Download July 15, 2003 - eSocialSciences

Predicting Calendars (and saving money if you
want to)
S.Srinivasan*1
How calendars are repeated in some years can be understood by doing simple calculations.
*e-mail: [email protected]
For the purpose of this article, we will refer only to the Julian calendar system or what is
called in India the “English” Calendar System.
How many different calendars do you need so as to not to be able to buy calendars every
year? Or if you want to be ‘green’ and save paper? One can easily calculate. Every year
starts on one of seven days, Sunday, Monday, etc. Therefore for every year of 365 days we
will have 7 different calendars. Then there are leap years – 366 days in the year. Each of
these too can start on seven different days. Therefore seven more calendars. So totally we
have 14 calendars. If we understand the sequence and periodicity of these fourteen
calendars we can in principle not buy a calendar every year!
Let us explore a bit more.
Suppose Year 1 starts on a Sunday. And let us also suppose it is a non-leap year. Let us
assume Year 2 and 3 are also non-leap years. Since 365 divided by 7 leaves a remainder of
1, the 365-day year begins and ends on the same day. And Year 2 will start on Monday,
end on a Monday. Year 3 will start and end on a Tuesday. But Year 4, a leap year, will
start a Wednesday and end on a Thursday. And year 5 will start on a Friday.
Let us denote the non-leap year starting on Sunday as Calendar 1 year. Non-leap year
starting on Monday will be Calendar 2 year and so on. We will also denote a leap year
starting on Sunday by Calendar 8 and a leap year starting on Monday by Calendar 9. Leap
Year starting on Saturday will be Calendar 14. The following table gives how the calendar
years repeat themselves.
Calendar Years of 2001-2100
2001 – 2
2002 – 3
2003 - 4
2004 - 12
2005 – 7
2006 – 1
2007 - 2
2008 - 10
2009 – 5
2010 – 6
2011 - 7
2012- 8
2013 – 3
2014 – 4
2015 - 5
2016 - 13
2017 – 1
2018 – 2
2019 - 3
2020 - 11
2021 – 6
2022 – 7
2023 - 1
2024 - 9
2025 – 4
2026 – 5
2027 - 6
2028 - 14
2029 – 2
2030 – 3
2031 - 4
2032 - 12
2033 – 7
2034 – 1
2035 - 2
2036 - 10
1
I thank my brother Ramesh for putting me on to these patterns. The omissions however are mine.
eSS Current Affairs, Srinivasan, Calenders
November 2011
2037 – 5
2041 – 3
2045 – 1
2049 – 6
2053 – 4
2057 – 2
2061 – 7
2065 – 5
2069 – 3
2073 – 1
2077 – 6
2081 – 4
2085 – 2
2089 – 7
2093 – 5
2097 – 3
2038 – 6
2042 – 4
2046 – 2
2050 – 7
2054 – 5
2058 – 3
2062 – 1
2066 – 6
2070 – 4
2074 – 2
2078 – 7
2082 – 5
2086 – 3
2090 – 1
2094 – 6
2098 – 4
2039 - 7
2043 - 5
2047 - 3
2051 - 1
2055 - 6
2059 - 4
2063 - 2
2067 - 7
2071 - 5
2075 - 3
2079 - 1
2083 - 6
2087 - 4
2091 - 2
2095 - 7
2099 - 5
2040 - 8
2044 - 13
2048 - 11
2052 - 9
2056 - 14
2060 - 12
2064 - 10
2068 - 8
2072 - 13
2076 - 11
2080 - 9
2084 - 14
2088 - 12
2092 - 10
2096 - 8
2100 - 6
Source: http://www.accuracyproject.org/perpetualcalendars.html
Thus 2001 is a Calendar 2 year, 2002 will be a Calendar 3 year and so on. 2004 will be a
Calendar 12 year. And 2005 will be a Calendar 7 year and 2007 will be a Calendar 2 year.
So if you think calendars repeat every 6 years you may be accused of jumping to
conclusions.
The behavior goes as follows:
Within a century (1900-1999) by and large the following holds (the exceptions are
discussed later):
 Leap year plus 1: then that year’s calendar will repeat after 6 years, e.g., 2001 calendar
is repeated in 2007.
 Leap year plus 2, or 3: then that year’s calendar will repeat after 11 years, e.g., 2002
calendar will repeat in 2013.
 Leap years: Calendars of leap years generally repeat after 28 years (ha! LCM of 4 and
7). Year 2000 will repeat in 2028, 2056, 2084, etc.
In fact any non-leap year calendar will also generally repeat after 28 years. But before that
it will have repeated in the sequence 6-11-11 years – e.g., 2001, 2007, 2018, 2029, 2035,
2046 and so on.
Leap year calendars follow this sequence generally (or see fourth column in table above):
Calendars 13, 11, 9, 14, 12, 10, 8, 13…
Matters would have rested here and this article could have ended here. But …
When you cross a non-leap century year, calendars repeat differently!
eSS Current Affairs, Srinivasan, Calendars
November 2011
The above facts are true at the minimum only for years within a century that is say within
2001-2094 at the most. Among the century years, only those years divisible by 400 are
leap years. Thus 2000 is a leap year but 1900, 2100, 2200, 2300 are not. The next century
leap year will occur in year 2400.
Calendars similar to leap year 2000 between 1800 and 2099
1820
1848
1876
1916
1944
1972
2000
2028
2056
2084
In the above, since 1900 is not a leap year, the 1876 calendar repeats after 40 years (28 +
6+6?) in 1916. But that is not the end of the matter. The 1892 leap year calendar repeats
after a mere 12 years!
1808
1904
2016
1836
1932
2044
1864
1960
2072
1892
1988
Here in general this is how the leap year calendars repeat across 1900 (the number in
bracket indicates the gap across century non-leap year 1900, that is between the years in
bold):
1844, 1872, 1912, 1940 ... (40 years)
1848, 1876, 1916, 1944 … (40 years)
1852, 1880, 1920, 1948 … (40 years)
1856, 1884, 1924, 1952 … (40 years)
1860, 1888, 1928, 1956 … (40 years)
1864, 1892, 1904, 1932 ... (12 years)
1868, 1896, 1908, 1936 ... (12 years)
Patterns for Non-Leap Years across a Non-Leap Century Year
But one more twist to the tale! The patterns of repetition across a non-leap century year
are different for non-leap years too. Instead of following the pattern of 11-11-6 years of
gap between like calendar years, across 1900 it is different: say for example the similar
calendar years 1866-1877-1883-1894 -1900 -1906 -1917-1923-1934, the gaps are 11-611-6-6-11-6-11.
For the same calendar years 1867, 1878, 1889, 1895, 1901, 1907, 1918, 1929 it is 11-11-66-6-11-11
For the same calendar years 1869, 1875, 1886, 1897, 1909, 1915, 1926 - here the year
1897 repeats after 12 years! So does 1898 – it next repeats in 1910.
But 1899 repeats only after 6 years – it is 1882, 1893, 1899, 1905, 1911, 1922 etc.
eSS Current Affairs, Srinivasan, Calendars
November 2011
Readers are invited to find a general rule behind these patterns.
Also of interest it is assumed that if the leap year were 3 or 5 years – how will these
patterns work out? Or suppose the week was of 10 days.
How many different calendars will be there if we used India’s National Calendar system?
[For easy access to same calendar years, please see:
http://www.export911.com/ref/cale807300G.htm]
Calendar Math
MAY
M
7
14
21
28
T
1
8
15
22
29
W
2
9
16
23
30
T
3
10
17
24
31
F
4
11
18
25
S
5
12
19
26
S
6
13
20
27
In a typical calendar like the one above, there are some small arithmetic curiosities.
Firstly, take any rectangular array like the one shaded above.
The diagonally corner numbers add up to the same sum: 25 + 8 =22 + 11.
Numbers along the column increase by 7 as they should do every week.
Suppose you take a 3x 3 array of numbers in the above. Like what you see below.
8 9
15 16
22 23
10
17
24
They add up to 144. How? Take the central number and multiply by 9. Can you say why?
(Incidentally the sum of the corner numbers is twice the central number: 24 + 8 = 22 + 10
= 32 = 2 x 16. So is the sum of the middle numbers: 15 + 17 = 9 + 23 = 2 x 16. This
should tell you why the sum of all the numbers is 9 times
16.)
Now you can try the sum of this array. And actually add and verify it.
15
22
29
16
23
30
17
24
31
Now draw a 5x4 box around any 20 numbers in the calendar. Say like this:
eSS Current Affairs, Srinivasan, Calendars
November 2011
1
8
15
22
2
9
16
23
3
10
17
24
4
11
18
25
5
12
19
26
They add up to 270. How? Add the smallest and biggest numbers in the array and multiply
by 10. Can you say why this is so?
This works for other general array of numbers. Try and discover some as also many other
patterns in the calendar array itself.
A Small Puzzle
Ask your friend to add any 4 numbers that form a square on the calendar and tell you the
sum. Like
10 11
17 18
The sum is 56. She tells you the number 56. You have not seen the 4 numbers. But you
can tell her the four numbers!
This is how: Divide 56 (the sum) by four and then subtract 4. That gives you the first
number. You add 1, 7, and 8 to get the other numbers.
Did you know?
4/4, 6/6, 8/8, 10/10 and 12/12 always fall on the same day of the week in any year.
eSS Current Affairs, Srinivasan, Calendars
November 2011