Archimedes Plutonium wrote:
....
So, after each year I look at 31Dec and in my other 6 calendars I fish
out the one that harmonizes 31Dec with the correct 1January. If that
year happens to be a leap year containing a 29Feb which none of my 7
calendars possess, then I look at my backpage footnote to remember if
it is a leap year
....
But I am not certain of that method, Ken. I am not certain that if I
were to collect 7 calendars all having nonleap years and all having a
different day of the week for 1Jan, whether those 7 will cover all
bases for a 29Feb and 1Mar
switch.

Would you know Ken, whether 7 calendars is the Minimum number of
calendars to cover all future years, provided those 7 have a double
switch in leap years? You see, I am certain that 14 will cover all
bases, but will 7 cover all bases if I make a double switch in leap
years.
....

I don't know Ken, but I do know that any set of 7 non-leap-year
calendars that start with 7 different days is enough. If a year
starts on a Monday -- eg, 1996 and 2007 -- on 1 Jan of that year
use a calendar that starts on a Monday. If the year is not a leap
year, leave the same calendar up all year. If the year is a leap
year, put a postit note in place of 29 Feb, and on 1 March put up
the calendar that starts Tuesday. And so forth for other days of the
week. If you only have leap-year calendars, the plan is much
the same, except that in leap-years we leave the calendar up all
year, and in non-leap-years, cover up 29 Feb with a postit note to
change to the calendar that starts the year a day earlier in the week.

However, no mix of m leap-year and 7-m non-leap-year calendars,
with 0 m 7, will suffice for all possible years. The mix of
calendars must be able to represent 1 Jan and 1 March on any day
of the week. (For example, 1969, 1973, 1977, 1981, 1985, 1989, and
1993 all start on different days of the week and have 1 March on
different days too.) Second, the mix must be able to represent leap-
year 1 Mar on any day of the week. (For example, see 1968, 1972, 1976,
1980, 1984, 1988, and 1992 calendars. On linux you can use the command
for ((i=1968;i1993;i+=4));do cal $i|head;done|less for this.)
Third, in a non-leap year, 1 March is 3 days later in the week and in
a leap year, 4 days later in the week. Hence a mix of 7 calendars,
some leap-year and some not, cannot represent all 1 March days of the
week. (Order the calendars by 1 Jan day-of-week and note that when a
leap-year calendar follows a non-leap-year calendar, or vice versa,
some 1 March day is doubly represented, leaving another unrepresented.)

It is simple to show that if you have any 13 or more calendars and no
more than 2 of them start on a given day of the week, then all 1 Jan
and 1 March possibilities are covered. But 12 is not necessarily
enough: eg, if you have calendars of 1968, 1976, 1980, 1984, 1988,
1992, 1969, 1973, 1977, 1981, 1985, and 1993, you will have 1 or 2
calendars for each possible 1 Jan, and 2 for most 1 March days,
but none for Wednesday.
-jiw