Maths Item of the Month – November 2012
When dates align
10th November 2012, i.e. 10/11/12, is an 'AP date': a date whose numbers are in arithmetic
progression.
Some questions:
When was the first AP date (dd/mm/yy) in your lifetime?
When did/will your birthday fall on an AP date?
What is the shortest gap between two AP dates this century?
What is the longest gap between two successive AP dates this century?
What is the probability that somebody who is currently under 18 was born on an AP
date?
Solution
To find the first AP date of our lifetime, we will first look at the upper limits of the individual
values d, m and y. The upper limits are defined by m, as this has the lowest maximum value
of 12. The highest possible value of d is generated by setting y to zero and m to 12. This tells
us that d cannot exceed the value of 24. Likewise, the highest possible value of y is found to
be 23, by setting d to 1.
Assuming that you were born after 1923, the first AP date in your lifetime must therefore be
the first AP date of the 21st Century. To find this, we simply check for an AP date in the first
month of the century, as there can be no more than one AP date each month.
This results d = 2. The first AP date of most people’s lives is therefore 02/01/00.
Finding an AP date given two parts of a date, such as a birthday, is as simple as finding the
next term in an Arithmetic Sequence. If your birthday falls on the 5th July, i.e. 05/07, then the
AP date will be 05/07/09.
It is possible, however, that this technique will result in a negative value of d or y. For a
month to contain a valid AP date it must adhere to some rules which prevent it from returning
a negative day or year. We find that m must be more than half of y (and at least half of d).
If we use this rule to plot a table of months containing valid AP dates, it shows a pattern:
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00
01
02
03
Year:
04
05
06
07
08
Jan (01)
Feb (02)
Mar (03)
Apr (04)
May (05)
Jun (06)
Jul (07)
Aug (08)
Sep (09)
Oct (10)
Nov (11)
Dec (12)
This pattern is important when examining the distances between AP dates. Looking at a year
internally, the space between two consecutive AP dates is always a month and two days. So
from 02/01/00, the next AP date is 04/02/00. Without a change of year then, the space
between AP dates varies between 30 and 33 days, depending on the month.
The important differences to examine are those in the crossover between years. Between
Year 00 and Year 01, the first and last AP dates respectively are 24/12/00 and 01/01/01.
This is a difference of 8 days; far less than the others!
Using the pattern shown above, we see that from Year 00, this crossover slowly increases in
size, with a month being added every two years. Between 03 and 04 for example, the
difference is 72 days (remember 04 is a leap year).
This tells us that the shortest time between AP dates is indeed the 8 days between 00 and
01.
To find the longest difference, we must find the latest gap between years. As the latest year
containing an AP date is Year 23. The biggest gap must therefore be between Year 22 and
Year 23. This comes to 364 days!
Finding the probability of a person being born on an AP date is a simple exercise of dividing
the number of possible AP dates, by the total number of possible dates.
Assuming today’s date to be October 31 2012, the first day that someone could have been
born and currently be under 18 is the 1st November. The number of possible days between
now and then for the person to be born is 6575 (remember leap years!). The number of AP
dates can be found using a table like above, and equals 120.
The probability, P, equals
=
= 0.0181 (3 s.f.).
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