Anecdotally or statistically, an unhealthy number of mathematicians and math educators around the globe are chain-smokers. Out of habit or addiction, they need to puff out before any proof.
Likewise, it wouldn’t be surprising that a (smaller) percentage of them would also need to drink before they derive any mathematical result, or prove or disprove any conjecture, which is worth gracing the pages of a reputable journal or periodical.
Strong Zero
On a visit to a local supermarket two years ago, I spotted Strong Zero, which led me to tweet about it:
Strong Zero: Something to reward yourself with at the end of a long fruitful day indulging in mathematical proof to destress yourself with fellow boozers.
Then, I had in mind to “derive, then drink,” rather than the other way around. The choice is yours! Know your limits!
If you drink, don’t derive! Really?
So, a little boozing and smoking (in moderation) might debatably be an unspoken (inexpensive?) boost or catalyst to experiencing an aha! When the product of two negatives produces a positive!
A Quickie from Russia
Since we’re on the topic of drinking alcoholic beverages or liquor to boost mathematical productivity, let’s end with a mathematical quickie from Putinland, which was pre-Xed (or tweeted) slightly less than a dozen years ago:
A man and his wife drink a keg of kvas in 10 days. He alone can drink it in 14 days. How long will his wife take to drink a keg?
Challenge: Try solving the above proportion problem in a nontraditional way (with or without a drink)! Better still, use a bar model to do it.
On May 23, 2023, @OrwellNGoode tweeted the rate question below, with the “teacher” marking the student’s answer of 20 minutes wrong, and showing why it’d be 15 minutes instead.
Who the heck takes 10 minutes to saw a board?
I’ve no idea what grade level this math question was being assigned to. Although questions on ratios and rates are formally introduced in grades 5–6 in most parts of the world, however, it’s not uncommon to spot these types of mathematical quickies in grades 1–4 Singapore math olympiad papers to trap the unwary.
Assuming that the word problem didn’t come from a bot or from ChatGPT, the teacher’s intuitive reasoning may be said to be ratio-nally sound but rationally incorrect in solving this pseudo-proportional math question.
Logically or mathematically speaking, few math teachers would disagree that the student was right and the teacher was wrong.
This arguably “badly worded” or “ill-posed” math question provides a fertile ground for a number of possible (valid or creative?) answers, probably much to the annoyance of most math teachers and editors, who often feel uncomfortable or jittery about questions with more than one possible answer.
Indeed, there is no shortage of supporters to defend a “15 minutes” answer. For example, since there is zero mention that the length of each board is of equal length (and we can’t assume it to be so), or as each sawing might take place in a different direction, the “logical” answer of 20 minutes can’t be taken as mathematical gospel truth.
That three pieces need two cuts or sawings is unanimous among problem solvers. The bone of contention is the assumed length of the second cut. Say, if the second cut was half as long as the first one, then it’d take half the time of the first cut, in which case the answer of 15 minutes would be practically plausible.
A creative solution from @A_MGregory
It looks like we’re only limited by our imagination or creativity to rationalize why the answer can’t in practice be 15 minutes or any other duration, by using a different (creative) reasoning from the flawed one provided by the “teacher.”
Like most artificial or impractical word problems in school math, this rate question debatably falls short of design thinking and is thus open to different interpretations or assumptions, which might also weaponize some “anti-woke” math educators to ban or censor these types of “confusing or tricky” math questions.
Ironically, this is why injecting a dose of realism or creativity to these oft-ill-posed or contrived math questions would help open up the minds of uncritical or unquestioning math educators.
Don’t just answer the questions, question the questions.
For the majority of us who aren’t born or blessed with a mathematical or symbol-minded brain, but nevertheless appreciate the austere beauty of mathematics, writing about mathematics and math education is the second best thing we’d do to console ourselves that we needn’t be first-rate mathematicians to enjoy the language of science and technology, or to appreciate the science of patterns.
Some mathematicians write novels under a pseudonym to avoid any suspicion from their faculty bosses; others compose limericks and haikus as a creative outlet to showcase their hidden poetic talents. And for the rest of us who are neither novelists nor poets, maybe submitting some definitions to Urban Dictionary, by coining new mathematical words, or redefining old ones, could be the first step to activating that atrophied right part of our brain, which is allegedly responsible for creativity.
On this Pi Day, let me share with fellow math educators eleven approved definitions related to the irrational and transcendental pi. Don’t ask me how many times I got rejected and needed to resubmit some of these definitions again, before the Urban Dictionary editors decided to approve them.
Pre-Pi Day
Pre-Pi Day seems to have been serially downvoted and subsequently deleted to prevent digital abuse, because the approved entry can no longer be accessed.
Rejection isn’t failure. We keep refining or redefining any rejected definitions until the editors have zero excuses to reject the resubmitted entries. I wished I’d share some recipe for these approved pi definitions, but any attempt to offer some tips to increase a math educator’s chances of getting these math words or terms approved would probably be futile, to say the least.
Over time, although I’ve managed to reduce the odds of rejection, however, some submissions inevitably end up in the little red book of the mean editors—maybe these word doctors had a bad day, or simply because I was submitting some “mathematical crap” that caused me to receive emails like the following:
Urban Dictionary – Pi-rated was not published
Thanks for your definition of Pi-rated!
A few volunteer editors read your definition and decided to not publish it. Don’t take it personally!
Pi-rated The term to describe any faux facts about the irrational number pi.
On Pi Day, our teacher tricked us with some pi-rated math: pi is a rational number (22/7); pi has a different value on the moon that on earth; pi has a value of three in the Bible.
It’s never too late to be mathematically playful, by playing your part in submitting some irreverent mathematical definitions to enliven your math lessons.
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
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.
Math educators, especially stressed [often self-inflicted] local teachers in Singapore, are always on the look-out for something funny or humorous to spice up their oft-boring math lessons. At least, this is the general feeling I get when I meet up with fellow teachers, who seem to be short of fertile resources; however, most are dead serious to do whatever it takes to make their teaching lessons fun and memorable.
It’s often said that local Singapore math teachers are the world’s most hardworking (and arguably the world’s “most qualified” as well)—apparently, they teach the most number of hours, as compared with their peers in other countries—but for the majority of them, their drill-and-kill lessons are boring like a piece of wood. It’s as if the part of their brain responsible for creativity and fun had long been atrophied. A large number of them look like their enthusiasm for the subject have extinguished decades ago, and teaching math until their last paycheck seems like a decent job to paying the mortgages and to pampering themselves with one or two dear overseas trips every other year with their loved ones.
Indeed, Singapore math has never been known to be interesting, fun, or creative, at least this is the canned perception of thousands of local math teachers and tutors—they just want to over-prepare their students to be exam-smart and to score well. The task of educating their students to love or appreciate the beauty and power of the subject is often relegated to outsiders (enrichment and olympiad math trainers), who supposedly have more time to enrich their students with their extra-mathematical activities.
A prisoner of war in World War II, Sidney Harris is one of the few artists who seems to have got a good grasp of math and science. While school math may not be funny, math needn’t be serious for the rest of us, who may not tell the difference between mathematical writing and mathematics writing, or between ratio and proportion. Let Sidney Harris show you why a lot of things about serious math are dead funny. Mathematicians tend to take themselves very seriously, which is itself a funny thing, but S. Harris shows us through his cartoons how these symbol-minded men and women are a funny awful lot.
Angel: “I’m beginning to understand eternity, but infinity is still beyond me.”
Mathematical humor is a serious (and dangerous) business, which few want to invest their time in, because it often requires an indecent number of man- or woman-hours to put their grey matter to work in order to produce something even half-decently original or creative. The choice is yours: mediocrity or creativity?
Humorously and irreverently yours
References
Adams, D. S. (2014). Lab math. New York: Cold Spring Harbor Laboratory Press.
Harris, S. (1970). What’s so funny about science? Los Altos, Ca.: Wm. Kaufmann, Inc.
Check out an inexpensive (but risky) way to make a Singapore math lesson less boring: The Use of Humor in Mathematics. The author would be glad to visit local schools and tuition centers to conduct in-service three-hour math courses for fellow primary and secondary math teachers, who long to bring some humor to their everyday mathematical classrooms—as part of their annual 100 hours professional upgrading. Please use his e-mail coordinates on the Contact page.
A classic elementary math problem that folks from a number of professions, from psychologists to professors to priests like to ask is the following:
A bat and a ball cost $1.10 in total. The bat costs a dollar more than the ball. How much does the ball cost?
For novice problem solvers, the immediate, intuitive answer is 10 cents. Yet the correct response is 5 cents. Why is that so?
If the ball is 10 cents, then the bar has to cost $1.10, which totals $1.20. Why do most of us jump to the wrong conclusion—that the ball costs 10 cents?
We should expect few students to bother checking whether the intuitive answer of 10 cents could possibly be wrong. Research by Professor Shane Frederick (2005) finds that this is the most popular answer even among bright college students, be they from MIT or Harvard.
A few years ago, I included a similar question for a grade 2 supplementary title, as it was in vogue in some local text papers. See the question below.
Recently, while revising the book, I saw that the model drawing had been somewhat modified by the editor. Although a model drawing would likely help a grade 2 child to better visualize what is happening, however, a better shading, or the use of a dotted line, would have made the model easier to understand. Can you improve the model drawing?
Try to solve the above question in a different way, using the same model drawing.
Interestingly, I find out that even after warning students of the danger of simply accepting the obvious answer, or reminding them of the importance of checking their answer, variations of the above question do not seem to help them improve their scores. I recently tickled my Fan page readers with the following mathematical trickie.
Two cousins together are 11. One is 10 years older than the other. Find out how old both of them are.
Let me end with this Cognitive Reflection Test (CTR), which is made up of tricky questions whose answers tend to trap the unwary, and which may be suitably given to problem solvers in lower grades.
1. If it takes 5 machines 5 minutes to make 5 bearings, how long would it take 100 machines to make 100 bearings?
2. In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?
3. A frog is climbing up a wall which is 12 m high. Every day, it climbs up 3 m but slips down 2 m. How many days will it take the frog to first reach the top of the wall?
4. A cyclist traveled from P to Q at 20 km/h, and went back at 10 km/h. What is his average speed for the entire journey?
5. It costs $5 to cut a log into 6 pieces. How much will it cost to cut the log into 12 pieces?
In his creativity book, The Forgotten Half of Change, Luc De Brabandere shares that it took about half a century to move from sailing ships to steamships. Resistance to change from sail manufacturers only led to ships with more and more masts. During that time, hybrid ships took to the waves; for example, the Sphinx was equipped with three masts but also with a funnel for a steam boiler.
The Evolution from Sails to Steam
In 1833, if you were an onlooker seeing that ship sail by, how would you perceive its double source of energy? On one hand, you might see the engine as something that could be useful on a day with no wind; on the other hand, you could think of the sails as something that could come in handy should the engine break down. This means one can view the same boat from two angles; as you see more and more of these steamships, then one day, you realize that this is no longer an option, thus resulting in a break in your perception.
Or, what about an abaculator—an abacus that comes with a four-function calculator? Or, a book with an attached CD (containing an electronic version of the book)? Change is not an option—the options are what and when. As De Brabandere remarked, “You have to change twice: perception and reality.”
Change is not an option—the options are what and when. As De Brabandere remarked, “You have to change twice: perception and reality.”
© Sidney Harris There is nothing new under the mathematical sun! 
