Question

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)?

Commentary

The lesson of this exercise: know your modules!

Python

#!/usr/bin/env python
months = [
    31,
    28,
    31,
    30,
    31,
    30,
    31,
    31,
    30,
    31,
    30,
    31
]

week = ['sun',
        'mon',
        'tue',
        'wed',
        'thu',
        'fri',
        'sat'
       ]

leapyears = [year for year in range(1, 101) if year % 4 == 0]

day = 0 # 1 Jan 1901
month = 0 # January
year = 1 # 1901
weekday = 2 # Tuesday

days = {day: week[weekday]} 

since_last_month = 0
since_last_year = 0

first_of_months = {}
while year <= 100:
    #print "%s/%s/%s - %s" % (month+1, since_last_month+1, year+1900, week[weekday])
    # if it's the first of the month, make a note 
    # of the weekday.
    if since_last_month == 0:
        first_of_months["%s/%s/%s" % (month+1, since_last_month+1, year+1900)] = week[weekday]

    # increment the counters..
    day += 1
    since_last_month += 1
    since_last_year += 1
    
    # check what year it is
    days_of_year = 365
    if year in leapyears:
        days_of_year += 1
    if since_last_year >= days_of_year:
        year += 1
        since_last_year = 0

    # check what month it is
    days_of_month = months[month]
    if month == 1 and year in leapyears: # february
        days_of_month += 1
    if since_last_month >= days_of_month:
        month += 1
        since_last_month = 0
        if month > len(months)-1:
            month = 0

    # check what day of the week it is
    weekday += 1
    if weekday > len(week)-1:
        weekday = 0
    days[day] = week[weekday]

def date_sort(date):
    date_str = date[0]
    month,day,year = date_str.split('/')
    return int(month)*int(day)*int(year)

print(len([weekday for weekday in list(first_of_months.values()) if weekday == 'sun']))

$ time python3 first-sunday.py
real   0m0.066s
user   0m0.059s
sys    0m0.008s

Python

#!/usr/bin/env python
from calendar import monthrange; from itertools import product
print(len([(year, month) for year, month in product(list(range(1901, 2001)), list(range(1, 13))) if monthrange(year, month)[0] == 6]))

$ time python3 calendar-module-first-sunday.py
real   0m0.030s
user   0m0.022s
sys    0m0.008s

Ruby

#!/usr/bin/env ruby
require 'date'
puts Date.new(1901,1,1).upto(Date.new(2000,12,31)).find_all { |d| d.mday == 1 && d.wday == 0 }.count

$ time ruby first-sunday.rb
real   0m0.053s
user   0m0.045s
sys    0m0.008s