This is a question from a hungarian math contest from the year 1948.
It was Saturday on the 23rd October, 1948. Can one conclude that New Year falls more often on Sundays than on Mondays?
Well, I really have no idea how to start on this. It was on a contest and I am unsure what the contestants could use as aid, but I assume basically nothing. The question doesn´t say when there are leap years, altough the year 1948 WAS a leap year so contestants may be aware of the fact.
I have solution to the question in the book but I prefer to solve these problems on my own so I am preferably looking for some HINTS.
Any help is appreciated.
The number of years in a 400-year period that start on a given day of the week is seventy-one minus the number of leap years that start the day before (since each leap year causes a day of the week for January 1 to be skipped).
From 1601 to 2000, the leap years that started on Saturday were 1628, 1656, 1684, 1724, 1752, 1780, 1820, 1848, 1876, 1916, 1944, 1972, and 2000. (total=13)
Also, again from 1601 to 2000, the leap years that started on Sunday were 1612, 1640, 1668, 1696, 1708, 1736, 1764, 1792, 1804, 1832, 1860, 1888, 1928, 1956, and 1984. (total=15)
Hence, $71-13=58$ out of every 400 years start on Sunday, while $71-15=56$ out of every 400 years start on Monday. So, the quote in the title is indeed correct.