heh, yes... very good point. I've always been amused (and often quite frustrated) at my own ability to overlook the most obvious logic when my mind becomes focussed on one particular aspect of a problem. Something to do with trees and forests, I think :)

However, I should point out that in order to see the full pattern that emerges it is necessary to test for ~400 years. This makes sense of course, given the (Gregorian Calendar) Leap Year Rules.

The output from the following (modified from my original) code gives an indication of the pattern:

#!/usr/bin/perl -l
use strict;
use warnings;
use Date::Calc qw/Days_in_Month Day_of_Week/;

my @days_of_week = qw/Mon Tue Wed Thu Fri Sat Sun/;
my $cnt;
my @cnt;

for my $year (1 .. 2500) {

    my %days_of_month;

    for my $month_of_year(1 .. 12) {
        for my $day_of_month(1 .. Days_in_Month($year,$month_of_year))
+ {
            my $dow = Day_of_Week($year,$month_of_year,$day_of_month);
            $days_of_month{$day_of_month}{$days_of_week[--$dow]}++;
        }
    }

    for my $day_of_month (sort {$a <=> $b} keys %days_of_month) {
        next if scalar keys %{$days_of_month{$day_of_month}} == 7;
        my @missing_days;
        for (@days_of_week) {
            push (@missing_days, $_) if !defined $days_of_month{$day_o
+f_month}{$_};
        }
        $cnt++;
        if ($missing_days[0] eq 'Sun') {
            push @cnt, $cnt;
            if ($cnt == 11) {
                print "$year:@cnt";
                @cnt = ();
            }
            $cnt = 0;
        }
    }
}
[download]

However, it's not so much the pattern that I found interesting - but the fact that every year it's _always_ the same day of the month (31st) that doesn't occur on every day of the week.

Cheers,
Darren :)