In some of our recent Insights puzzles, we’ve tackled questions related to time: How much time does it take for a gene’s rate of evolution to slow to a virtual standstill? How many half-lives does a pound of radioactive material have? What can an LCD display teach us about time’s arrow? This month, let’s take a look at a different aspect of time — how we keep track of it.

Our subject is the calendar and that oft-repeated question when a date is specified: “What day of the week is that?” Sometimes, if the date isn’t too far into the future, we can figure it out in our heads, but often we need to check the calendar. Calendars, like culture and language, are unruly things that have arisen organically by happenstance. They try to balance many competing demands, such as astronomical compliance (“A year must have 365 days — well, except in leap years” or “A month should be as long as a lunar cycle — kind of”), social/religious conventions (“A week should have seven days”), historical accidents (“Why is the 10th month called October — shouldn’t that be the eighth month?”) and regional and religious identity (e.g., Chinese, Hebrew, Islamic and Hindu calendars, among others). Most calendars do have some elements of logic and order, but these are often afterthoughts. Every calendar has its version of “Are there 30 or 31 days in April?” And if you add the superstitious fear of Friday the 13th to the chaos, it gives us plenty of fodder for our puzzle today.

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?

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 from 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?

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.

If nothing leaps out at you, here are a couple of hints. (Click on Hint 1 and Hint 2 to make them appear.)

Hint 1: To appreciate the relationship between the two dates that form a pair, you need to look at them from both sides of the pond.

Hint 2: Are there other ways to write a date than the one you are used to?

Such puzzles could not have existed if we had used top-down logic to reform our unruly calendars. There are many such proposals, all of which try to render obsolete the pesky question “What day of the week is that?’’ Such reformed calendars would not need to be printed every year just to let us know what day of the week a specific date falls on. One such suggestion involves having 13 months of 28 days each, with one or two additional extracalendar holidays that could be called something like “End-of-Year Day” or “Leap Day.” These days would be intercalary, or off-calendar, days. They would not be assigned to any month and would not be assigned a day of the week, either — they would be super-Sundays! With this scheme, you wouldn’t even need separate monthly calendars: You could just figure out the day of the week from the numerical date, no matter what month it is.

The above system is logical, but it suffers from one flaw: It has 13 months, and 13 is not just considered unlucky, but also isn’t divisible by four. This means that the division of the year into four quarters, or four seasons, of about three months each is not preserved. So let’s get creative. 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? You can designate up to 7 days off-calendar holidays that do not have to belong to any month or week or both. It’s always nice to give yourself extra holidays!

In the spirit of dreaming up logical solutions that have no chance of being followed in the real world, let me tell you about my own suggestion for calendar reform. It concerns the base of the calendar — the 0th year from which we define the present year. Different calendars handle this differently, based in a parochial way on mythical or religiously significant events, such as the birth or enlightenment of a religious leader. Could we ever agree on a universal scientific justification to start counting years from a specific date when a particular intellectual realization emerged? Which idea would you choose? By my calendar, we are currently living in year 158 YI (Year of the Insight).

Readers are welcome to weigh in with alternative suggestions.

*Editor’s note: The reader who submits the most interesting, creative or insightful solution (as judged by the columnist) in the comments section will receive a *Quanta Magazine* T-shirt. ( Update: The solution is now available here.) And if you’d like to suggest a favorite puzzle for a future Insights column, submit it as a comment below, clearly marked “NEW PUZZLE SUGGESTION” (it will not appear online, so solutions to the puzzle above should be submitted separately).*

*Note that we may hold comments for the first day or two to allow for independent contributions by readers.*

How the Calendar Should Be Organized

• Every year should start out on Sunday, January 1 and end on Saturday, December 28.

• In non-leap years, there will be one extra day called “the Extra Day” that is for yearly review and planning.

• On leap years there will be another extra day called “Another Eextra Day” that is for 4-year review and planning.

• Each season will be 13 weeks long, starting at the beginning of the year for winter.

• Each 13-week season will be broken into 3 months of 4 weeks each, plus an extra week called “an Extra Week” (An Extra Week Winter, An Extra Week, Spring, …) that is for seasonal review and planning.

• With this calendar, every date of every month would always be on the same day of the week.

It's a fact that the 13th is more likely to be a Friday than any other day of the week; see, for instance, S. R. Baxter, The Mathematical Gazette Vol. 53, No. 384 (May, 1969), pp. 127-129 for a proof submitted, appropriately, by a 13-year-old.

Dear Sir,

The flaw you point out in the 28 day 13 month calendar that 13 is not divisible by 4 is Not a flaw if a financial Quarter is redefined to be 5 weeks rather than 4. Note that 13/4=3.25 or 4 weeks + 1 week. All such Quarters would have precisely 91 days unlike the current system where in 2017 Q1 has 90 days. Q2 91,

Q3 92, Q4 92. The 13 month or Lunar Calandar avoids this by putting all adjustments off the page at the years end.

Regards, Dave Prentiss

In computer science it is common to refer to epoch time as January 1, 1970. Please see: https://en.wikipedia.org/wiki/Unix_time

An alternative nearby suggestion is July 20, 1969, when man landed on the moon. Leaving the Earth and walking on another celestial body is a significant milestone for humanity.

Question 1:

Assign each day a number in Z7: Monday = 0, Tuesday = 1, … Sunday = 6

It follows that on a non-leap year, if the nth of January is the day D, the rest of the months have the day on the nth:

January = D

February = D + 3

March = D + 3

April = D + 6

May = D + 1

June = D + 4

July = D + 6

August = D + 2

September = D + 5

October = D

November = D + 3

December = D + 5

Which can be figured by taking the number of days mod 7 in one month and adding it to the next month. E.G 31 = 3 mod 7, and thus as the nth of January is D, it will be D + 3 on the nth of February.

On a leap-year:

January = D

February = D + 3

March = D + 4

April = D

May = D + 2

June = D + 5

July = D

August = D + 3

September = D + 6

October = D + 1

November = D + 4

December = D + 6

The maximum number of days that are the same on any particular nth of a month is 3, and thus there is a maximum of 3 Friday the 13ths. The minimum is 1. This comes from the frequency of the days in the above results.

Question 2:

Again assigning the nth of January as D, we obtain:

January = D

February = D + 3

March = D + 3

April = D + 6

May = D + 1

June = D + 4

July = D + 6

August = D + 2

September = D + 5

October = D

November = D + 3

December = D + 5

By removing a day from December and adding it to August we obtain:

January = D

February = D + 3

March = D + 3

April = D + 6

May = D + 1

June = D + 4

July = D + 6

August = D + 2

September = D + 6

October = D + 1

November = D + 4

December = D + 6

And here D + 6 appears with a frequency of 4, and thus there can be 4 Friday the 13ths in a normal year. A similar result can also be obtained by removing a day from August and moving it the March or February. I'm also fairly sure that the maximum is also 4 is we are allowed to do this to two pairs of months, such as moving a day from December to August and moving a day from January to February.

https://archive.org/details/KpH1966Calendar

. . . here above is given a URL for a perpetual calendar on a single page, along with an algebraic expression for computational purposes . . .

It's time for mankind to abolish the seven-day week, especially when fewer and fewer businesses and states recognize a sabbath. The seven-day week came into existence when Pharaoh ruled and Moses was his secretary of labor. The time is long since gone. If I had a business, or an institution, that had to be operating every day, I would organize my labor into six-day cylces, where on any given day two thirds of my employees are working and one third is off and in any six-day period each employee has two days off.

Each quarter to have 3 months of 4, 5 and 4 weeks (each week having 7 days). The twelfth month to have 5 weeks every 5 or 6 years (a leap week – this will also have benefit of keeping the Abrahamics happy). The northern winter solstice to fall in the last week of the year, hence triggering the leap week. The change-over to occur for the year 2042 – which will become 424242 so that all of human history, but obviously not all of human evolution, will have a positive number (I know this is a daft idea, but Douglas Adams would have approved, and that's good enough for me).

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/yy).

Three additional date pairs obey this rule for the current year (Jan 7, 11 and Feb 8), but the original three are unique in that they work for all years (including leap years, and yrs divisible by 100 and not by 400).

You can derive these dates by solving for Zeller's congruence https://en.wikipedia.org/wiki/Zeller%27s_congruence

Apart from that curious fact, the other thing that is common to those date pairs is that they are all fine days to get drunk.

May 9th is National Moscato Day

July 11th is National Mojito Day, and

December 12 is National Ambrosia Day..

Hic!

There is a different mathematical formula that explains the fact of the rotation of time!

I have thought often about how we can improve the calendar. I suggest that the new year begins on the Sunday after the winter solstice and each month has 30 days divided into 5 6-day weeks. We get rid of Tuesday. At the end of December, we add a 3 day midwinter holiday and a 2 day midsummer holiday at the end of June, the latter becoming 3 day holiday during the leap year. The names for these holidays need to be chosen. Birthdays that now fall on the 31st are moved to the 30th.

While these suggestions for calendar alteration are well-beyond my mathematical abilities, I do have one insight that seems important.

The Summer Solstice (around June 20th) and the Winter Solstice (around December 20th) are arbitrary due to the seasons in the northern and southern hemispheres.

The Vernal Equinox (March 21 more or less) and the Autumnal Equinox (September 21 more or less) are days in which our whole world experience 12 hours of day and 12 hours of night. They could be designated Equinox North (when the sun's rays are perpendicular heading to the Tropic of Cancer) and Equinox South (when the sun's rays are perpendicular heading to the Tropic of Capricorn).

The best choice for a "new years day" might be the Perihelion in January when the Earth is closest to the Sun, about 2 weeks after the "Winter" Solstice. I opine this because the onset of seasonal weather rarely coincides with our calendar or with the Earth-Sun relationship (e.g."first day of winter" or "last day of summer"). Simply put, the "shortest"day, the day with the least aount of possile daylight, generally, in all places around the world, may hardly ever be experienced as the coldest or "Wintry-est" day. The Seasons usually lag due, in part, to the predictable incoming solar radiation budget (which has little to do with the Perihelion or the Aphelion!).

Thanks for allowing me this opportunity to think like a geographer !