Treebeard's Stumper Answer

Five Fridays
I write these stumpers for the Dunn Middle School Friday Newsnote, and I keep collections of past newsnotes and stumpers on the Web. I noticed last week that there were five Friday newsnotes in the month of October 1999. I believe this is the first time that this has ever happened, though there have been other months that could have had five newsnotes if not for school vacations and other interruptions. This suggests a tricky stumper. Ignoring vacations, what are the most (and least) months with five Fridays that can occur in a single calendar year (Jan.  Dec.)? How uncommon is this event, and what hidden patterns does it reveal?
Unlucky Strikes
98.6 in the Shade
World Party
Too Many Ancestors
Ollie Up
1 Oct 99
8 Oct 99
15 Oct 99
22 Oct 99
29 Oct 99
Five Fridays in October 1999.
Every year has at least four months with five Fridays, but this year there are five. The first four weeks of each month account for 4 x 7 x 12 = 336 days. That leaves 365  336 = 29 extra days that will become four fiveday months for each day of the week, plus one day left over. A year is 52 weeks plus a day (or two in a leap year). So a year that begins on a Friday like 1999 (or a Thursday in a leap year) will have five fiveFriday months. This result is quite general. January 1, 2000 is a Saturday, so there will be five fiveSaturday months next year.
Notes:
There are too many fives in this stumper!
Whew! Hyphens aren't enough for this, but brackets might help on that last one, for LISP programmers at least:
 October was a fiveFriday month,
 1999 is a fivefiveFridaymonth year,
 There is a fivefivefiveFridaymonthyear year cycle that I will explain below!
Let me explain...
 There is a {five [five (five Friday month) year] year} cycle!
I thought of this stumper when I first noticed there were five Friday Stumpers in October, but I didn't know if it would go anywhere. I checked a calendar and discovered that in 1999 there are four fiveMonday weeks, four fiveTuesday weeks, four fiveWednesday weeks, and so on for every day of the week, except there are five fiveFriday weeks! I knew I was on to something, so I made a bunch of copies of a perpetual calendar from an old almanac, and I started marking fiveday months with a highlighter.
This year 1999 begins on a Friday as shown, and it's not a leap year. Everything else is determined. There are five fiveFriday months (January, April, July, October, and December) shown in yellow, but only four fiveMonday months (and every other day), as shown in cyan.
Leap years like the coming year 2000 are different in that they have two fiveday months:
The year 2000 begins on a Saturday, and it is a leap year. (The year 1900 wasn't a leap year! Why not?) There are five fiveSaturday months shown in yellow, and five fiveSunday month shown in green. There are only four fiveFriday months (and every other day), as shown in cyan.
Eventually, I found this simple pattern:
 Whichever day of the week a nonleap year begins on, there will be five months that year that have five of that first day, and only four months that have five of every other day of the week.
 Leap years are different. There will be five months that have five of the first day, and there will also be five months that have five of the next day of the week. A leap year that begins on a Saturday (like the year 2000) will have five months with five Saturdays and five months with five Sundays, but there are only four months that have five of every other day.
Cool pattern! Then I started thinking about it, but Graybear beat me to it:
I'm kicking myself right now because I immediately tried to solve this by looking at a calendar and determining, day by day (Sunday, Monday, etc.) how many months had five of that day (That's how I had solved the Friday the 13th stumper on November 13, 1998 a year ago). Then I saw that it would have been more exciting to solve it in my head. Therefore, here is the 'cerebral solution':Exactly right, and to the point. I had to read this over and over to really understand it. There are too many fives, which makes it confusing!Every month has either 4 or 5 Fridays  because each month has at least 28, and less than 35, days in it. If there were exactly 52 weeks in a year, eight would have 4 Fridays, and four would have 5 Fridays. Think of it this way  twelve months of 4 Fridays equals 48 Fridays, so the remaining 4 Fridays must mean that 4 months each have an extra. In other words (or numbers), ( 8 * 4 ) + ( 4 * 5 ) = 52. Since a year is actually 52 weeks plus one day, if a year starts on Friday, (Thursday or Friday in a leap year) there will be five months with five Fridays.
The almanac's perpetual calendar is based on the fact that there are only 14 different possible calendars, depending on which day of the week the year begins, and whether it's a leap year. These different calendars repeat in a sequence of 28 years called the Solar Cycle. Each day of the week is the start of three normal years and one leap year in this 28 year sequence:
Year First day
of yearLeap
Year?FiveFriday
MonthsMonths with
Five Fridays
1972 Saturday Yes 4 Mar, Jun, Sep, Dec 1973 Monday 4 Mar, Jun, Aug, Nov 1974 Tuesday 4 Mar, May, Aug, Nov 1975 Wednesday 4 Jan, May, Aug, Oct 1976 Thursday Yes 5 Jan, Apr, Jul, Oct, Dec 1977 Saturday 4 Apr, Jul, Sep, Dec 1978 Sunday 4 Mar, Jun, Sep, Dec 1979 Monday 4 Mar, Jun, Aug, Nov 1980 Tuesday Yes 4 Feb, May, Aug, Oct 1981 Thursday 4 Jan, May, Jul, Oct 1982 Friday 5 Jan, Apr, Jul, Oct, Dec 1983 Saturday 4 Apr, Jul, Sep, Dec 1984 Sunday Yes 4 Mar, Jun, Aug, Nov 1985 Tuesday 4 Mar, May, Aug, Nov 1986 Wednesday 4 Jan, May, Aug, Oct 1987 Thursday 4 Jan, May, Jul, Oct 1988 Friday Yes 5 Jan, Apr, Jul, Sep, Dec 1989 Sunday 4 Mar, Jun, Sep, Dec 1990 Monday 4 Mar, Jun, Aug, Nov 1991 Tuesday 4 Mar, May, Aug, Nov 1992 Wednesday Yes 4 Jan, May, Jul, Oct 1993 Friday 5 Jan, Apr, Jul, Oct, Dec 1994 Saturday 4 Apr, Jul, Sep, Dec 1995 Sunday 4 Mar, Jun, Sep, Dec 1996 Monday Yes 4 Mar, May, Aug, Nov 1997 Wednesday 4 Mar, Jun, Aug, Nov 1998 Thursday 4 Jan, May, Jul, Oct 1999 Friday 5 Jan, Apr, Jul, Oct, Dec
2000 Saturday Yes 4 Mar, Jun, Sep, Dec 2001 Monday 4 Mar, Jun, Aug, Nov (etc.) Count the years with five fiveFriday months and you'll see that there are five years with five fiveFriday months every 28 years. It's almost too much to mention that there are usually five fourFridaymonth years between successive fiveFridaymonth years. Like I said at the start, there is a {five [five (five Friday month) year] year} cycle!
There's another unexpected pattern revealed here too. FiveFriday months occur in January, April, July, October, and December in a normal year or January, April, July, September, and December in a leap year. When any day of the week occurs five time in a month, five times in a year, it is always in those months!
The 28 year solar cycle breaks on century years that are not leap years according to the revision of the Gregorian Calendar by which century years are only leap years if they are divisible by 400. The year 2000 is a leap year, but 1900 and 2100 aren't. Does this effect the distribution of fiveFridaymonth years in the long term?
I tried my usual Web searches on this stumper and came up empty. Maybe this is an original?
 I did find Web several references to the Five Fridays of Lent, and the beautiful Greek Orthodox Akathist Hymn. Is this coincidence?
 The Calendar FAQ has much information on the oddities of our calendar.
 My stumper on Friday the 13th (13 Nov 1998) is about another odd calendar pattern that involves the solar cycle.
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