49th Kürschák 1948 Problem 1 Knowing that 23 October 1948 was a Saturday, which is more frequent for New Year's Day, Sunday or Monday?
This is hard. The rules are that a year divisible by 4 is a leap year unless it is divisible by 100 but not 400. So 1948, 2000 are leap years, but 1900 is not. There are 365 = 52·7 + 1 days in a non-leap year and 366 = 52·7 + 2 days in a leap year. Suppose that NY day in year 0 is day 0 and that year 0 is the year after a leap year. Suppose that we do not encounter any centuries. So NY day in year 1 is day 1, NY day in year 2 is day 2 etc. We get: Leap * * * * * * * Year 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Day 0 1 2 3 5 6 0 1 3 4 5 6 1 2 3 4 6 0 1 2 4 5 6 0 2 3 4 5 0 This is periodic with period 28 years. Note that in a complete period we have just 4 of each day. Now 100 = 3·28 + 16, so if NY day 2001 is day 0, then NY day 2101 would be day 6. But since 2100 is not a leap year, it is actually day 5. Similarly, NY day 2201 is day 3 and NY day 2301 is day 1. However, NY day 2401 is not day 6, but day 0, because 2400 is again a leap year. So we have a period of 400 years. Note that in years 0-15 above we have 2 of each day except 1 and 3 which both have 3 days. Thus in the 100 NY days starting with day 0 in 2001, we an excess of 1 each for days 1 and 3. 2001-2100 (NY day 2001 is day 0) excess of 1 each for day 1 and day 3 2101-2200 (NY day 2101 is day 5) excess of 1 each for day 6 and day 1 2201-2301 (NY day 2201 is day 3) excess of 1 each for day 4 and day 6 2301-2400 (NY day 2301 is day 1) excess of 1 each for day 2 and day 4 Thus over the whole 400 year period we find that if day 0 has n NY days, then we get day 0 has n day 1 has n+2 day 2 has n+1 day 3 has n+1 day 4 has n+2 day 5 has n day 6 has n+2 Note that 1 Jan = 32 Dec = 62 Nov = 93 Oct = 23 Oct + 70, so NY day 1949 is a Saturday. Hence NY day 1977 also. So NY day 2000 (23 years later) is also a Saturday (referring to the table above). So NY day 2001 is a Monday (2000 was a leap year). So day 6 is Sunday. Hence every 400 years NY day occurs more often on a Sunday than a Monday. 49th Kürschák 1948 © John Scholes jscholes@kalva.demon.co.uk 29 Oct 2003 Last corrected/updated 29 Oct 03