# Given a day of the week and the day of the month, what is the range of time within which it will uniquely specify a single date?

In other words, given "2 Tues", (e.g. today, 2 July 2013), for how long must I wait until it is Tuesday on a 2nd of the month again?

How does this interval change for each week? Is it constant or does it fluctuate? If it fluctuates, is there a minimum bound on this duration of "day/day-of-week certainty"?

P.S. I actually have a real reason to use the answer for. You see, in my current zsh shell command prompt on my terminal on my computers, I print out the day of the week (so I can reasonably tell how recently I did something), and also the day of the month (because something more than 7 days old would be ambiguous without at least this info). The goal was to produce a format that is easy to parse and helpful for quick relative calculations (e.g. "Tues" is more helpful than "7/1/2013", if for no other reason than the fact that I'm more apt to know the day of the week than the day of the month). I can't think of a practical reason not to also stick the month in there (as it'd take up only two or three extra characters), so clearly the practical importance is low, but it did get me thinking about whether adding the month really would reduce the uncertainty period by a factor of exactly 12 or not.

Edit: You know, I'm not sure how e.g. 28 days in February (and how that is divisible by 7) slipped past me. Feeling stupid right about now.

• For example (this was the first thing I tried): "1 Friday" happens three times in 2013, including February 1 and March 1. – Jonathan Jul 2 '13 at 4:53
• You shouldn't feel stupid. Lots of things are obvious once they're pointed out. – MJD Jul 2 '13 at 5:13
• I like to hold myself to high standards. I like to imagine questions as interview questions. For some reason. – Steven Lu Jul 2 '13 at 5:14

Note that (except in leap years) February 1 and March 1 must obviously fall on the same day of the week, and so too will February $n$ and March $n$ for any $n$, since February is exactly 4 weeks long.

Similarly, September $n$ and December $n$ always fall on the same day of the week, because September + October + November adds up to 30+31+30 = 91 days = exactly 13 weeks.

Similar analysis, or staring at a calendar, shows that in the same way, April and July's dates always fall on the same days of the week (as do January's in leap years), and also March and November (and Feburary, in non-leap years), October and January in non-leap years, and August and February in leap years.

May and June, however, are perfectly safe. If I tell you that I got married last year on Saturday the 26th, you know it must be May, because May is the only month last year whose 26th day fell on a Saturday. May dates are always unambiguous in this way, and so are June dates. Additionally, February and October's dates are unambiguous in leap years, and August's in common years.

So it is quite ambiguous. A given date will be ambiguous a little over ¾ of the time.

The basic problem is that there are twelve possible months that a particular date could fall in, and the seven days of the week cannot give enough information to disambiguate these. Or put another way, there are twelve months and only seven weekdays, so you know some of the months must be doubling up.

• Ah, so the lower bound there is 28 days due to February. Much worse a result than I imagined it would be! – Steven Lu Jul 2 '13 at 5:03