#### Problem Of the week: 15th May 2023

#### Dylan Guo

You want to create a family tree, but you are having trouble tracking down your grandmother’s birthdate. The only clue you found was a letter written from your great grandfather to your great grandmother on the day you grandmother was born. Unfortunately, some of the characters were smudged out, represented here with a “___”. (The length of the line does not reflect the number of smudged characters.)

“Dear Elizabeth,

Little did I know when I headed to work this Monday morning, that by evening we would have a beautiful baby girl. And on our wedding anniversary, no less! It makes me think back to the day on that incredible weekend, J___ 27th, 19___, when we first shared our vow to create a family together, and, well, here we are! Happy eighth anniversary, my love.

Love, Charles”

Using this information, can you deduce when your grandmother was born?

Maybe have a look at John Conway’s Doomsday Algorithm.

Every four years there is a leap year (366 days in the year) except on multiples of 100. However, if the year is also a multiple of 400, then it is still a leap year (e.g., 1100 is not a leap year but 1200 is).

John Conway’s Doomsday Rule is his ingenious method to quickly map any date in history with its day of the week. He practiced it daily for years and got so good that he could compute the day of the week for 15 dates in his head in under 10 seconds. However, his method is not required to solve this problem. Instead, all that you need to know is a few things about the calendar:

- There are 365 days in each non-Leap Year.
- There are 366 days in each Leap Year.
- Leap Years occur every four years except in years that are divisible by 100, and except-except in years that are divisible by 400.

Thus, 1900 is the only year in the 20th century that is divisible by 4, but not a leap year, and since 365 divided by seven days of the week has a remainder of 1, any date will move forward one day of the week after every non-leap year. So, if July 31st is a Friday in 2020, we know it will be a Saturday in 2021. And since 366 divided by 7 has a remainder of 2, any date will move forward two days of the week when it passes a leap day. From the letter, we can surmise the following three things:

- The month in which they were married was January, June, or July.
- They were married on a weekend day.
- Eight years later, their anniversary was on a Monday.

Almost all eight-year periods involve passing over six non-Leap Years and two Leap Years. This means you’d advance by one day of the week for each of the non-Leap Years, and two days of the week for each of the two Leap Years. In total, you’d advance by 10 days of the week, which is the same as advancing by 3 days of the week. This means that if the wedding date was on a Saturday, eight years later should be a Tuesday, and if their wedding date was a Sunday, eight years later should be a Wednesday. But we know eight years after the wedding was a Monday.

This can only occur if the eight-year period includes only one Leap Year, which means the eight-year period in question must have started in 1900! Furthermore, they must have been married before February 28th, as dates after February 28th in 1900 would still pass two Leap Days over an eight-year period. So, since the month started with J, we now have the complete answer: the marriage must have been on January 27th, 1900, which means your great-grandmother must have been born on January 27th, 1908!