Modular counting: Of election dates and other patterns
Since the 1995 elections I’ve been involved in election campaigns. Preferring to stay behind the scenes, I invariably started out doing desk and paper work, like scheduling, data banking and bookkeeping. To put color in that six-week daily grind, I volunteered to compose campaign jingles as well.
My candidate then was a municipal councilor newly elected in 1992 and going for a second second term in 1995. In updating the campaign jingle he used in 1992, I remember wondering if we could just edit the voice-over, part of which exhorted the listeners to vote for so-and-so candidate on election day. That went something like, “Mga kababayan, pagdating ng Mayo onse, iboto si…” My initial idea was to simply replace the date “onse” with whatever date the day of elections would fall on, like “otso” or “katorse,” and to replace “iboto” with “ibalik.” These days, with accessible high technology, that could have easily been done. A dozen years ago, armed with just cassette tapes, it was an entirely different story. Come jingle-playing time, one’s best option was a tape deck which could do auto-reverse.
National and local elections fall on the second Monday of the month of May. In 1992, election day fell on May 11; in 1995, on May 14; in 1998, on May 11. Just when I was beginning to think that we could save campaign money in a third-term campaign in 1998 by either deploying the 1992 jingle again or, technology now allowing, replacing “katorse” with “onse” in the 1995 jingle, my candidate decided to run for the position of municipal mayor instead. But let’s not go into my candidate’s political forays. Let’s, you and me, go instead on a mathematical foray.
I will not discuss the mathematics found in a campaign’s budgeting, bookkeeping and forecasting. The budgeting is simple arithmetic, the bookkeeping is basic accounting, and the forecasting can be easily rendered futile or inutile by the intricacies and convolutions of campaign dynamics. Through this article I intend to introduce you to modular counting.
Modular counting is best described by the operation – division. Modular counting assigns to a number the remainder when that number is divided by a given fixed number, called the modulo or simply, mod. Consider the positive integers:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 …
If I divide each of the numbers above by 2, the remainder is either 0 or 1, respectively, depending on whether the number is even or odd. Assigning to each number above the remainder when it is divided by 2, we get
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 …
Hence, we say 1 mod 2 = 1, 2 mod 2 = 0, 3 mod 2 = 1, 4 mod 2 = 0, and so on. Evidently, odd numbers are assigned the value 1 and even numbers are assigned the value 0 in mod 2 counting. The alternation of values is crucial; it is impossible to have two consecutive 0’s or 1’s – this means having two consecutive even numbers or two consecutive odd numbers! Think of the cadence of a march, and drill commanders shouting “Left! Right! Left! ...” Is it not impossible to march properly to “Left! Left! Left! …”? It is likewise impossible to turn a switch on and then on again; it must be switched off in between. Even for relationships, is not the term “on-again, off-again” and not “on-again, on-again”?
Let’s take a look at more illustrations. Consider mod 3 counting. This means we divide each of the positive integers listed above by 3 and assign the respective remainders:
1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 …
Assigned “1” are 1, 4, 7, and so on. Assigned “2” are 2, 5, 8, and so on. Assigned “0” are 3, 6, 9, and so on. Those who play mahjong will recognize the numbers in a puro in the up-and-down or escalera configurations, where the aim is the completion of consecutive trios. Non-mahjong players may more clearly understand mod 3 counting in the setup of a traffic light: green, yellow, red, or equivalently, go, slow down, stop. Needless to say, the sequences in mahjong and traffic lights can never be jumbled.
For mod 4 counting, our country’s fondness for beauty pageants makes me choose quarter turns as a good example. Unlike military drills which can make cadets/ettes turn right or left at random, a beauty pageant standard quarter turn is usually done to one’s left, and woe is the candidate who turns right! Being quarter turns, these are done four times until the candidates face front again. I am not sure whether candidates are still made to do these, but obviously these allow judges to view candidates from all sides – front and back, left and right. Just as inappropriate as a candidate turning right would be any bungling of the sequence of numbers in mod 4 counting. Observe, division of the positive integers above by 4 gives:
1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 …
Of course, 0, 1, 2 and 3 are the only possible remainders when dividing by 4. Equivalently, “1” as the first quarter turn has the candidate facing her left, “2” has the candidate facing back, “3” has the candidate facing her right, and “0” returns the candidate to the original position.
Here are some more familiar and everyday examples of modular counting. For each example verify that elements of the sequences can never be arranged differently.
Mod 7 is seen in the days of the week, taking Monday as 1, Tuesday as 2, and so on until Saturday as 6 and Sunday as 0. It is also represented by notes on a scale, with Do as 1, Re as 2, Mi as 3, and so on until Ti as 7, “which will bring us back to Do” as 0, as Julie Andrews sang in The Sound of Music.
Mod 12 is easily illustrated by the months of a year, with January as 1 and December as 0, and the numbers on a clock or watch face, with
Mod 26 is already being practiced by millions as early as high school, albeit unwittingly. Consider the English alphabet and assign 1 to letter A, 2 to letter B, and so on until 25 to letter Y and 26 (or 0) to letter Z. A high school student who accidentally bites her tongue excitedly asks her girlfriends for a number and immediately converts this to a letter corresponding to that number, according to the assignment mentioned. For example, the number 18 would correspond to the letter R, and the girlfriends would squeal, “Uuy, naalala siya ni Rodney!” Or, the number 28, in mod 26, would correspond to the letter B, and point to a Bobet. Sometimes, bigger numbers are requested so that the correspondence is less obvious (and intentional). Hence, the number 81 would correspond to the letter C and be interpreted as Caloy, and number 105 would correspond to the letter A and be interpreted as Ariel. That’s kilig care of mod 26!
Then there was the big Y2K scare at the turn of the millennium when all programs were only configured until year 1999, so that at
Let’s slow down a bit and go to mod 28 and mod 84. Why these modulo’s? Because in my mind’s meanderings about election dates, my paper and pencil gave me these numbers of years for the second Mondays of May to completely repeat their pattern of dates.
Year Date of 2nd Monday Year Date of 2nd Monday
of May of May
1989* 8 2073 8
1992 11 2076 11
1995 8 2079 8
1998 11 2082 11
2001 14 … …
2004 10
2007 14
2010 10
2013 13
2016 9
2019 13
2022 9
2025 12
2028 8
2031 12
2034 8
2037 11
2040 14
2043 11
2046 14
2049 10
2052 13
2055 10
2058 13
2061 9
2064 12
2067 9
2070 12
*assuming elections were held this year
Indeed, it’s not as obvious as our previous examples. We can play this in two ways. First, we may assign the number 1 to 1989, 2 to 1992, and so on until 27 for 2067 and 0 for 2070 and come up with mod 28. More accurately, we assign the number 1 to 1989, 2 to 1990, 3 to 1991, 4 to 1992, and so on until 83 for 2069 and 0 for 2070 and come up with mod 84. If you have the time and inclination, I encourage you not only to fill in the blanks in the table above, but better yet to expand the table and include the years in between, like 1990 and 1991, to be able to see the entire list and pattern. Of course, expect only to find entries ranging from 8 to 14, as these are the only possible dates for a second Monday (or any second — day, for that matter).
For mod 28 also, we can play a birthday game. Find out on what particular day you were born and verify that your 28th birthday will fall, or fell, on the same day. Of course it can happen before then, like my day of birth was the same for my fifth, 11th and 22nd birthdays, but the pattern of days your birthday will fall on (for all years from the first through the 27th), will repeat completely beginning on your 28th birthday, as if you’re a newborn again and back to age 0, a la mod 28. This is exactly the essence of modular counting.
For the true numerophile, or the brave and daring, I recommend mod 3300. This is the number of years that our calendar year will be in synch with the solar year, or the time it takes the Earth to complete its orbit around the Sun. A calendar year is actually about a quarter-day short of a solar year, and leap years are meant to take care of that. All the calculations involved – think mod 4, mod 100 and mod 400 – will show that it will take 3,300 years before the calendar and solar years will again be out of synch (look up websites such as infoplease.com or wikipedia.org).
You may question the utility of this, since we most probably won’t be around anymore for the turn of the century, more so the next millennium. I do not need it anymore as a campaign worker. I have since given up on fund-saving efforts with respect to campaign jingles. You see, just last May my candidate ran for provincial board member so coming up with a campaign jingle has become much less simple than a mere change of dates in the voice over. But hey, you can always ask your child to do mod 3300 counting and, lo and behold, put him or her to sleep with minimum effort. Now isn’t that a good thing?
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Cayen Arceo is an associate professor of Mathematics in UP Diliman. Her research areas include partial differential equations and operations research. A current interest is General Education Mathematics which she has been teaching for the past several semesters. E-mail her at [email protected].
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