Leap Monday
Can you find the next time that the 29th of February will fall on a Monday?
Age
11 to 14
Challenge level
Problem
In 2016, the 29$^{\text{th}}$ of February was a Monday. When will be the next leap year when the 29$^{\text{th}}$ of February is a Monday?
This problem is adapted from the World Mathematics Championships
Student Solutions
Answer: 2044
Listing the days of the week of 1$^\text{st}$ March
365$\div$7 = 52 remainder 1
29 Feb 2016: Monday
1 March 2016: Tuesday
364 days later is Tuesday again so
1 March 2017: Wednesday
1 March 2018: Thursday
1 March 2019: Friday
1 March 2020: Sunday (leap year)
2021: Monday
2022: Tuesday
2023: Wednesday
2024: Friday (leap year)
Follow the leap year pattern
2028: Wednesday
2032: Monday
2036: Saturday
2040: Thursday
2044: Tuesday (so 29th February will be Monday)
Finding a pattern of the days of the week of the 29$^{\text{th}}$ of February
It will be the 29$^{\text{th}}$ of February again after 365$\times$3 + 366 = 1461 days.
1461 $\div$ 7 = 208 remainder 5, so 1461 days is the same as 208 weeks and 5 days.
The next 29$^{\text{th}}$ of February after 2016 will be in 2020, and it will be 5 days after a Monday - a Saturday.
Similarly, the 29$^{\text{th}}$ of February 2024 will be 5 days after a Saturday - a Thursday. And the 29$^{\text{th}}$ of February 2028 will be 5 days after a Thursday - a Tuesday. Similarly, the 29$^{\text{th}}$ of February 2032 will be a Sunday, the 29$^{\text{th}}$ of February 2036 will be a Friday, the 29$^{\text{th}}$ of February 2040 will a Wednesday, and the 29$^{\text{th}}$ of February 2044 will be a Monday.
Interesting aside to think about: 7 is a prime number, so whatever remainder we get after dividing by 7 (providing it is non zero), we will always have to go for 7 lots of the leap years to get back to Monday. So the answer will be $7 \times 4 = 28$ years time if the remainder is not zero. This is an example of Fermat's Little Theorem.
Using Lowest Common Multiples
Every 7$^{\text{th}}$ day is a Monday, so 7 days, 14 days, 21 days, ”¦ after 29$^{\text{th}}$ February 2016 will all be Mondays.
It will be the 29$^{\text{th}}$ of February again after 365$\times$3 + 366 = 1461 days, and again after 2$\times$1461 days, and again after 3$\times$1461 days, and so on.
This means the number of days between Monday 29$^{\text{th}}$ February 2016 and another Monday 29$^{\text{th}}$ February will be a multiple of 7 and a multiple of 1461. So the number of days until it next happens will be the lowest common multiple of 7 and 1461.
If 1461 were a multiple of 7, then the lowest common multiple would be 1461. However, you can tell by dividing 1461 by 7 that it is not a multiple of 7. So, since 7 is a prime number, the lowest common multiple of 7 and 1461 must be 7$\times$1461. That is after 7 sets of 4 years - so it will next happen 7$\times$4 = 28 years later, in 2044.