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Month equals day of month

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In the Gregorian calendar, call a date matching if its day of the month equals its month number, for example the $3$ rd of March or the $11$ th of November. Weekends are Saturdays and Sundays. How many matching dates in the years $2027$ , $2028$ , $2029$ , $2030$ and $2031$ together fall on a weekend?

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Hint

There are $12$ matching dates per year. Check the weekday of the dates in each year separately, and remember that $2028$ is a leap year.

Solution

The matching dates $\frac{4}{4}$ , $\frac{6}{6}$ , $\frac{8}{8}$ , $\frac{10}{10}$ and $\frac{12}{12}$ of any year share one weekday, so within a year they are all on a weekend or none is.
Checking the weekday of every matching date year by year gives $5$ weekend dates in $2027$ , $3$ in $2028$ , $5$ in $2029$ , $4$ in $2030$ and $1$ in $2031$ .
Adding the $5$ years gives $5 + 3 + 5 + 4 + 1 = 18$ .

Answer: $18$

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