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Singapore Math

Wednesday, October 31, 2012

Halloween Math

© 3vdesign.com
A decade ago, celebrating Halloween in Singapore was considered a foreign import among locals, just like it was with Valentine's Day a quarter century ago. Today, local community centers and tourist attractions like the Singapore Science Centre and Night Safari (which is part of the Singapore Zoo), couldn't resist organizing their horrors nights to meet citizens' interest in the occult. As a result, many parents and teachers are faced with the dilemma as to whether to allow their children and students to participate in celebrations that had their roots in sorcery, witchcraft, and satanism.

© savagechickens.com
For the rest of us who teach teenagers, who often like to dabble in, or experiment with, things occultic or satanic, how can we make use of some symbolism of Halloween, such as black cats, snakes, and broomsticks, to teach some creative problem solving in mathematics? How can we bewitch or bedevil them with "dark math," by entertaining them with some "wicked thoughts" about the most disliked subject in school?

Below are a baker's dozen of dark math problems we could pose to kids, who could get an intellectual kick out of solving them. Who says math and fear can't positively mix? Didn't someone say that God created the natural numbers, and all the rest is the work of man—and of the devil?

© christmastimeclipart.com

1. Dracula's Pets

Count Dracula collected an even number of insects (beetles and spiders) in his coffin. He counted 54 legs in all. How many spiders did he have?

13 is a "Lucky" number!
© ps3trophies.com

2. Friday the 13th and Halloween

Look at the following pattern:

13 × 13 = 169 and 31 × 31 = 961

Is this a mere palindromic oddity?

3. The Fewest Number of Coffins in a Day

On which day of the year do fewest people die?

© cartoonbarry.com

4. Cemetery Math

© 123rf.com
On a tombstone is written the following:

Jasper Rice
Died: 31/10/85
Buried: 30/10/85

If there was no error or crime involved, explain this apparently impossible situation, which took place in October 1985.  

5. The Secret Lives of Vampire Numbers

© christmastimeclipart.com

Numbers such as 1435 are called vampire numbers (or just vampires) because they are formed when two parent numbers (also called fangs), 35 and 41, are multiplied together (35 × 41 = 1435). Another vampire number is 2197, because it contains digits of its fangs, 27 and 81 (27 × 81 = 2187).

(a) Are there any two-digit vampire numbers?
(b) How many 4-digit vampires are there?

6. A Deadly Will 

A dead man's will states that seven bats were to be divided among his three ungrateful sons. The eldest was to receive one half of the bats, the middle one, one-fourth, and the youngest, one-eighth.

© holidaysville.com
The three wicked sons had nightmares as they couldn't inherit their shares without cutting bats into parts.

They sought the help of a wise priest who suggested that he would lend them one bat so that a “fair” allocation could be possible. The eldest son would then receive 1/2 × 8 = 4 bats, the middle one, 1/4 × 8 = 2 bats, and the youngest one, 1/8 × 8 = 1 bat. Thus, the sons would receive 4 + 2 + 1 = 7 bats, returning the borrowed bat to the holy man. Is the priest's solution “fair”?

© l3ak.com
7. Halloween Falls on a Frightday!

In a complete period of 400 years (after this, the calendar repeats another similar cycle), which day of the week is Halloween mostly likely to fall?

8. Legion's Number

In the Gospel of Mark, Jesus encounters a multitude of demons possessing a man. When the Son of Man asked them, "What is your name?" the man replies, "My name is Legion, for we are many."

The computer scientist, Clifford A. Pickover, defined the number 666⁶⁶⁶ as Legion's number of the first kind, and the number 666!⁶⁶⁶! as Legion's number of the second kind.

(a) What are the last 31 digits of 666!?
(b) What are the last ten digits of 666!⁶⁶⁶!?
(c) Using a computer, what are the last ten digits of 666⁶⁶⁶?

By definition, 666! = 1 × 2 × 3 ×⋯× 665 × 666

© www.coxandforkum.com

9. The Math of 9/11

Using the pattern and period of the calendar, and the day of the week September 11 falls this year, what day of the week was 9/11—the "Holy" 2001 War by the "fidels" on the Twin Towers?

The Pentagram
© werewolves.com
10. Is Phi Divine or Demonic?

An equation uniting the Number of the Beast 666 and the golden ratio ø is given by

ø = −{sin 666º + cos[(6)(6)(6)º]} 

Verify phi's apparent liaison with the satanic number.

11. A Grave Calculation

© hawaiidermatology.com

How long will it take before the whole 
world is covered in gravestones?

Hint: What is the total land area on Earth? 
How much area does each person's grave take up?

12. Triple 4 and Triple 6

The Chinese avoid the "death number" 444; the Westerners avoid the "satanic number" 666. The West like the number 7 and multiples of 7, while the East like even numbers but avoid odd numbers. In general, Asians' "lucky" numbers are Westerners' "unlucky" numbers.

Why do both the East and the West still have faith in numerology and astrology, in spite of literacy, numeracy, and technology?

© all4myspace.com
13. Santa versus Satan

It is said that Satan is a wolf in sheep's clothing: f(Santa) = Satan.

Prove that Oct. 31 = Dec. 25.


Pickover, C. A. (2002). The mathematics of Oz. Cambridge University Press.

Pickover, C. A. (1997). The loom of God: Mathematical tapestries at the edge of time. New York: London Plenum Trade.

Santos, A. (2009). How many licks? Philadelphia: Running Press.

Yan, K. C. (2011). CHRISTmaths: A creative problem solving math book. Singapore: MathPlus Publishing.

Selected Answers

1. 3 spiders and 5 beetles.
Spiders have 8 legs and beetles 6.
We need to solve 8s + 6b = 54 with s, b positive and s + b even.
Hence there are 3 spiders and 5 beetles.
The Fiery Pentagram
© shutterstock.com

3. On the same day that the fewest people are born—29 February.

4. The man dies on a Monday and his body was then flown across the International Date Line to a place where it was still Sunday for immediate burial.

5. (a) No 2-digit vampires exist.
(b) 7 four-digit vampires.

8. (a) They are all zeros.
The last digits of 666! are 0 because 510 and 210 both divide 666! The number 666! has well over 100 zeros.
(b) The last 10 digits of 666!⁶⁶⁶! are 0. 
(c) The last 10 digits of 666⁶⁶⁶ are 0,880,598,016.

Challenge: What are the first 10 digits of Legion's number of the second kind?

9. Tuesday.

11. More than a million years.
Hint: How many people die every year round the globe? 
Time taken = (Total area of the Earth) ÷ [(Number of dead people per year) × (Area per dead person)]

13. If "Oct." is the short form of "octal" and "Dec." for "decimal," then 31 (in base-8 notation) is equal to 25 (in base-10 notation): 31₈ = 25₁₀

© Yan Kow Cheong, October 31, 2012

Wednesday, July 25, 2012

Thank you, Math!

Journal writing in mathematics education is often perceived as a boring activity in getting students to reflect on their own mathematical thinking. Perhaps an appealing way to assess students' mastery of mathematical concepts is to encourage them to write "mathematical letters" to a buddy in the form of "thanks goodness for anything numerical or mathematical."

Let's look at a sample of these math-lite correspondences.

Dear Numbers

© mashable.com 
Thank you for being such a buddy I can always count on. You don't change because of geography or history. A while ago, you gave me an idea to blog on "Singapore Math by Numbers." Without you, the world would come to a standstill. You numbers don't just rule the world; today, you've been semi-elevated to a god—with all your metrics and analytics.

Numerically yours

Dear Calculus book

Thank you for letting me use you as a weight or support. Your thickness scares the hell out of so many otherwise intelligent mortals, especially those who have had an unpleasant experience with their high school math teachers. Besides, I use you as an inexpensive cure to help my friends and loved ones, who are suffering from insomnia—your deltas and epsilons are sufficient to hypnotize the hardcore sufferers.

Infinitesimally yours

Dear 1001
© shashindrasri.blogspot.com

You're a mirror to my numerical soul. Indeed, you provided me with so many oddities and idiosyncrasies about the Palindromic Family. In fact, I've started writing the first few chapters of "The Joy (and Pain) of Palindromic Numbers."

Palindromically yours


Thanks for being such a reliable companion in my dealing with irrational and illogical folks. Your cold, austere character has put off many egos, who would otherwise have trampled over the rest of us.

Logically yours

An i-protractor
© 1mobile.com

Dear Protractor

I hope you won't go the way of the abacus or slide rule, because I'm seeing lesser use of you these days. Anyway, I promise to keep you as a vintage math instrument for my grandchildren.

Dear Compasses & Unmarked Ruler

Thank you for frustrating and humbling thousands of our mathematical forefathers in their futile attempts to constructing a square with the same area of a circle. To revive your popularity, I'll be conducting an enrichment math lesson on The Lost Art of the Compass.

Product Details
Dear 25

Thanks for confiding with me some of your innermost secrets, which allowed me to write some parallelisms between the king of the public holidays and the queen of the sciences. Don't worry, I'll dedicate CHRISTmaths to you on its silver edition.

Joyfully yours

Dear Boring Math Teacher

You provide jobs to half a dozen tutors, as they try to teach the most disliked subject in school in a less parrot-like manner! And your boring test of artificial questions only serves as a filter to separate the nerd from the herd—the mathematical goats from the sheep.

© business.mcmaster.ca
Dear Probability

Thanks for mocking my intuition and flawed logic. The answers to your problems have humbled the mathematical brethren, as you tease them with your counter-intuitive results.

 © tower.com


Thank you for making me pronounce the first part of the word "factorize" to sound like the four-letter word. To test my hypothesis, I tried that trick on my French female teacher many moons ago, and the class then couldn't control themselves but burst into laughter*. 

Fact-ally yours

* Try this trick on your French-speaking friends and fans who speak English as a second or third language.

Dear Math Editor

Thank you for introducing me to so many conceptual and linguistic faux pas—I couldn't have fabricated them without your poor grammar and punctuation, and shaky grasp of some difficult mathematical concepts. Some are jaw-achingly funny. My collection of your blunders is growing, and I couldn't wait to compile them in a book for green editors. If only I knew earlier that "poor writes" could open up the doors to many freelance assignments. 

© 123opticalillusions.com

Dear Dice

Thank you for feeling so good in my hand and for showing me so many different possible outcomes. You've taught me a lot about chance and probability, and you look pretty sitting on my book shelves. That's where you'll stay. Love, Leah.

Dear Sex

Thank you for revealing to me your intimate liaison with the most boringly taught and disliked subject in school. And for giving me enough materials to writing and getting "Mathematics And Sex" published.  Indeed, MATH is a four-letter word! 

© mathandchess.net
Dear Chess

Thank you for using me to beat those older and smarter chaps. Mentally satisfying to "humble" them! But I can't stand you when you used the same trick on me years later—when those young geeks refused to let me win or draw even one game.

Dear Calculators

Although you had atrophied my mental calculation abilities, however, you're a God-sent tool to working out those tedious and mindless computations. I'm totally dependent on you for my everyday numerical chores—from calculating the waiter's tip to finding the amount I owe to the credit card companies. On the negative side, you've upped my innumeracy index by a dozen points. On a positive note, I wished you'd upgrade yourself to becoming an instrument of desire, just like the iPad. 

Dear Amazon.com

A trick-and-tricky math book 
Thank you for showcasing our math titles on your site, but we hate that 55 percent you're charging us every time there is an order for our books. When we factor in the time we need to dispatch them, and the shipping freight, most of us are only operating at a negative profit! Plus, your payment in US dollars, and your reluctance to link with PayPal, makes authors living outside the US hesitant to promote their titles on Amazon.

© spaceadventures.com

Thanks for sharing with me how you're once marginalized by your fellow numerals, and how you're even ostracized by the Catholic Church for calling you the "infidel zero." Your forgiving spirit towards the Pope and his bishops, and your revelation to Newton and gang regarding some of your oddities and idiosyncrasies, had earned you a numerical place in the Mathematical Hall of Fame. However, your critics never cease to call you the "troublesome zero."


Pictures of Nothing: Abstract Art since Pollock (Bollingen)Folks always mistake you for the numeral zero—both of you are like faux twins. Zero is nothing, and nothing is zero. In fact, you're anything but nothing—you're something! Philosophers and mystics curse or deify you, with their ever-convoluted arguments—they've almost elevated you to the status of an atheistic religion. 

There's almost an alphabet for your name: abyss, chasm, emptiness, hole, pit, void, nothingness, nada, naught, …, nil, nix, zilch.


Dear Empty Set

Thank you for causing so much confusion among school kids. Do you exist or not? Are you really a set? Are you like a club with no members? Or, a bag without items? You're like something with zero nothings.

© shop.cafepress.com 
Dear Imaginary Numbers

Once upon a time, they denied your existence, just like your rejected, dejected, and ejected cousins, the negative numbers. But today, like zero, you're having the last laugh; in fact, we often take you, guys, for granted, albeit you prefer to keep a low profile. I wished school teachers won't tell the kids that you're complex, when you're really simple to manage.

Dear May 1

This is the only date in the year, when working folks don't have to labor. A day of protests and political rallies, you remind the masses of Karl Marx's words that workers of the world must unite—that they have nothing to lose but their chains.

© eoagh.com

Dear Black Hole

It's beyond my reasoning ability to understand the math that predicted your existence. Thank God, He produces a few Hawkings among His creation every century or so to reveal your galactic existence and importance. 

 © abqjournal.com
Dear Sudoku

You give me hope that in my golden years my chances of dying from schizophrenia or dimentia are significantly reduced if I indulge myself to playing with you more often.

Dear Numerology

© geocaching.com 
From the simple idea of digital root, you manage to bluff tens of millions of numerically challenged folks into believing that they can predict their good and bad days, and that they can take control of their future from their birthdates. Indeed, a pseudo-equation of yours is: f(birthdate) = lucky/unlucky days 

Dear iPhone

Thank you for saving me the trouble of having to carry a scientific calculator around. Yes, by a mere 90-degree turn, you magically turn yourself from a four-function to a scientific calculator. I look forward to seeing you functioning as a graphing calculator in a-not-too-distant upgrade. Hope you can strike a deal with Texas or Casio, just like you did with HP.

Dear Statistics

Thank you for revealing to me all your curves and standard deviations. However, I'm still confused about your three musketeers: mean, mode, and median. Which mean do you mean? Now I understand why some folks call you "Sadistic Statistics" for torturing hundreds of thousands of students every day.

Dear 13

© sabotagetimes.com 
I know you didn't deserve that awful publicity surrounding your name, especially when those superstitious folks wickedly try to associate you with some black Friday. You're just an innocent-looking prime who had suffered so much—far worse than 666 and 4444.

Thank you for providing me with rich materials to write about, every time you fall on a Friday. We, recreational math enthusiasts, feel a bit like "numerical hypocrites," when we can't help but gossip about the numerical quirks of Friday 13.

Un-triskaidekaphobically yours

Dear Death

Thanks for being the great equalizer. You won't let the rich and the powerful, or the arrogant and the haughty, live beyond their undeservedly numbered days. Three scores and ten, or a billion odd heartbeats—that's your deadline to them, all!

It's now time to write and e-mail some math-lite letters to the rest of us to read and enjoy!

Mathematically yours

Dieterich, L. (2011). thx thx thx: thank goodness for everything. Kansas City, Missouri: Andrews McMeel Publishing, LLC. thxthxthx.com

© Yan Kow Cheong, July 24, 2012