 14, 7, 4, 3, 4, 5, 2, 15, 5, 12, 1, 2, 10, 20, 13, 7, 11, 3, 6, 4, 18, 2, 5, 19, 17, 15, 11, 7, 13, 7, 8, 3, 12) = 1040
100d20 ⇒ (4, 15, 12, 14, 14, 4, 14, 11, 12, 16, 20, 12, 8, 18, 5, 13, 3, 13, 4, 9, 2, 14, 13, 5, 13, 13, 5, 3, 6, 10, 11, 10, 3, 18, 16, 10, 2, 13, 14, 6, 9, 18, 11, 2, 2, 11, 9, 9, 1, 9, 10, 11, 18, 8, 8, 1, 15, 18, 15, 6, 5, 18, 11, 20, 19, 1, 9, 8, 10, 8, 3, 7, 9, 12, 8, 11, 15, 8, 7, 3, 7, 1, 3, 15, 9, 19, 19, 11, 1, 3, 3, 18, 8, 9, 11, 15, 19, 1, 18, 17) = 998
100d20 ⇒ (9, 19, 17, 1, 6, 20, 10, 4, 14, 8, 10, 19, 16, 14, 9, 11, 19, 7, 17, 17, 19, 11, 4, 1, 5, 1, 11, 13, 5, 2, 4, 13, 1, 5, 11, 4, 10, 6, 9, 9, 4, 17, 15, 13, 8, 6, 7, 7, 13, 12, 5, 2, 13, 15, 19, 9, 13, 7, 7, 20, 12, 18, 6, 20, 2, 6, 2, 20, 20, 13, 14, 5, 13, 5, 6, 2, 18, 20, 10, 5, 11, 17, 16, 18, 19, 14, 14, 2, 14, 3, 14, 7, 4, 3, 12, 9, 13, 1, 6, 1) = 1018
100d20 ⇒ (19, 9, 15, 9, 19, 2, 7, 4, 4, 1, 16, 19, 4, 8, 2, 13, 17, 20, 8, 6, 8, 8, 5, 18, 12, 13, 6, 1, 2, 3, 19, 19, 17, 7, 7, 1, 13, 18, 6, 1, 13, 20, 4, 17, 5, 7, 18, 4, 15, 14, 17, 6, 2, 4, 16, 14, 10, 19, 13, 15, 16, 5, 8, 10, 5, 14, 11, 12, 9, 14, 10, 4, 13, 14, 14, 1, 13, 15, 2, 1, 6, 12, 5, 13, 9, 10, 7, 9, 13, 13, 20, 4, 7, 13, 7, 19, 3, 10, 1, 7) = 998
100d20 ⇒ (20, 6, 4, 13, 20, 14, 11, 13, 9, 16, 20, 10, 12, 8, 9, 1, 13, 17, 17, 13, 5, 16, 3, 20, 20, 4, 18, 1, 18, 12, 19, 6, 1, 2, 20, 8, 2, 20, 18, 13, 5, 19, 13, 2, 6, 20, 20, 11, 8, 11, 4, 1, 10, 3, 3, 19, 11, 14, 14, 8, 20, 7, 16, 8, 12, 2, 2, 5, 19, 1, 11, 7, 11, 1, 17, 9, 5, 16, 11, 15, 9, 10, 10, 17, 4, 14, 19, 20, 3, 7, 10, 16, 20, 20, 5, 18, 2, 2, 1, 5) = 1081
10000:
100d20 ⇒ (11, 8, 2, 12, 18, 13, 7, 15, 7, 3, 16, 20, 2, 3, 12, 17, 3, 18, 19, 11, 13, 19, 8, 1, 18, 6, 12, 8, 15, 17, 10, 7, 14, 10, 20, 11, 12, 10, 8, 17, 1, 9, 17, 11, 11, 18, 1, 17, 11, 6, 3, 2, 20, 2, 10, 12, 14, 14, 16, 5, 10, 5, 17, 14, 4, 14, 20, 18, 17, 4, 8, 13, 4, 15, 17, 7, 5, 8, 19, 1, 20, 7, 8, 7, 17, 15, 13, 7, 16, 3, 17, 2, 8, 17, 18, 12, 5, 4, 2, 2) = 1073
100d20 ⇒ (6, 11, 3, 13, 10, 16, 17, 2, 16, 3, 8, 10, 6, 8, 9, 3, 11, 4, 1, 16, 20, 1, 20, 14, 20, 4, 5, 8, 1, 5, 7, 11, 15, 6, 1, 19, 17, 16, 16, 2, 11, 11, 1, 7, 9, 2, 2, 9, 15, 19, 13, 18, 18, 17, 4, 14, 11, 7, 5, 19, 20, 12, 2, 20, 15, 6, 9, 17, 7, 16, 2, 3, 6, 2, 2, 20, 18, 4, 11, 6, 12, 10, 6, 9, 6, 10, 13, 18, 4, 19, 15, 11, 6, 18, 15, 18, 10, 8, 9, 15) = 1023
100d20 ⇒ (18, 4, 7, 6, 6, 11, 2, 17, 15, 2, 9, 10, 7, 15, 16, 5, 12, 8, 20, 10, 8, 18, 11, 14, 20, 2, 10, 12, 18, 10, 3, 11, 13, 19, 4, 11, 11, 13, 2, 18, 13, 17, 15, 12, 19, 19, 7, 18, 8, 20, 15, 17, 7, 5, 5, 7, 4, 20, 8, 16, 16, 17, 3, 18, 3, 9, 3, 17, 12, 14, 6, 3, 12, 20, 14, 1, 17, 6, 7, 9, 14, 2, 8, 16, 4, 4, 12, 6, 7, 1, 12, 11, 6, 1, 15, 5, 2, 14, 3, 2) = 1032
100d20 ⇒ (18, 4, 18, 17, 5, 7, 8, 5, 9, 8, 7, 17, 11, 17, 15, 2, 19, 15, 13, 15, 5, 20, 10, 16, 4, 18, 5, 7, 1, 11, 9, 11, 20, 17, 14, 2, 18, 7, 15, 15, 18, 1, 9, 1, 13, 9, 18, 12, 13, 9, 16, 14, 15, 2, 7, 1, 1, 20, 6, 15, 11, 16, 3, 19, 3, 15, 5, 16, 16, 16, 14, 16, 4, 1, 3, 19, 13, 18, 7, 18, 14, 20, 5, 19, 7, 18, 10, 19, 10, 3, 14, 4, 10, 7, 8, 16, 10, 3, 1, 10) = 1096
100d20 ⇒ (4, 12, 18, 12, 16, 16, 1, 10, 3, 5, 1, 11, 17, 7, 8, 15, 2, 5, 7, 9, 10, 15, 1, 19, 13, 15, 20, 15, 18, 13, 8, 16, 4, 2, 1, 18, 2, 9, 13, 11, 20, 6, 9, 12, 10, 17, 4, 20, 1, 2, 14, 17, 10, 19, 18, 5, 8, 17, 4, 15, 14, 13, 18, 6, 18, 19, 1, 19, 17, 6, 1, 8, 6, 4, 9, 13, 2, 7, 6, 20, 10, 17, 4, 12, 18, 9, 15, 15, 7, 11, 19, 13, 7, 11, 9, 1, 12, 13, 6, 3) = 1049
100d20 ⇒ (17, 17, 6, 7, 6, 15, 15, 18, 5, 10, 18, 6, 3, 14, 8, 1, 5, 3, 10, 19, 5, 13, 13, 5, 13, 15, 11, 7, 3, 15, 2, 17, 20, 4, 20, 11, 17, 11, 19, 14, 10, 16, 20, 3, 10, 7, 14, 10, 3, 18, 11, 19, 16, 1, 10, 18, 3, 4, 17, 18, 4, 2, 11, 6, 11, 5, 3, 3, 1, 17, 15, 17, 14, 6, 8, 1, 1, 5, 10, 8, 5, 14, 8, 10, 7, 14, 3, 13, 1, 20, 15, 1, 6, 5, 12, 18, 17, 13, 10, 10) = 1016
100d20 ⇒ (12, 5, 13, 9, 4, 12, 9, 13, 3, 2, 9, 18, 15, 20, 7, 16, 18, 5, 3, 10, 16, 1, 1, 8, 16, 13, 2, 8, 7, 16, 11, 1, 8, 18, 12, 4, 17, 14, 8, 16, 11, 5, 6, 17, 16, 2, 13, 6, 17, 4, 18, 20, 12, 17, 14, 7, 4, 17, 7, 5, 8, 6, 12, 5, 11, 12, 13, 14, 8, 8, 5, 5, 7, 7, 14, 17, 4, 1, 6, 10, 12, 15, 1, 4, 8, 2, 4, 3, 9, 2, 1, 20, 13, 15, 6, 15, 15, 4, 18, 17) = 975
100d20 ⇒ (14, 13, 3, 16, 14, 3, 7, 5, 4, 8, 2, 14, 19, 1, 2, 18, 20, 9, 5, 14, 10, 6, 20, 6, 17, 3, 20, 13, 10, 1, 17, 3, 11, 19, 16, 13, 3, 18, 7, 6, 20, 5, 13, 8, 2, 18, 16, 1, 1, 2, 5, 10, 13, 4, 2, 11, 10, 14, 4, 9, 2, 3, 14, 8, 5, 5, 6, 4, 20, 11, 9, 16, 19, 7, 2, 10, 2, 19, 12, 2, 19, 17, 10, 10, 17, 5, 9, 3, 16, 19, 8, 14, 17, 7, 7, 14, 7, 18, 6, 10) = 987
100d20 ⇒ (12, 2, 10, 18, 10, 9, 10, 3, 1, 11, 8, 13, 9, 14, 18, 10, 2, 19, 12, 13, 12, 12, 9, 16, 14, 12, 2, 3, 15, 3, 8, 13, 19, 16, 1, 5, 16, 17, 5, 4, 8, 6, 19, 1, 20, 1, 2, 6, 3, 14, 20, 14, 6, 7, 14, 12, 20, 19, 6, 7, 13, 1, 7, 7, 12, 12, 10, 1, 14, 13, 6, 11, 7, 5, 13, 7, 11, 2, 14, 18, 6, 1, 3, 1, 9, 5, 17, 12, 9, 7, 7, 20, 10, 14, 5, 7, 9, 6, 1, 13) = 957
100d20 ⇒ (7, 18, 11, 18, 6, 3, 18, 3, 7, 4, 6, 11, 2, 8, 11, 11, 9, 15, 4, 6, 4, 16, 10, 12, 14, 15, 4, 17, 17, 7, 7, 14, 16, 5, 10, 7, 14, 9, 19, 8, 19, 1, 2, 15, 3, 16, 12, 20, 11, 7, 20, 5, 1, 19, 2, 15, 12, 16, 17, 17, 16, 11, 19, 20, 20, 12, 20, 12, 17, 18, 19, 3, 4, 3, 5, 20, 13, 20, 4, 16, 19, 6, 9, 17, 15, 4, 15, 6, 13, 13, 3, 3, 18, 9, 5, 18, 4, 8, 8, 12) = 1110

(Edit to fix an error in my code and add chi-squared test results.)

[Count Buggula]

Ok, Paizo must be trolling you. They caught on to what you were doing and fudged the rolls. I mean, statistics are never that perfect!

[Aniuś the Talewise]

Sorry about the thread necromancy, I just find this very interesting and thought I'd write my own reflection, since Oladon did not write a conclusion about their own results.

By the way I have no real background in statistics & probability, or computer science. I'm actually an anthropologist/historian who hungers for knowledge of all strains, thus I know lots of pseudo-random (lol) interesting tidbits.

--

Computers are not natively capable of true randomness. All a computer can do is do math with the input it has been given. As I understand it, a computer can simulate pseudo-randomness and the resulting numbers appear to be random enough for most purposes (gaming being one of them), but this is not true randomness and there will always be a little bit of bias in the results.

Now on the other hand: quantum fluctuation of particles are truly random--thus, one can get truly random numbers by measuring over and over the fluctuations of some quantum activity. For example, roll20.net uses the fluctuations of a beam of light.

This page analyzes the results of d20 rolls for the past hour or so in real time.

According to the roll20 page as I look at it now, 10.5084 is the average 20 roll. Oladon's average roll is also 10.5 for their results, but they rounded to the nearest tenth so I wouldn't know if it is slightly more or less without doing the math myself. Anyway, the mean being 10.5 is a good sign. 10.5 is the sum of all possible d20 values (210) divided by the number of possible values (20).

And then here's where it gets interesting. On the same load of the page that gives us 10.5084, roll20 shows us that each value of the d20 gets approximately the same number of rolls. This doesn't appear to be true at all for paizo's roller, as the chart for Oladon's results shows us that the number of rolls for each value varies wildly.

However, that may only be appearances, as roll20's statistics are based on 872,160 rolls (on the same load of the page I've been quoting in this post) while Oladon's statistics are only based on 10,000. Even on the roll20 graph, there still appears to be some variation in the number of rolls of each value, though it is on a much smaller scale relative to the number.

If we were to repeat Oladon's experiment but with a number of rolls much more comparable to that of roll20, would we see a graph similar to roll20's chart, or would we still see a graph with a very wide range of variation?

Anyway, this was fun to analyze and think about.

[Cort Odekirk]

We considered basing our RND generator on quantum fluctuations but building a particle accelerator under the building proved problematic from a permit fees perspective.

We tried throwing grapes really hard through some circular pvc piping, but it just wasn't the same.

[Hawkmoon269]

Cort, have you considered just building multiples of these for the various size dice? I'm sure the loud humming and churning noises would help keep away all those raptors and other creatures that seem to plague the Paizo warehouse.

[Aniuś the Talewise]

Hawkmoon269 wrote:

Cort, have you considered just building multiples of these for the various size dice? I'm sure the loud humming and churning noises would help keep away all those raptors and other creatures that seem to plague the Paizo warehouse.

this is a thing of exquisite madness and beauty. I love it.

[Cort Odekirk]

Aniuś the Talewise wrote:

Hawkmoon269 wrote:

Cort, have you considered just building multiples of these for the various size dice? I'm sure the loud humming and churning noises would help keep away all those raptors and other creatures that seem to plague the Paizo warehouse.

this is a thing of exquisite madness and beauty. I love it.

LOL, that's fantastic and the noise would probably drive everyone out of the building.

But I want one.

[Orfamay Quest]

Aniuś the Talewise wrote:

Computers are not natively capable of true randomness. All a computer can do is do math with the input it has been given. As I understand it, a computer can simulate pseudo-randomness and the resulting numbers appear to be random enough for most purposes (gaming being one of them), but this is not true randomness and there will always be a little bit of bias in the results.

Bias isn't usually the problem. Predictability is. It's fairly simple to build a pseudorandom generator that will create a truly uniform distribution, or any specific distribution you are interested in. The problem is that if I know the last several numbers, I can possibly predict the next one.

[Aniuś the Talewise]

Orfamay Quest wrote:

Aniuś the Talewise wrote:

Computers are not natively capable of true randomness. All a computer can do is do math with the input it has been given. As I understand it, a computer can simulate pseudo-randomness and the resulting numbers appear to be random enough for most purposes (gaming being one of them), but this is not true randomness and there will always be a little bit of bias in the results.

Bias isn't usually the problem. Predictability is. It's fairly simple to build a pseudorandom generator that will create a truly uniform distribution, or any specific distribution you are interested in. The problem is that if I know the last several numbers, I can possibly predict the next one.

I see!