Counting Sundays

You are given the following information, but you may prefer to do some research for yourself.
1 Jan 1900 was a Monday.
Thirty days has September,
April, June and November.
All the rest have thirty-one,
Saving February alone,
Which has twenty-eight, rain or shine.
And on leap years, twenty-nine.
A leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400.
How many Sundays fell on the first of the month during the twentieth century (1 Jan 1901 to 31 Dec 2000)?

We will count the number of Sundays that fell on the first day of the month from 1 Jan 1901 to 31st Dec 2000.

Firstly, note that the days of the week repeat every 7 days and the sequence of “first day of the month” repeats every 28 years in a calendar that includes a leap year every 4 years.

However, the statement “A leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400” breaks the 28-year cycle every 100 years, but then reinstates it every 400 years. Therefore, for the purpose of this problem, we can consider that the sequence of “first day of the month” repeats every 100 years, as the years 1901-2000 do not include the year 2000 itself.

We will now count the number of Sundays on the first day of the month for the years 1901 to 1930 (a 28-year cycle) and then multiply by the number of cycles (100/28).

1 Jan 1901 was a Tuesday. So let’s start with January, 1901 and look at what days of the week the first of each month will land on.

1. January 1, 1901: Tuesday (3 days after Monday)

We add the number of days in the current month (mod 7, because the 7 days of the week repeat) to find the first day of next month.

2. February 1, 1901: Friday (31 days after Tuesday)

3. March 1, 1901: Friday (28 days, because 1901 is not a Leap Year)

4. April 1, 1901: Monday (31 days)

Note: Each time we find a Sunday, we add one to our total count.

Let’s jump to 1904, because it’s a Leap Year:

1. January 1, 1904: Friday

2. February 1, 1904: Monday (One Sunday)

March 1, 1904, would be 29 days after a Monday, which is a Wednesday (not a Sunday).

When we do this for each month from 1901 to 1930, we get 34 Sundays. Then we consider the fact that this 28-year pattern would repeat 3 times in the 100 years from 1901 to 2000.

So, we need to multiply 34 Sundays by 3: 34 Sundays * 3 = 102 Sundays.

Now, there will be 16 remaining years from 1985 to 2000. When we do the same calculation for these remaining years, we get 8 additional Sundays.

Finally, add 102 Sundays and 8 Sundays to get your answer: 110 Sundays fell on the first of the month during the twentieth century (1 Jan 1901 to 31 Dec 2000).

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