a When you add two numbers modulo 7, this means that you add them, but then just write down the remainder when you divide the answer by 7.

Thus 19 + 12 (modulo 7) = 3, because 19 + 12 = 31 = 4 ´ 7 + 3, so leaves remainder 3 when divided by 7. Also, 19 + 16 (modulo 7) = 0 because 19 + 16 = 35 is a multiple of 7.

If 13th of January is day 0, then 13th February is day 0 + 31 = 3 (modulo 7). If it isn’t a leap year, then 13th March, April, etc are days 3, 6, 1, 4, 6, 2, 5, 0, 3, 5 modulo 7. Thus in a non-leap year, there can be at most three Friday 13th’s: if February 13th is a Friday (“day 3”) then so will March and November 13th be. In a leap year the sequence is 0, 3, 4, 0, 2, 5, 0, 3, 6, 1, 4, 6, so again at most three Friday 13th’s: if January 13th is a Friday (now “day 0”) so will April and July be. The next year with three Friday 13th’s is 2026.

b If 1st October is Monday, then so will 8th, 15th, 22nd and 29th; similarly, if 2nd or 3rd October is a Monday, the fifth Monday in the month will be on 30th or 31st October. If there are only four Mondays in the month, we cannot have 1st October on Monday, or Sunday (else 2nd would be a Monday) or Saturday (else the 3rd would be a Monday). If 1st, 2nd or 3rd October is Friday, there will be five Fridays in the month, in the same way, so 1st October cannot be Friday, or Thursday (else 2nd would be a Friday) or Wednesday (or 3rd would be a Friday). Thus the only possibility is for 1st October is Tuesday: then 4th, 11th, 18th and 25th will be Fridays, and 7th, 14th, 21st and 28th will be Mondays. The next year when this happens is 2024, though there is nothing special about October, except that it has 31 days: the next month where this happens in a 31-day month is December 2020, though there are also four Mondays in September 2020, which is less surprising; 30 day months can have four Mondays and four Fridays if the 1st of the month is Monday, Tuesday or Wednesday.

Challenge 242: Determining Days