August 13 - A Special Day

Here in the United States, dates are commonly written in a month-slash-day format. April 7th would be written as 4/7 with April, the forth month, followed by a divided-by sign, then 7, for the seventh day.  As you look at the date, it can also be viewed as a fraction, 4/7th.

I wondered if there were any days where the representation of the day, month-slash-day, would correspond to the portion of the year that had passed. I hadn't run across anyone who had looked into that before. Ultimately, I coded some Excel formulas and found the days and found that 8/13, August 13, contained the moment which was 8/13ths of the way through a non-leap year.

8/13

But before I did that, I thought through some scenarios:

Improper Fractions
If the answer turned out to be 6/10 or 3/9, I would not consider that a good answer because something special like a day that represents the fraction of the year that has passed can't be "well, if you arrange it as an improper fraction, ...". If I didn't find any days, I'd scrap the analysis.

At most 12 Special Days
Within the domain of a single month, the fractions would proceed n/1, n/2, n/3, n/4, ... until n/31. Each successive day is a smaller number, yet, with each successive day, a greater portion of the year has gone by. It was unlikely (provably so) that no month could have more than one special day.

Had I known at this point that leap years mattered, I should have concluded that there are at most 24 special days with this quality.

Casting out Days in January
My first case was the month of January. Thinking through the 31 days of January, I was able to exclude the dates 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11 right away because as a fraction, they would represent a date too far after the end of  January. Likewise, 1/15 was about 1/24th the way through the year, so a conditionally concluded that 1/12, 1/13, 1/14, and 1/15 would likewise represent a date too late, but also, 1/24, 1/25, 1/26, through 1/31 could be eliminated, because as a fraction, they would be earlier than their date representation.

Casting out Days in Other Months
After working through what could be eliminated in January, I concluded that for any month N, days N/1, N/2, N/3, ... N/N could be eliminated because the fraction was greater than 1, and thus after 12/31. With each passing month, there are fewer days that can be considered as a possible solution.

Leap Year
I wasn't sure at first if Leap Year would play a role. Ultimately, my brute-force analysis in Excel verified that it did matter.

Brute Force
Excel provides the ability to perform date constructions and numerical calculations. I generated two columns, month, and day, and populated the columns with a formulasMOD(ROW(),12)+1 and MOD(ROW(),31)+1. Since 12 and 31 are relatively prime, I knew that 372 rows would cover all possible date combinations.

I needed a reference year, which I assigned to an anchored cell. Then using VALUE() I generated all possible dates, and all fractional increments from year, to year+1. Excel coughed on values such as 2/30 or 11/31 which are not valid dates, but for the rest, I could determine if the fractional offset occurred between the start and end of the corresponding date. After all the dust settled, only one day survived.

Balancing Paper Reasoning with Brute Force
As a student of mathematics, I would have preferred to arrive at the solution strictly by paper analysis. As I looked at the problem, I was realizing that the "theory" didn't really help work through the texture of the days. Some months are longer than others based on political decisions, and this type of background doesn't lend itself to purely theorem-level assumptions.

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