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Calendar of a distant planet

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On the planet Kest the week has $6$ days, numbered $0$ to $5$ . A year has $4$ months of $23$ , $19$ , $23$ and $19$ days in that order. A year divisible by $3$ but not by $12$ is long: its last month has $20$ days instead of $19$ . Day $1$ of month $1$ of year $1$ is weekday $0$ , and the weekday number goes up by $1$ (back to $0$ after $5$ ) with every day, continuing across month and year boundaries. In years $1$ to $60$ , how many months begin on weekday $0$ ?

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Hint

Work out the weekdays of the $4$ month starts in a year as offsets from the weekday of its first day, and see how the year start moves from year to year.

Solution

In any year starting on weekday $w$ , the month starts are $w$ , $w + 23$ , $w + 42$ and $w + 65$ , which are $w$ , $w - 1$ , $w$ and $w - 1$ modulo $6$ , so a year contributes $2$ months starting on weekday $0$ if $w$ is $0$ or $1$ , and none otherwise.
An ordinary year has $84$ days, a multiple of $6$ , so the next year starts on the same weekday; a long year has $85$ days and shifts the start by $1$ .
The long years up to $60$ are the multiples of $3$ other than $12$ , $24$ , $36$ , $48$ and $60$ , so the start weekday of year $y$ is the number of long years before $y$ , modulo $6$ .
This puts the start on weekday $0$ in years $1$ to $3$ , $22$ to $27$ and $46$ to $51$ ( $15$ years), and on weekday $1$ in years $4$ to $6$ , $28$ to $30$ and $52$ to $54$ ( $9$ years).
These $24$ years contribute $2$ months each, so the total is $2 \times 24 = 48$ .

Answer: $48$

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