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I have a friend who receives his social security check on the fourth wednesday of every month. So, based on the concept of one month, he thought that each check was four weeks apart, regardless of dates. A month is a month. Given the randomness--periodicity over long periods of years notwithstanding [400 years]--of what day of the week might be what date of the month, I did a cursory glance at the math and accepted it, although it didn't feel right, those straggling days at the end must count for something. This month--November--he has to wait 5 weeks between checks. So, my intuition told me the waiting time varied between 4 and 5 weeks. But, I can't let stuff like that hang in mid-air, so when I got home, I took pen and legal pad and did the math, got my head around it. And what I discovered was that my intuition was wrong.

Now, this may be common knowledge, I don't know, but the waiting time between checks is either exactly four seven-day weeks or exactly five seven-day weeks, the minimum and the maximum. No four weeks plus so many days. Here's the skinnies:

The date of the first occurrence of any day of the week, let's take wednesday, to be specific, varies from the first to the seventh. And, the fourth wednesday of every month falls between the 22nd and the 28th inclusive. Those are the arithmetic constraints. Given a month of 31 days, if wednesday falls on the first--its fist occurrence--and we add 21 days--three weeks--it will put the fourth occurrence of wednesday on the 22nd of that month. Now, if wednesday falls on the first, that means--for a 31-day month--that the fourth wednesday of the previous month fell on the 25th [25 + 7 = 32, the first of the next month]. So, waiting 22 days in this month plus six in the previous gives us 28 days or four seven-day weeks. This is true for dates 25 to 28 inclusive.

Pick another one--the 27th. The next occurrence of wednesday will be on the 34th, or the 3rd of the next month [x-date modulo 31]. Add 21 days [three weeks to get to the fourth wednesday], and you get the 24th. The previous month had four days left--28 to 31 inclusive--plus 24 yields a 28-day [four seven-day weeks] waiting time. This is true when the first occurrence varies from the first to the fourth inclusive. Take the fourth, we add 21 days to have the fourth wednesday on the 25th. But for the first occurrence to be on the fourth means that the previous month's fourth occurrence of wednesday was on the 28th [28 + 7 = 35 or the 4th], leaving three days [29, 30, 31] of a 31-day month [25 + 3 =28]. Okay.

But now look at when the first occurrence is the 5th, 6th, or 7th. If it's on the 5th, the fourth wednesday is 21 days later on the 26th. The first occurrence on the fifth puts the previous month's fourth wednesday on the 22nd because the next wednesday falls on the 29th, and, 29 + 7 = 36, the fifth. Adding the 30th and 31st gives us nine days remaining in the previous month after the last check. So, 26 days plus 9 equals 35 days [26 + 9 =35], or five seven-day weeks. This is true for all three. If it falls on the 7th, it means the previous fourth wednesday was on the 24th [24 + 7 =31]. Seven plus 21 equals 28, and to that add a full seven days waiting time in the former month and we get 35 days or five full weeks.

Nothing in between as I had intuitively misconstrued, no partial weeks, the math don't lie; it is not a continuum. It's deceptive, the idea of a regular event like 'the fourth wednesday of every month' [or of any day, of course] being separated by four weeks sounds rational, almost mathematical. My intuition suspected a varying number between four and five weeks. But to find out that it's either exactly four weeks or exactly five weeks surprised me. Did someone at the SSA figure that out or did they just decide on it for bureaucratic reasons unknown and it just happened to turn out that way? Are there more four-week waiting periods than five? Do they average out over time?

In summation: if a check comes on the 22nd, 23rd, or 24th (of a 31-day month), you have exactly five weeks to wait for the next one. If it occurs on the 25th through 28th inclusive, you have exactly four weeks to wait. For a 30-day month, the 24th moves into the four-week waiting set [24 + 7 = 31 or the first of the next month]. Add 21 and the fourth occurrence falls on the 22nd to which we add 6 days from the previous month [22 + 6 = 28].

We used for our example 'the fourth wednesday of the month.' But the arithmetic works for any week, any day, and in either direction. Going forward, for a 31-day month, if the first occurrence of the day a person receives his check falls on the 1st through 3rd inclusive, he has to wait five weeks for his next one. And if it falls on the 4th through 7th inclusive, he has to wait four weeks. For a 30-day month, similarly, if the first occurrence of your day falls on the 3rd, your wait moves into the four-week period. In other words, if a person receives his check on the 3rd, the 10th, the 17th, or the 24th, he has to wait five full weeks for the next one unless the current month has 30 days. In which case, the interval of waiting is four weeks.

As far as February goes, if it only has 28 days, regardless of what day or week you receive your check, you only have to wait four weeks for the next one. There's one exception: if you get paid on the same day as the first of the month in a February of 29 days, you have to wait five weeks for your next check.

It seems I overcomplicated this whole thing, but, that's the inductive method in action.

Stating the rule: If, in the first week of any month, the date of your day to receive a social security check, plus 28, is less than or equal to the number of days in that month [31, 30, 29, 28], then the time interval to your next month's check will be exactly five weeks. If the sum is greater than the number of days in that month, then next month's check will be exactly four weeks away.

There, coherence and relative simplicity.

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