Monthly
Maths
I s s u e
What effect does
a leap year have
on the dates in
the calendar?
Because seasons
and astronomical
events do not
repeat in a whole
number of days, a
calendar that has
the same number of
days in each year
would, over time,
drift with respect to
the event it was
supposed to track.
By occasionally
inserting (or
intercalating) an
additional day or
month into the year,
the drift can be
corrected. A year
that is not a leap
year is called a
common year.
For example,
in the
Gregorian
(solar)calendar
February in a leap
year has 29 days
instead of the usual
28, so the year lasts
366 days instead of
the usual 365.
Similarly, in the
Hebrew (lunisolar)
calendar a 13th
lunar month is
added seven times
every 19 years to
the twelve lunar
months in its
common years to
keep its calendar
year from drifting
through the
seasons too rapidly.
Innovators in
mathematics education
www.mei.org.uk
1 2
Corresponding years
There are seven possible days on which a
year can start, and leap years (or intercalary
or bissextile years) will alter the days of the
week after February 29. This means that
there are 14 configurations that a year can
have. All the configurations can be
referenced by a Dominical letter, where a
letter is
assigned
from A
through to G
to each day
of the year.
In this system, the "dominical letter" for a
year is the letter which corresponds to the
Sundays of that year. In a leap year,
February 29 does not have a distinct letter.
This causes all subsequent Sundays to be
associated with a different dominical letter
than those at the beginning of the year, so all
leap years get two dominical letters.
For example, 2011 was a common year
starting on Saturday, meaning that 2011
corresponded to the 2005 calendar year.
2012, on the other hand, is a leap year
starting on Sunday, meaning that the first two
months of the year begin as they did in 2006
(i.e. January 1 is a Sunday and February 1 is
a Wednesday) but because of leap day the
last ten months will correspond to 2007
(i.e. March 1 is a Thursday, etc.)
Click here to read more
Click here to read about dominical letters
Calculating the days of the week
Zeller's congruence is an algorithm devised
by Christian Zeller to calculate the day of the
week for any calendar date, the months
being numbered from 3 for March to 14 for
February. The year is assumed to begin in
March.
J a n u a r y
2 0 1 2
Zeller’s congruence
Zeller’s congruence for the Gregorian
Calendar is:
d=(n+int[(m+1)×2610]+y+int[y4]+int[C4]−2C)mod 7
where:
d stands for the day of the week,
numbered 0 to 6 for Saturday - Friday.
n is the day of the month, numbered 1 to
28, 29, 30 or 31 depending on the
month in question.
m is the month, numbered from 3−12 for
March - December; m=13 in January
and m=14 in February.
y is the last two digits of the year, or the
last two digits of the previous year if
m=13 or 14.
C is the first two digits of the year.
'int[x]' is a function which rounds x down
to the nearest integer or returns x if x
is already an integer.
'mod 7' means return the remainder when
divided by 7.
Nrich offers a two part investigation into
Zeller’s algorithm here There is also a general
investigation into algorithms, including a warm
up exercise, teachers’ notes and extensions
here
Investigate how you can work out what day of
the week your birthday will be on next year,
and the year after... this problem gives an
insight into modular arithmetic without
worrying too much about notation, by looking
at the concept of remainders. Click here
An introduction to the notation and uses of
modular arithmetic is hereAn investigation into
‘Zeller’s birthday’ is here and into modulo 7
and modular fractions here
Read more about Zeller’s congruence here
and here
Useful links
Which is the safest
New Year's resolution?
MEI Maths Item
of the Month
Beat Blue Monday on January 16th
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New Year’s
Resolution
Riddle
On Jan 1, 10
friends make a new
years resolution.
The 1st person is to
swim every day,
the 2nd person is to
swim every 2nd
day....etc and the
10th person every
10th day. How
many days go by
until all 10 friends
swim on the same
day?
(Answer below)
The solution lies in
finding the lowest
common multiple.
Read more here
and here. A
powerpoint is
available here
We require the
lowest common
multiple of 1, 2, 3,
4, 5, 6, 7, 8, 9, 10
Writing each
number as prime
factors we get:
1 => 1, 2 => 2, 3 =>
3, 4 = 2^2, 5 => 5,
6 => 2 x 3, 7 => 7,
8 => 2^3, 9 => 3^2
and 10 => 2 x 5
so, L.C.M is 2^3 x
3^2 x 5 x 7
=> 8 x 9 x 5 x 7 =
2,520
Corresponding months
Corresponding months are those months
within the calendar year that start on the
same day. For example, September and
December correspond, because September
1 falls on the same day as December 1.
Months can only correspond if the number of
days between their first days is divisible by 7,
or in other words, if their first days are a
whole number of weeks apart.
For example,
February
corresponds to
March because
February has
28 days, a
number
divisible by 7,
28 days being
exactly four weeks. In a leap year, January
and February correspond to different months
than in a common year, since February 29
means each subsequent month starts a day
later.
An interesting investigation would be to
calculate and record which months
correspond with which in common years,
in leap years and in all years.
(See answer below)





Common year
January and October.
February, March and November.
April and July.
No month corresponds to August.





Leap year
January, April and July.
February and August.
March and November.
No month corresponds to October.




All Years
September and December.
April and July.
No month corresponds to May or June.
The Calendar Trick
All you need is a calendar which has the
dates lined up under the days of the week,
so the numbers are arranged something like
this:
.
.
6
7
13 14
20 21
27 28
1
8
15
22
29
2
9
16
23
30
3
10
17
24
31
4
11
18
25
5
12
19
26
Ask a student to draw a 3x3 box around ANY
nine of the numbers on the calendar. Almost
immediately you can say what the nine
numbers all add up to!
e.g.
.
.
1
2
3
4
5
6
7
8
9 10 11 12
13 14 15 16 17 18 19
20 21 22 23 24 25 26
27 28 29 30 31
All you do is look at the number in the middle
and multiply it by 9. In this case the middle
number is 22 and 22 x 9 = 198
You can also do this trick with a 20 number
box, (although it won’t always be possible to
put a box round 20 numbers in February)
e.g.
. 1
2
3
4
5 6
7
8
9 10 11 12 13
14 15 16 17 18 19 20
21 22 23 24 25 26 27
28 29 30 31
Add together the smallest and the largest
numbers in the group. Multiply the answer by
10, e.g. 2 + 27 = 29. 29 x 10 = 290
Some things you could investigate:
1. Easy ways to multiply by 9, e.g. multiply
the number by 10 (by just adding a ‘0’) and
then subtract the number.
2. Other number patterns inside the boxes what are they? Can you explain them?
(A clue for one pattern: Look at the pairs of
opposite numbers)
Nrich offers more investigations using
calendars on its ‘Calendar Capers’ page
When can you
have two New
Years in the
same year?
1. The Islamic New
Year occurs on 1
Muharram. Since the
Muslim calendar is
based on 12 lunar
months amounting to
about 354 days, the
Muslim New Year
occurs about eleven
days earlier each year
in relation to the
Gregorian calendar.
Hence, two Muslim
New Years occurred
in Gregorian year
2008.
2. This year the
Chinese New Year
falls on January 23. In
2011 it fell on
February 3. A Chinese
year normally consists
of 12 months where a
month corresponds to
one lunar cycle. Each
month starts on the
day of the new moon.
Since the cycle is not
an even number of
days, a month in the
lunar calendar can
vary between 29 and
30 days and a normal
year can be 353, 354,
or 355 days.
3. Easter Style dating
was used all over
Europe, but especially
in France, from the
eleventh to the
sixteenth century. The
New Year started on
Easter Saturday or
sometimes on Good
Friday.
A disadvantage of this
system was that
because Easter was a
movable feast, the
same date could occur
twice in a year; so the
two occurrences were
distinguished as
"before Easter" and
"after Easter”.
When can you see in the New Year
twice in the same day?
The circumnavigator's paradox
Three
people
meet at
the
same
place
along
the
equator; one travels around the globe to the
east; one around the globe to the west; the
third remains in the place where the three
met. The two travellers return to the spot
where the third waits, and each believes a
different number of days has elapsed.
Each measured the journey by the
movement of the sun. The sun's position and
movement were different depending upon
the direction each person travelled; the
person who didn't move also experienced the
rising and setting of the sun differently. Thus,
each person calculated the excursion in
different ways. The person who travelled
west had the shortest trip, while the one who
travelled east had the longest trip.
Greenwich Observatory
was set up in 1675 by
King Charles II to study
a means of fixing
longitude. The establishment of the Prime
Meridian as a line running through
Greenwich Park, London, England, as 0
degrees longitude was agreed upon in 1884
by delegates to the International Meridian
Conference in Washington. The International
Date Line sits on the 180º line of longitude in
the middle of the Pacific Ocean, and is the
imaginary line that separates two
consecutive calendar days.
View video clip explaining how the
International Date Line works
The adoption
of this location
for the Prime
Meridian meant
that the logical
course for the
International
Date Line was
180 degrees to
the west.
Conveniently, that location cut mainly through
open sea rather than land masses.
Immediately to the left of the International
Date Line the date is always one day ahead
of the date (or day) immediately to the right of
the International Date Line in the Western
Hemisphere. Travelling west across the
International Date Line results in a day being
added.
View video showing crossing the
International Date Line ceremony
Without the International Date Line, people
who travel west around the planet would
discover that when they returned home, it
would seem as though an extra day had
passed. Because of the division of the globe
into time zones, the New Year moves
progressively around the globe as the start of
the day welcomes in the New Year.
The first time zone to see in
the New Year is just west of
the International Date Line. At
that time the time zone to the
east of the Date Line is 23 hours behind, still
in the previous day. The central Pacific
Ocean island nation of Kiribati claims that its
easternmost landmass, uninhabited Caroline
Island, is the first to usher in the New Year.
View interactive map showing New Year
2012 around the world
Tonga and Samoa have the same time but
are one day apart, as Samoa is in the
Western Hemisphere, on the opposite side of
the International Dateline from Tonga.
Travelling from Samoa to Tonga would
enable you to see the New Year in twice.
Read more and view map here
View NASA video explaining how
astronauts know how time works
on an international space station.
The Tabular or
arithmetic Islamic
calendar (hisabi) is
determined by
arithmetic rules
rather than by
observation, and is
used by more than a
billion Muslims
worldwide to
determine the main
days of observance
in the Islamic
religious year.
The number of days
in the 12 months
alternates between
30 and 29 days. In
order to keep the
calendar in step with
the lunar phases,
every 2 or 3 years an
extra day is added at
the end of the year
to the last month,
resulting in a
calendar year of
355 days. 11
intercalary days are
added every
30 years.
The definition of a
leap year in the
Islamic calendar
varies. All agree on a
30 year cycle,
however which years
within the 30 are
leap varies by the
leap year pattern.
The 16-based leap
year pattern
algorithm is the most
commonly used, and
is the default: 2, 5, 7,
10, 13, 16, 18, 21,
24, 26, 29. Microsoft
uses the 15-based
pattern, and calls it
the 'Kuwaiti
algorithm'.
Read more here
Although the date for New Year's Day is not
the same in every culture, it is always a time
for celebration and for customs to ensure
good luck in the coming year. The New Year
begins on January 1 only for cultures that
use a 365-day solar calendar.
From the earliest times
in Europe, winter
festivals have been held
around or just after the
winter solstice
(December 21). In the
very earliest Roman
calendars there were no months of January
or February at all. The ancient Roman
calendar had only ten months and the New
Year started on 1 March.
The Julian calendar was
established by Julius Caesar in
45 BC, with January as the first
month of the year, followed by
February. In order to
synchronize the calendar with
the sun, Caesar had to let the
previous year drag on for 445
days. The average length of a year in the
Julian calendar was slightly longer than the
actual length of a solar year. Thus, by the
1700s, the official dates of the 4 equinoxes
had moved about 10 days from the days on
which they actually fell. A correction to the
date had to be made when England changed
over to the Gregorian calendar in 1752.
Hence Wednesday, September 2 was
followed by Thursday, September 14.
The Julian and Gregorian calendars are solar
calendars. Some cultures have lunar
calendars. A year in a lunar calendar is less
than 365 days because the months are
based on the phases of the moon. The
Islamic calendar system is a lunar calendar
based on observation, meaning that a new
month can only be declared based on human
observations of the moon, something which
can vary and which is unsuited to computer
calculation, unlike the Tabular or arithmetic
Islamic calendar. (See left column)
Lunisolar calendars
Most lunar calendars are in fact lunisolar
calendars, where the months reflect the lunar
cycle, but leap or ‘intercalary’ months are
added to bring the calendar year into
synchronisation with the solar year. Such
calendars have a variable number of months
in a year, because a year is not evenly
divisible by an exact number of lunations.
Without the addition of intercalary months the
seasons would drift each year. This results in
a thirteen-month year every two or three
years.
The Chinese calendar is
an example of a
lunisolar calendar,
where the first day of a
month is the day when
an astronomical new
moon occurs in a
particular time zone.
7 times in a 19 year cycle, an extra leap
month (runyue) is added to the year to bring it
back into line with the longer solar year. A
complete cycle takes 60 years and is made
up of five cycles of 12 years each. The year
can begin anywhere between late January
and the middle of February, marking the first
of 15 days of celebration.
In Chinese tradition, each
year is dedicated to a specific
animal: Dragon, Horse,
Monkey, Rat, Boar, Rabbit,
Dog, Rooster, Ox, Tiger,
Snake, or Ram. Each of these
animals is thought to bestow
its characteristics to the people born in its
year. 2012 is the Year of the Dragon, which
signifies success and happiness.
For other lunisolar calendars,
such as some Hindu calendars,
each month begins on the day
after the full moon or the new
moon, while others are based on
the first sighting of a lunar
crescent, such as the Hebrew calendar.
Read more