# Solution: ‘Friday the 13th’

## Introduction

Our Insights questions this month were based on the vagaries of the modern calendar and that eternal question about any specified date: “What day of the week is that?” Our first two questions concerned the frequency of Friday the 13th’s, which some consider an unlucky day.

## Question 1:

The year 2017 began with a Friday the 13th in January, and another one is due in October. What’s the maximum and minimum number of Friday the 13th’s that there can be in a Gregorian calendar year?

This question can, of course, be solved by brute force methods, but can you find an easy way to answer it that you could conceivably even do in your head?

The maximum number of Friday the 13ths in a year is three, and the minimum is one, as Cameron Eggins explains. Here’s a way to think about it: If two given months start on the same day of the week, then the day of the week for the 13th of both months will also be concordant. So all we need to do is figure out which months will start from the same days of the week, for both nonleap and leap years.

Let’s map the days of the week to the numbers 0 through 6, and assign the base number 0 for January. We can now find the offsets to this base number for each subsequent month by casting out complete weeks. Basically, we perform modulo arithmetic: Take the number of days in the month, divide by 7 and add the remainder to the previous month’s number, and reduce the sum to a number below 7, if necessary, to get the number for the next month. January has 31 days, which is four complete weeks plus three days, so February’s base number is 0 + 3, which is 3. Continuing this way, we obtain the numbers 0, 3, 3, 6, 1, 4, 6, 2, 5, 0, 3, 5, which give the offsets for the 12 months in a nonleap year. The number 3 occurs three times, for February, March and November, and this is the maximum number of occurrences of all the numbers. If any one of these three months has a Friday the 13th, so will the other two months. Hence three is the maximum number of Friday the 13ths that it is possible to have in a nonleap year. Now notice that each of the seven numbers occurs at least once, which means that all starting days of the week are represented, so you cannot avoid having at least one Friday the 13th in a nonleap year. Doing the same procedure for leap years does not change our maximum and minimum, which remain 3 and 1 respectively.

Incidentally, if you memorize this string of numbers that represent the months — 0 3 3 6 1 4 6 2 5 0 3 5 — you can figure out the day of the week for any date using simple addition and casting out 7s. Let’s take Oct. 13, 2017. Map the weekdays to the digits 0 through 6 such that 0 = Sunday and 6 = Saturday. Add the number of years since 2001 to the number of leap years since 2001: 16 + 4 = 20 = 6 (mod 7). Now add the date and the offset number for the month, giving 6 + 13 + 0 = 19 = 5, which is a Friday. You can do it pretty quickly in your head with some practice. For dates in the 1900s use the number of years plus number of leap years since 1900.

## Question 2:

Suppose that instead of being spooked by Friday the 13th, you consider it to be your lucky day, and you want to maximize the number of Friday the 13th’s in a year. You are allowed to tamper with the monthly distribution of days in a normal nonleap year in the following way: You can take away one day from any month of the year and add it to any other. For instance, you could, like Robin Hood, rob the day-rich December of one day, reducing it to 30 days, and bump up February’s quota to 29 days. Or you could, like a kleptocrat, decree that January has 32 days while poor February has just 27. What’s the maximum number of Friday the 13th’s you could create in this way in a single year? What if you could do the above procedure for two pairs of months, without using any month twice?

The answers to the two questions are 4 and 5.

You can solve this by inspecting the string of numbers we obtained above: 0 3 3 6 1 4 6 2 5 0 3 5. There are already three 3s, so it is simplest to try to maximize them. Notice that there is a 4 and a 2, and we can convert both to 3s by performing the day-borrowing procedure described on adjacent months. By taking a day from May and giving it to June, we can surgically alter the 4 to a 3, thus adding a fourth 3 and potentially creating a fourth Friday the 13th. Similarly, by borrowing a day from August and giving it to July, we can change the 2 to a fifth 3, without changing any of the other offsets. So our string of offsets is now 0 3 3 6 1 3 6 3 6 0 3 5, which includes five 3s. This means that if February has a Friday the 13th, so will March, June, August and November. Lucky you!

Pete Winkler pointed out that the 13th is in fact more likely to be a Friday than any other day of the week, something, he said, that was proved by a 13-year-old! Just knowing this fact, it is possible to conclude that there is an integral number of weeks in a time period of 400 years. Do you see how?

## Question 3:

On the subject of days and dates, there is something special about the 12th of March, the 9th of May and the 11th of July that requires a global perspective to appreciate.

Each of these dates has a “twin” date with which it shares a special property. Can you figure out what it is? Note: There are some other pairs of twin dates (how many?) that have a similar property, but the three that are mentioned above (with their twin dates) possess it to a degree that is ahead of the others by leaps and bounds.

This question was correctly answered by amrith raghavan:

“The 12th of March, the 9th of May and the 11th of July share the property that they fall on the same day of the week even if the month and day of the dates are switched. That is, these dates would fall on the same day of the week if written the American way (mm/dd/yyyy) or the British way (dd/mm/yyyy).”

Yes, indeed! These dates fall on the same day whether they are written in the globally more common European/imperial format (DD-MM-YYYY) or in the American format (MM-DD-YYYY). These three pairs of dates are unique in that they fall on the same day in both nonleap and leap years:

12-03/03-12

09-05/05-09

11-07/07-11

There are six other pairs of dates that share this property, but three of them work only in nonleap years (01-07/07-01, 01-11/11-01 and 02-08/03-02) or leap years (01-06/06-01, 02-03/03-02 and 02-12/12-02). Several years ago, I constructed a science fiction adventure based on this fact.

In the Insights column, we also discussed calendar reform. In response to one suggestion that involves having 13 months of 28 days each, with one or two additional extracalendar holidays, I commented that the above system does not preserve quarters of three months and had asked, “Can you think of a way that preserves weeks of 7 days, has months of about 30 days (let’s say 30 plus or minus no more than 5 days), preserves equal quarters and equal seasons, and does not need a new calendar every year?” However, after reading the comment by David Prentiss, I agree that there is no need to try to preserve four-month quarters, as it would require the insertion of extracalendar days four times in the year. Instead, it is much easier to redefine the quarter to be exactly 13 weeks (three 28-day months plus one week, a very small adjustment). The beauty of the 13th-month calendar, as David Prentiss stated, is that it avoids this by putting all adjustments (which are limited to just one or two days) at the year’s end. So I rescind the question: I think there is no doubt that the 13-month calendar is most logical, and we should move to it. Of course, there is little chance of that happening in the near future, if ever. Calendar reform faces a great deal more entrenched resistance than just from triskaidekaphobes!

Finally, I asked for readers’ views on what the base year for a universal human calendar should be. Michael Ahern suggested 1969, when human beings first landed on the moon. That’s definitely a good candidate. But for me, no other choice can come close to the year when Charles Darwin published what the philosopher Daniel Dennett has called “the greatest idea that anyone ever had” — the theory of evolution by natural selection. For the first time in our history, we as a species glimpsed our true origins. The year 1859 marks the emergence of our species from its intellectual childhood, from the realm of magic and fantasy into the world of rational thought.

As usual, it was not easy to decide the winner of the *Quanta* T-shirt this month. I’ve decided to give it to amrith raghavan, based on his answer to Question 3. Considering how he ended his comment, I hope he is sober now! Cameron Eggins just misses out. See you next month for new insights.