On what day of the week does the 13th of a month fall most often during a Gregorian 400 year cycle?

Number Theory Level 3

This problem's question: On what day of the week does the 13th of a month fall most often during a Gregorian 400 year cycle?

If a year number is not divisible by four evenly, then it is a 365 day year (ordinary), otherwise if a year number is not divisible by 100 evenly, then it is a 366 day year (leap), otherwise if a year number is not divisible by 400 evenly, then it is a 365 day year (ordinary), otherwise it is a 366 day (leap) year.

January 1, 2001, was a Monday.

Wednesday Monday Friday Saturday Thursday Sunday Tuesday

2 solutions

Kyle T
Jun 4, 2019

Just confirming, I got the same counts as you
Friday happens once more than any other day
(Friday the 13th oooooh spooky)

<?php
$results = array();
$days = array('sun','mon','tue','wed','thu','fri','sat');
for($i=0;$i<=6;$i++){
    $results[$days[$i]] = 0;
}
for($y=2001;$y<=2400;$y++){
    for($m=1;$m<=12;$m++){
        $time = mktime(0,0,0,$m,13,$y);
        $results[$days[date('w',$time)]]++;
    }
}
echo '<pre>'.print_r($results,true).'</pre>';
/*
Array
(
    [sun] => 687
    [mon] => 685
    [tue] => 685
    [wed] => 687
    [thu] => 684
    [fri] => 688
    [sat] => 684
)
*/
?>

Actually, considering that my development of the counts was done in Wolfram Mathematica 12, our methodologies are very similar.

A Former Brilliant Member - 2 years ago

A Former Brilliant Member
Jun 4, 2019

The counts:

$\begin{array}{lr} \text{Monday} & 685 \\ \text{Tuesday} & 685 \\ \text{Wednesday} & 687 \\ \text{Thursday} & 684 \\ \text{Friday} & 688 \\ \text{Saturday} & 684 \\ \text{Sunday} & 687 \\ \end{array}$

On what day of the week does the 13th of a month fall most often during a Gregorian 400 year cycle?

2 solutions

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