A grade 3 question with right answer marked wrong.
A grade 3 Singapore math question made some waves among netizens in yesterday’s local paper. https://bit.ly/3VOzpHP
Debatably, the root of the discussion centers around the “right” way to do multiplication.
Process v. Product
In Singapore, it’s not uncommon for elementary school teachers and tutors to witness a student’s incorrect method of solution to a word problem that produces the correct answer. This frustrating situation arises far too often than what many of them would want to admit.
Due to a shortage of time, most school teachers pay lip service to the method of solution, pretending not to see that the students’ workings fall short of what is expected of them. These time-starved civil servants would simply look at the children’s answers and conveniently mark them “correct.”
In some cases or circles, it’d not be surprising too that some teachers or tutors themselves are oblivious of the correct method or procedure to solve a (routine or nonroutine) math question.
Personally, I’d want to give these school teachers the benefit of the doubt that they’re merely “lazy” rather than because they’re conceptually blind to the mathematical stain on their students’ worksheets.
Getting students to master a math concept with understanding requires time and effort (and also patience and pain), and most stressed teachers can’t afford either one in the name of having to “rush to complete the syllabus.”
For example, students’ or parents’ parroting that “multiplication is repeated addition” gives teachers and tutors quasi-zero clues whether they’ve understood the multiplication concept or not.
With regard to this grade three routine question that made the headlines, writing the correct procedure, without the teacher explaining to the student why he or she went wrong, only solves half the problem.
Understandably, some parents would argue that teachers shouldn’t be too rigid or radical about their children’s half-baked solutions to avoid dampening their self-esteem. For mathematically puritan math educators, the process is more important than the product.
Procedural proficiency with little understanding of math concepts would only produce elementary school drill-and-kill math graduates—the whys are as important as the hows.
An early penalty is better than a later one when it comes to a child’s learning of basic mathematical concepts in their formative years, which must be accompanied by a teacher’s or tutor’s explanation of the incorrect solution.
And in this grade three routine multiplication question, order matters.
Today is the last day of the annual 15-day Chinese New Year (or Lunar New Year) festival in China and Chinese communities around the world.
The Lunar New Year is so-called because the dates of celebration follow the phases of the moon—the new moon could fall on dates between January 21 and February 20, which is similar to Easter that could take place between March 22 and April 25.
Due to its “movable” date, the Lunar New Year (which is unspokenly steeped in superstition and divination, but unquestionably or expectantly celebrated by a billion-odd mainland Chinese and the forty-plus million faithfuls in the Chinese diaspora as part of Chinese tradition) serves as a rich recreational math or calendrical activity for teachers or educators worldwide.
In the Year of the Ox (or “Covidox”), which ushered in a palindromic date (12/02/2021), I pondered: “Any sexy formula that tells us when the Chinese New Year falls in a given year? Not calendrical recipes meant for symbol-minded geeks, but one for the majority of us, the simple-minded folks who’d key in the year and out come the CNY date & day of the week.”
With superstitious couples unfairly or irrationally treating baby tigers and bulls as “inauspicious,” but don’t mind baby bunnies, could supposedly conservative or puritan “fine” Singapore with a frightening low fertility rate of 1.2—below its replacement rate of 2.1, which could see its population heading the way of the dodo sans selective immigration and baby bonus cash incentives—expect a mini-baby boom in the Year of the Rabbit?
With few Covid restrictions still in place, would Singaporeans and permanent residents (and tax fugitives fearing political persecution or prison) be more excited this year to play their part in producing an above-average number of newborns-bunnies? And with Valentine’s Day around the corner, could the nation expect an overbooking of hospital beds or single wards in November?
I completely forgot that I wrote A Dozen Numerical Deeds for the Chinese New Year eight [sounds like a numerologically Sino-auspicious number?] years ago. If you want to keep the spirit of giving alive in the new bunny year, help yourself with some of the suggested gifts to bless others, Chinese and non-Chinese.
Pi recently took a page from Singapore’s only ruling party’s decades-long successful template how to stay in power perpetually, with quasi-zero public protests, strikes, and riots.
Will Pi win her viral legal case against the World Health Organization (WHO)? Millions of math teachers worldwide are unspokenly hoping that the WHO won’t alienate or betray Pi, just as it did for the marginalized and unrecognized variant Xi.
Would Pi be vindicated for unfairly being discriminated by those who’re playing politics to prevent her from being admitted as a bona-fide corona variant worthy of WHO’s recognition?
Imagine how excited math educators globally would be if Pi won her legal case against the WHO.
Come Pi Day or Pi Approximation Day, if the 16th letter of the Greek alphabet were officially declared a WHO-approved variant by then, schools and universities would be electrified in conducting an eclectic mix of pi talks and lectures to promote Pi’s ambivalent “murderous” new status.
I, for one, have started to work on Murderous Pi—the title of a math e-book to celebrate Pi’s evolution from a humble mathematical constant to a proud corona variant, which would be plagued with devilish or wicked problems on mathematics’s most beloved irrational number.
Are you looking forward for Pi’s long-overdue recognition by WHO’s health professionals in the Lunar New Year of the Rabbit?
Until she’s virally credited for her fatally mixed impact on earthlings before she’d rest in peace to make way for Rho and Sigma, who’re lining up to be WHO-recognized, Pi’s rejection would be perceived as an irrational [mathematical & viral] injustice or betrayal, whose aftermath paranoiac behaviors among math educators and virologists worldwide could potentially be unpeaceful, to say the least.
Like the “infidel” Zero, who’s ostracized by the Church Fathers of yesteryear, may the “agnostic” or “devilish” Pi be vindicated as she longs to gain formal approval to be in the company of fellow corona variants.
The advent of cryptocurrencies like Bitcoin and Ether has provided math educators worldwide with fertile resources to indulge themselves in creative mathematical posing and solving.
Unfortunately, the negative perception that cryptocurrency or crypto is a vector for serious organized crime and money laundering has led millions of half-informed or risk-averse folks to adopt a wait-and-see attitude vis-à-vis Bitcoin transactions.
Who/What Is Satoshi Nakamoto?
Nobody knows the identity of Satoshi Nakamoto. If the name isn’t a he or she or it, could the name be a covert group of cryptographers and mathematicians?
Like the modern-day equivalent of the Bourbaki group—the collective pseudonym of a group of predominantly French mathematicians in the 1930s, who tried to axiomatize mathematics to make it more rigorous?
So far, the few suspects—digital-currency addict Nick Szabo, Japanese mathematician Shinichi Mochizuki, and Co.—all have denied being the founder of Bitcoin, except for Australian computer scientist who loudly but unprovenly claimed that he is Nakamoto.
Crypto Math
Posing fertile crypto math questions is only limited by our imagination. Thanks to Bitcoin or cryptocurrency, NFTs, and the Metaverse, I’ve toyed around with a number of crypto math questions.
Below are a sample of Bitcoin-related questions that I hope would make their way into a math booklet fit for publication in a-not-too-distant future.
Bitcoins are divided into Satoshis: one hundred million Satoshis in each Bitcoin. At the current Bitcoin price, what fraction of a U.S. cent is worth the smallest fraction of a Bitcoin?
In 2010, a pizza restaurant agreed to accept ten thousand Bitcoins in exchange for two large pizzas. At today’s exchange rate, how much would each pizza be worth?
In September 2021, El Salvador approved Bitcoin as a secondary currency; in April 2022, Central African Republic followed suit. Which rogue or war-torn nation in Asia or the Middle East would be the first one to make Bitcoin its official currency? Or would it be “fine” city Singapore that would lead the way in becoming SE Asia’s crypto hub?
Crypto Apocalypse: What are the odds that due to hyperinflation (or a possible WW3 in the aftermath of the senseless Ukraine-Russia war) people would start losing faith in Bitcoin to the point that it suffered the same fate as the Zimbabwean dollar bills?
A golf resort is rumored to have been gifted with 13.257 ETH and 12.5 bitcoin from a Middle Eastern prince. How much did the shady resort receive in cash donation from their criminal donor?
Crypto Winter Is Coming!
With news of a crypto winter in the horizon, let’s hope that the mathematics of Bitcoin or cryptocurrency wouldn’t deter math educators globally from getting involved in creative mathematical thinking and problem solving.
I don’t know about you, but I’m waiting for Bitcoin to drop under $10,000 as my next buy alert. The future lies in Bitcoin—or in blockchain.
In the aftermath of a Singapore permanent resident who was bitten 26 times by a romp of otters in 10 seconds at the Singapore Botanic Gardens, which attracted mixed reactions from both locals and foreigners, I entertained the idea how math teachers could use this once-in-a-lifetime incident to indulge in some problem posing for elementary math students.
As natural swimmers, Singapore’s smooth coated otters, once thought to be extinct, have made a comeback to the island-state. They are believed to have swum across the Straits of Johore and made their homes here. Their present “locations of choice” are: Bishan Park, the Kallang River estuary, Marina Bay, and Singapore Botanic Gardens—and the Singapore Zoo.
Singaponacci Numbers and the C-Word
Mathematically or theoretically speaking, without culling, Singapore’s otters could exponentially grow as fast as Fibonacci’s rabbits, although otter sexperts think there are alternative humane ways of family planning.
An “otterman” who was furious that netizens had uttered the C-word against the cute-looking furry creature blamed them for spreading white lies or fake news about these “critically endangered animals.”
The Fear Factor
These “urban pests” seem to be better protected under the Wildlife Act than we humans against the Delta and Omicron variants. They’ve the SPCA behind them, but we’ve the POFMA* and FICA** over us to police our online behaviors.
Can you distinguish an adult otter from a baby seal?
Co-Existing with Covid and Otters
Last week, in the light of Singapore’s otter saga making the headlines overseas, I tweeted the following:
Otter Math: If the present population of 150 otters doubles every leap year, while Singapore’s annual fertility rate continues to head south, with zero immigration allowed, when would otter babies outnumber human babies? https://lnkd.in/gVfqjEZ8
Animals’ cruelty against humans: Otters bit a Singapore Botanic Gardens visitor 26 times in 10 seconds. What are the chances that such a similar incident happening again are less than the odds of someone being abducted by an alien? https://bit.ly/3oH93IO
Canals, Condos & Churches
Singaporean river otters are notorious in devouring dear ornamental fish found in condominiums and churches. If the probability of rich people disliking otters is x folds higher than that of poor folks, guesstimate x.
What are the odds that someone from a middle-income family in Singapore is more likely to cross paths with an otter family thanks to a greening Singapore, which provides an ideal milieu for otters to breed stresslessly, compared to Singaporeans, who often can’t even find a little space to do their private business?
Otter: Singapore’s Unofficial Mascot
Think of Singapore as an otter, and its frenemies-neighbors as crocodiles and monitor lizards. When threatened by its natural resources-rich, but militarily weaker, neighbors, Singapore would hit back to defend itself. Sure, they’d use low-cost warfare tactics like forest haze and arsenic-polluted rainwater to frustrate their little-red-dot neighbor—a form of asymmetric warfare to neutralize the enemy by natural and man-made means.
How not to mismanage Singapore’s otter population
Thou Shalt Not Catch an Otter!Fine: S$X
How much is the fine for someone caught capturing or trapping otters in the “fine” city of Singapore? What are the odds that a repeated offender could get jailed or/and caned for their illegal activity?
What If? Exotic Singapore Cuisine
Does otter meat taste more like chicken, beef, or pork? What if Singapore started farming otters to satisfy the desire of adventurous gourmets? Like crocodile, deer, or frog meat, could the legalized consumption of otter meat be a lucrative food business for both locals and aliens, who are looking for an exotic dish (halal, kosher & vegetarian), if there were a mismanaged otter population control in future?
An Otters Naming Contest
In the past, Singapore had the famous or notorious romp of otters called the “The Marina 10.” Now, we’ve local groups like the Zouk Family. Has the raft of otters that bit the über-unlucky British man 26 times at the Singapore Botanic Gardens been christened? If not, why about having a national contest to name them following the recent biting incident, which could have potentially led to Singapore’s first case of “death by otters”?
Otter Math for Mature Students
Let’s end with a grades 1–2 otter math question to tickle actress Sandra Bullock’s two children and their peers, who’ve been jabbed (or maybe even boosted) with a dose or two of the Singapore math vaccine to protect them against the plague of innumeracy:
After seeing an otter family of twelve devouring the fish in their condominium pond, a number of children were traumatized for weeks. Five children had recurring nightmares about the attack. There were three fewer children having nightmares than those needing counseling.
a) If two boys and a pair of twin sisters met up with a school counselor, how many children were altogether affected by the otter invasion?
b) Guesstimate how many fish were eaten up by the hungry otters in a few minutes before they made their way to the swimming pool.
Answer: (a) 13 children (b) Hint: Think about the demographics of the condo residents.
A grades 4–6 Singapore math question
Answer: 1. 8 koi fish. Can you solve the question in more than one way without algebra?
* POFMA: Protection from Online Falsehoods and Manipulation Act ** FICA: Foreign Interference (Countermeasures) Act
While the world is anxiously waiting what “Christmas (or Cliff-mas!) gift” North Korean dictator Kim Jong Un has reserved for President Trump, with the two rogue leaders now having revived their name-calling game, and as the world witnesses the impeachment trial of the Liar-in-Chief, how can math educators nevertheless use the caustic and divisive political climate in the US to spice their enrichment or recreational math lessons? How can they generate some mathematical positives from the many political and moral negatives?
A few Christmases ago, I wrote Christmaths: A Creative Problem Solving Math Book, which looks at some parallels between the king of public holidays and the queen of sciences. And now thanks to Donald Trump, his narcissistic, racist, and supremacist behaviors and tweets continue to provide thousands of math teachers worldwide fertile fodder to link math to politics.
The Mathematics of Trump
For many of us living outside the US, who are agnostic about the political views of both Democrat and Republican politicians, it’s hard not to poke fun at American [Trump?] politics.
Below are a sample of published entries, which irreverently looks at some parallelisms between Trump and Math.
Meet the Pinocchio-in-Chief
From Trumpworthy Math, I went on to define two units for measuring falsehood: tru and Pinocchio.
Pinocchio
A unit for measuring how far or how much a politician, dictator, or businessperson is lying to an oft-gullible public—named after a marionette with a long nose, the picture of a dishonest person.
Trump’s denial of having sex with an adult entertainer, while his wife was still pregnant, is at least six Pinocchios.
by MathPlus October 31, 2018
Lies, White Lies, and Statistics
True or False
(a) Politifact: “76 percent of Trump’s statements were false or mostly false.”
(b) Politico: “Trump told a lie every three minutes and fifteen seconds.”
What is a Trump Number?
Below is one of the two published definitions of a “Trump Number,” which had evaded me for some time, because I just couldn’t nail down a description, without too much falsehood or exaggeration.
Based on your knowledge or readings about Donald Trump, how would you define a “Trump Number”?
The “C” Word Among Atheists and Islamists
Two Christmases ago, when President Trump boasted how he restored the spirit of Christmas among those who worship the King of kings, I coined War on Christmas.
War on Christmas
When conservatives and President Trump have had enough of the political correctness by Democrats that people should say “Happy Holidays” and not “Merry Christmas.”
President Trump is hell-bent on winning the war on Christmas, who bragged that under him Americans can now freely and proudly say “Merry Christmas”—they shouldn’t be hijacked by atheists and Islamists in sharing Christmas greetings to others.
by MathPlus December 25, 2017
And last Christmas, I christened the following:
Trumpmas
The name Donald Trump would have Christened Christmas if he could have his way—after all, every day, the media unfailingly have something un-Christian to write about him, since he miraculously became the US President.
Those working under him long for the day when Trumpmas would turn into Christmas—probably they’ve to wait until they resign or get fired by the president.
by MathPlus December 23, 2018
The Existence or Nonexistence of Santa Claus
Following the Christmas phone call between the US President and a lucky child last year, I coined the following:
And that presidential “child-unfriendly” phone call prompted me to pose the following Christmaths question:
Santa’s Proof
Santa Claus (🎅): “Children don’t exist.”
Prove or disprove whether Father Santa is correct or not.
Wrapping a Christmas Present à la Trump 🌲🎉🎊🎄
On November 13, 2017, I tweeted the following:
“Nothing” for Christmas
The Gift of Nothing: When you go shopping and could find nothing to buy as a gift for your best friend or loved ones, why not give them the gift of nothing, because nothing is not for sale? Put nothing in a big box and give it to them. #nothing #zero #gift
Last Christmas Eve, I coined Trumpgifting:
Immoral Math: Quid-Pro-Quo Math, Barr Math, and Mueller Math
With some exaggeration and irreverence, it’s not too difficult to coin some mathematical words or phrases based on how President Trump and his oft-ethically challenged administration misrule the country.
Green Math: Trump’s Climate Hoax
With Donald Trump dumping the Paris Agreement for his personal political agenda, math and science teachers could play their part in educating students about the long-term consequences of the president’s selfish unscientific decision.
Fake Father Christmas
On April 13, 2019, I tweeted the following:
Fake Christmas: What are the odds that President Trump’s hypothesis that global climate change is a hoax could speed up the first Christmas summer in the US or EU? Or when fake Xmas trees permeate the homes? #Christmas #Christmaths #weather #climate #Singapore #math #humor 🌲🔢
The Sharpie Pen and Christmas Cards 🎅 🎄 ✉️
Guesstimate how many Christmas cards (and their envelopes) you can use with a Sharpie pen.
Estimate how long a line you can draw with a Sharpie pen.
If each person sent an average of 30 Christmas cards, how long would the Sharpie pen last them? Three years?
Trump’s Greetings Cards
Last December, to resurrect the lost art of sending Christmas or greetings cards, I coined the following slogan:
Make America Greet Again
Bringing back printed greeting cards as in the good old days before the advent of e-cards —when sending personalized cards meant so much to both senders and receivers.
This festive season, Trump supporters want their president to pass an executive order to ban those working in the US government to send e-cards, as this virtual way of greetings is “robotic” and “dehumanizing”—they want to make America greet again.
by MathPlus December 02, 2018
Guesstimate how many millions of greetings cards would be sent every year in the United States if President Trump were to pass his executive order on banning Christmas e-cards.
BC & AD: Restoring Year Zero
Discuss the social, economic, and political implications if the world were to switch to a “Trumpian calendar.”
Pi Exposition
One way for math educators to poke fun at the disturbingly obscene number of ethically challenged senators in exposing their moral hypocrisy to condone or defend the wrongs of a morally bankrupt president is through the irrational and transcendental number pi.
Or, even evangelical Trump supporters voters who refused to acknowledge or right his moral and political wrongs shouldn’t be let off the hook for condoning or exonerating the sins of a “statistical president.”
Prove that the biblical value of pi, as mentioned in 1 Kings 7:23 and 2 Chronicles 4:2, is 3.
The Aftermath of Christmas
Estimate how many people each year suffer food poisoning after eating Christmas leftovers. About half a million?
Roughly how many unhappy people return their unwanted Christmas presents on Boxing Day (and come home with discounted items)?
A Christmas Gift for Trump
Let me end with two Trump-friendly entries I submitted two months ago:
A National University of Singapore’s brochure cover promoting math and art.
A fortnight ago, Center of Math (@centerofmath) tweeted the following picture:
Imagine if Donald Trump were your high school math teacher. How would he disruptively or irreverently use the above illustration to teach some “pop (or poop) calculus” to his math-anxious Liberal Arts students?
From Epstein to Trump
President Trump’s “fun buddy” of yesteryear, Jeffrey Epstein, was unverifiably a pretty good math teacher before he became a successful financier, whose sinful soul had since journeyed to that hot fiery place, on the other side of eternity, sooner than later.
And not too long ago the president who claimed to have a “genius IQ” boasted that his “favorite” daughter is very good at numbers, which indirectly implies that she must have inherited his (or his first wife’s) “mathematical gene.”
An e-card promoting the disruptive title, “If Trump Were Your Math Teacher” (2020).
A phrase that had made its way into Urban Dictionary, but had since mysteriously disappeared from cyberspace.
Trump’s Calculus
Based on the defined meanings of f(x), what could f′′′(x) represent?
In Singapore, the durian is officially the only tropical fruit that is banned inside a public train or bus—to critics, it smells worse than urine combined with a pair of used socks.
Presently, transport officials are likely to confiscate the notorious fruit should someone be found conspicuously with it, until recurring public complaints force politicians to implement a fine for those caught carrying one in forbidden places.
If anyone in Singapore can be fined for failing to flush a public toilet, it’s not far-fetched to expect a penalty in a-not-too-distant future for those who inconsiderately propagate the pungent aroma of durians among Singaporeans.
A proof that Singapore is a “fine” city.
Dubbed the “King of fruits” by locals, enjoying the durian is arguably an acquired taste; however, it may cause premature death when eaten together with some types of food or drinks—check this out with your doctor to avoid going to the other side of eternity sooner than later.
For math educators who can’t stand the pungent smell of durian, much less taste it, how can they creatively make use of this much-loved or much-disliked fruit in their mathematics teaching?
In the aftermath of a church in Sarawak, Malaysia erecting a Christmas durian tree, the following estimation questions crossed my mind:
1. Guesstimate the number of durians that were used to make the Christmas tree depicted below.
A durian Christmas tree at a church in Sarawak, Malaysia. Source: @nobisha by norizan sharif on 25/12/18
2. Estimate how much the durian business in Malaysia meant for the China market is worth every year.
3. Estimate how many durians a ten-hectare durian plantation could produce every year.
4. What percentage of the Asian population love to eat the pungent-smelly durian?
Singapore Math and Durian
Below are two irreverent tweets I posted to poke fun at the notoriety of the durian among fruit lovers, who are often tickled by durianians who wouldn’t think twice about forking out more than fifty bucks for one über-smelly durian.
Make a short trip to Malaysia or Thailand during the peak durian season. Try to get hold of a dozen-odd types of durian from the local market or some durian plantations owners. Compare their prices, weights, textures, pH levels, smells, or tastes; and make some conjectures based on nasal, oral, and tactile factors. Does the number of spikes of some durian type exhibit Fibonacci-like behaviors?
2. Death by Durian
Model how many “durian bombs” pseudo-jihadists planning a terrorist hoax in some public places like a college campus or shopping mall would need to simulate some panic or irrational fear among the undergraduates or shoppers.
What are the odds that one of Singapore’s neighboring frenemies could one day use the durian as a low-tech weaponry to neutralize her, just as man-made haze pollution from unfriendly neighbors could potentially be weaponized to suffocate an entire nation?
3. A “Fine” Durian
Imagine that you have been assigned to draft a set of rules that would penalize those caught with durians in forbidden public areas in Singapore. Model a “fines guideline” that wouldn’t unfairly punish those who selfishly insist on polluting their milieux with the nose-unfriendly smell of durians.
New Year, New Entries
On a more positive or non-apocalyptic note, for this new year, some of you might wish to redefine Durian Math or add a new twist to it, as you discover new ways to infuse the term in your math lessons.
A blessed New Year 2019 to all math educators around the world.
Esplanade Theatre—Singapore’s “The Durian.” Photo source: visitsingapore.com
Thousands of students around the world celebrate Pi Day today, but local math students in Singapore can only dream of being part of this annual mathematical event. Singapore math students, teachers, and parents don’t (and can’t) celebrate Pi Day, as long as they officially follow the British style of writing their dates (DD/MM/YY).
What makes matters worse is that this year, Pi Day falls on the first day of the one-week school break, which makes it almost impossible for hardcore math teachers, who want to buck the calendrical trend, to get their students excited about the properties and beauties of the number Pi.
Until Singapore switches to the American style of writing dates (MM/DD/YY), which may not happen, at least during my lifetime, however, this shouldn’t prevent us from evangelizing the gospel of Pi among the local student population.
Here are seven e-gifts of the holy Pi, which I started musing about 314 minutes ago on this Pi Day.
Vintage Christmas—Just like Baby Jesus two millennia ago!
Christmas is a golden and joyful opportunity for number enthusiasts and math geeks to sharpen their creative mathematical problem-solving skills.
Here are 12 CHRISTmaths cookies that may help you shake your brain a little bit in the midst of Christmas festivities.
Warning: Refrain from forwarding this post to relatives or friends living in countries, which are intolerant of Christmas and Christianity, such as Brunei, Saudi Arabia, and Somalia, as it’s haram for “infidels” to take part in any kind of Christmas celebrations. And I assume that includes reading any on-line materials deemed un-Islamic or un-Mohammedan, which might lead believers astray from the faith.
1. Unlucky Turkeys
Estimate the number of turkeys that make their way to the supermarkets every year.
2. A Xmas Candy
Mary wanted to buy a candy that costs 25 cents. A dated vending machine would take one-cent, five-cent, and ten-cent coins in any combination. How many different ways can she use the coins to pay for the candy?
Remember to scan your Christmas item!
3. The Dimensions of a Cross
A square of side 25 cm has four of its corners cut off to form a cross. What is the perimeter of the cross?
4. The Number of Crossings
Two lines can cross one time, three lines three times, four lines six times, and five lines ten times. If there are 25 lines, what would be the maximum number of crossings be?
5. An Eco-Xmas
If all instances of the word “CHRISTMAS” were replaced with “XMAS,” how much ink and paper (or Xmas trees) could you save every year? How much money could be channelled back to feeding the poor and the hungry during the festive season?
In an age of Xmas e-cards and video cards, how many Christmas greetings cards are still being sent worldwide? How many trees are being saved every festive season?
(a) Without a calculator, how would you verify whether the number 25! has precisely 25 digits or not.
(b) Which positive integers n (other than the trivial case n = 1) for which n! has exactly n digits?
GST (or VAT) with no thanks to Father Xmas
8. Xmas Trees
Guesstimate how big a forest would 25 million Christmas trees occupy.
9. Folding papers
Fold a single piece of paper perfectly in half, from left to right. How many creases will there be after the 25th fold, when you continue folding so that all the rectangles are folded into two halves each time?
10. Pre-Xmas Tax
Imagine Singapore were to implement a pre-Christmas tax on all kinds of Christmas marketing before the first week of December. Estimate how many extra million dollars would the Income Tax department collect every festive season.
