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Singapore Math

Saturday, December 17, 2011

25 Christmaths Toughies from Singapore

Here are 25 non-routine fertile questions for friends and relatives to solve during this festive season:

1. The Recurring 25

Convert 0.252525... to a fraction.

2. The Sides of a Polygon

If the sum of the interior angles of a polygon is 2525°, what is the maximum number of sides the polygon can have?

3. December 25, 2025

On what day of the week does Christmas Day fall in the year 2025?

4. Number of Diagonals in a 25-gon

What is the number of diagonals in a 25-sided convex polygon?

5. How Many Ways?

In how ways can you arrange the letters of the word CHRISTMASTIME?

6. Merry Xmas & Happy New Year

If the value of MERRY CHRISTMAS is 189, the value of A HAPPY NEW YEAR is _____.

7. Santa's Routes

Determine the total number of distinct shortest routes from point A to point B in the following diagram grid map.


8. A Sum of 25 Powers

Find the sum of S = 2 + 21 + 22 + 23 +⋯+ 225.

9. Writing 25

What is another way of getting the number 25 using the numbers 2 and 5 only once, other than equating it to 5²?

10. Time Needed

Determine how long it will take to return all the gifts mentioned in the song “The twelve days of Christmas” if the gifts are returned at the rate of one gift per day.

11. Join the Xmases Family

A mother who was born on December 25, married to a husband also born on December 25, gives birth to their first baby on December 25. What are the odds of the mother doing that?

12. Four Fours to Make 25

Using four fours, the four basic operations and, if necessary, √4 and/or 4!, form the number 25. [Note: 4! = 1 × 2 × 3 × 4]

13. The Largest 25-digit Integer

What is the largest 25-digit number that can be divided by 2 and 5 without any remainder?

14. Number of Zeros in 1 × 2 × 3 ×⋯× 24 × 25

How many zeros are there at the end of the product 1 × 2 × 3 ×⋯× 24 × 25?

15. The Chocolate Bar

Given a 25-piece square chocolate bar, how many snaps are required to break the bar into its individual pieces?

16. A Creative Thinking Question

It is late on Christmas Eve and little Noel is waiting the Christmas tree to be finished. At exactly what time will this happen?

17. Christmas and Halloween

Prove that Oct. 31 = Dec. 25.

18. Divisibility by 25

Show that, if n is a positive integer, 16n + 10n – 1 is divisible by 25.

19. A Pair of Christmas Alphametics

In each case, replace each letter with a digit, different letters being different digits.



20. A Sum of Squares

Find the value of 1² – 2² + 3² – 4² +⋯+ 25².

21. Number of Rectangles

A board has 25 squares. How many individual rectangles are there in all?

22. A Christmas Party

At a Christmas party, each child brought a present. Presents were put in a large basket. All presents were different but identically wrapped. Going home, each child randomly selected a present from the basket. What is the expected number of children who carry home their own presents?

23. How Many Lines?

Using only horizontal and vertical lines, what is the most number of straight lines you can draw between the dots in a 5 × 5 grid without lifting your pencil?

24. A Table of 25 Boys and 25 Girls

A total of 25 boys and 25 girls sit around a table. Show that it is always possible to find a person both of whose neighbors are girls.

25. All n's End in Zeros

For what values of n does 1 × 2 × 3 ×⋯× (n – 1) × end in 25 zeros?

A Merry CHRISTmaths to all!

Selected answers

1. 25/99     2. 16 sides.         3. Monday.      4. 275 diagonals.     5. 778,377,600 ways.
6. 158.       7. 252 routes.      8. 2^26 - 2      9. 5 ÷ .2 = 25          10. 364 days.
11. 364/1         12. 4! + (√4 + √4)/4             13. 999…99990     14. 6.         15. 624 snaps.

16. Midnight.       18. 600.       19. (b) 2 + 97,445 + 6,928          20. 325.         
21. 225 rectangles.                    22. 1.                 23. 34 lines.      25. 105, 106, 107, 108, 109

For full solutions, consult the title in the Reference.

Yan, K. C. (2011). CHRISTmaths: A Creative Problem Solving Math Book. Singapore: MathPlus Publishing.

© Yan Kow Cheong, Dec. 16, 2011.

CHRISTmaths: A Creative Problem Solving Math Book
Now available as an iPad app

Thursday, December 15, 2011

25 Things You Wished Were Untrue about Christmas

25. Toys are made affordable for millions of parents thanks to the appalling working conditions endured by under-aged children in Santa’s sweatshops, in China.

image photo : Fractal Christmas trees
Fractal Christmas Trees
© Dreamstime.com
24. Proud CEO: “My time in looking for a five-dollar present for my company Christmas party is worth more than US$2,525—I’d rather issue a check 100 times the amount for the gift.”

23. An e-card costs much less than 25 cents—maybe a mere 2.5 cents plus or minus a few bits and bytes.

22. There are fewer joyful people on Christmas Day than on any other day of the year. Do you know why?

21. Men dishonor Christ more in the 12 days of Christmas than in all the 12 months of the year—by their boozing and carousing.

20. Even staunch Christians may not be aware that Christmas is a pagan feast, originally meant to worship the Sun god, instead of the Son of God.

19. There may be more non-believers celebrating Xmas than believers every year.

© teachercreated.com
18. Singles, poor people, and entrepreneurs could not wait for Christmas to come and go—not a happy time to be alone or be lonely.

17. Many believers and non-believers celebrate Christmas without a genuine understanding of the meaning of the public holiday—probably the most celebrated and the least understood festival of the year.

16. Retailers rely on Xmas significantly on the festive sales to remain financially afloat.

15. People merry so much on Xmas Eve that they don’t have time to open up their Xmas gifts—Boxing Day often becomes a sleeping day!

14. Even believers forget that Christmas needn’t be merry to be meaningful—it is the Christ of Christmas that needs to be celebrated, not Christmas itself.

13. Believers and non-believers are more and more expensively anxious or stressed to buy something to please others—as a sign of showing off their wealth, or exhibiting misplaced generosity.

12. New Christians do not want to be excluded from Christmas as they await for their employers’ Christmas bonus—that universally revered figure.

11. Employees are willing to work under unreasonable bosses because of the Christmas bonus—many at the same place, for the same boss, for more than 25 years.

The Search

© www.sodahead.com
10. Believers from other faiths often feel morally obliged to celebrate Christmas with their Christian colleagues and neighbors, all in the name of religious harmony.

9. Religious fanatics intolerant of Christianity wished Christmas were celebrated every quarter of a century instead of every year.

8. Some people would buy off a Christmas gift for their children on eBay for several thousand times the original price.

7. People max out their credit cards to finance the gift storm, as compared to their grandparents who saved money for Christmas.

6. Some allegedly US-friendly Arab nations tend to treat Christians worse than those living under dictatorial Muslim rule.

5. A month after Christmas, the holiday is only halfway paid off—credit card debt resulting from a borrowed merry Christmas. And a third of the money borrowed for Christmas spending is still not paid off two months after the holiday.

CHRISTmaths: A Creative Problem Solving Math Book
A Christmas gift for numbers lovers
4. This Christmas over a billion people in the world are surviving on less than 70 pence a day.

3. Politically correct retailers have banished Jesus and Christmas from the mall to accommodate to the secular demands of agnostics and non-theists.

2. Christmas is celebrated today more like a sales frenzy than as the most important birth in history—the commercial mentality has robbed the message of giving.

1. The closer people get to Christmas, the pressure to give these (useless) unnecessary gifts goes up and they feel depressed and unworthy if they can’t give.

Merry X-W/g to all!

© Yan Kow Cheong, Dec. 15, 2011

Friday, November 11, 2011

11 Random Musings on 11/11/11

On this "auspicious date" for the superstitious, and "marriageable date" for those tying the knot, here are eleven random numerical thoughts for this year's most memorable date.

1. When Numerology Meets Biology
A possible aftermath for this year's hot date is to expect a slight increase in the birthrate this coming August—maybe some 11 percent points up for greying nations like Singapore and Japan, where the fertility rate is currently below its replacement level.

2. The Heck of a Date
Let's celebrate, or meditate on, a dozen 1's at 11 seconds past 11:11 on November 11, 2011: 11:11:11 11/11/11

3. Murder by Numbers
Health risks for both mothers and babies, resulting from induced births or Caesarean operations, as parents-to-be long for an auspicious birthdate—a union of superstition and innumeracy!

4. A Mental Stimulation
A mental calculation for lovers on a hot date: Find the value of 111111 × 111111.

Predict the patternful answer for 11111111111 × 11111111111.

5. Timely Birth, Untimely Death
What is the probability that one born on 11/11/11 will have one's last heartbeat on 12/12/12?

6.  Born on 11.11.11  Died on 11.11.11
What are the chances that someone born on 11/11/11 (2011) will also die on 11/11/11 (2111)—when he or she is a centenarian?

7. The 11:11 Phenomenon
Are you a superstitious, or faithful, believer in the numerology of the 11:11 synchronicity?

Do you wake up every night at the same time, sometimes 11:11 p.m., sometimes 1:11 a.m., 2:22 a.m., 3:33 a.m.? 

Your minor medical expenses from a minor accident: $11,111.
Your hospital room number: 1111
Date of accident: November 11

Your mileage read 111111!

8. Special Birth-dates
What percent of children (or twins or triplets) in the world would be celebrating their 11th birthday on 11/11/11?

9. A Mathematical Quickie
The product of the ages of Mr. and Mrs. Yan, and their daughters, is 111,111. If Mr. Yan is two years older than his wife, find the sum of the ages of their children. 

10. Expressing Oneself in Ones
Using 1's only and the four operations (+, —, ×, ÷), what are some ways of expressing 111111?

For example, 111111 = 11 × 111 × (111 - 11 - 11 + 1 + 1)

11. A Pseudo-Mathematical Toughie
Mrs. Ones bought a certain number of pens for $1111.11. She sold each pen at $1.21. If one pen costs more than $1.00, how much did she earn in total?

Selected Answers
9. 18      11. $200.20

© Yan Kow Cheong, November 11, 2011.

Thursday, November 3, 2011

20 Things You Probably Didn't Know about Singapore Math

The media love to paint a positive or negative picture of life's successes and failures, and Singapore's success in mathematics education is no different. Here are some unwritten, often undesirable, factors contributing to Singapore's mathematical success.

20. About sixty percent of students know how to differentiate and integrate un-pathological functions by grades 9 and 10—they read two-year "Additional Mathematics" plus four-year "Elementary Mathematics."

19. About ninety percent of students would have had a math tutor by the time they reach grade 6—private tuition is a multi-million-dollar business in Singapore because school teachers know tutors and anxious parents would eventually fill in the gap.

18. Some sixty percent of students complete their secondary education in four years, which includes reading calculus, trigonometry, and proofs in plane geometry.

Product Details
A title that promotes
the model method

Product Details
A Singapore wallet-friendly
 grade 4 title 
a six-book series 
17. On average, most students would practice three to four assessment [supplementary] math titles every year, up to grade six, mostly purchased by parents and recommended by tutors, because local textbooks ill-prepare them for school tests and exams.

16. An estimated 60% of students in every cohort dislike math, because it’s taught in a boringly sterile manner in schools, and often by boring math teachers who are simply teaching math to the test.

15. Statistical anecdotal evidence suggests that as high as 80% grades 1-6 teachers prefer to teach other subjects to math—the painful truth is that grades 5-6 math (with their share of challenging word problems) are harder to teach than grades 7-8 math.

14. Most K-6 math teachers are non-college graduates; interestingly, they’re also known to be better math teachers than their peers armed with a university degree.

A Formula for Singapore's Math Success = 20% Textbook + 30% Teacher + 30% Tuition + 20% Parental Involvement

13. The better math teachers and tutors aren’t teaching in the top schools, but rather in neighborhood ones, with far less-ideal facilities and resources.

Product Details
A dear pseudo-monograph
about the
Singapore model method 
12. Since the 2000s, the standard of math education in Singapore has dropped significantly, due to the recruitment of non-math majors—many have a degree in Accountancy, Engineering, or Computer Science. This means that many wouldn’t have been exposed to a rigorous treatment of college math (abstract algebra, topology, or complex analysis).

11. The majority of math teachers moonlight, often compromising their day-time jobs—the better ones teach in tuition or enrichment centers, or give private tuition, often charging obscenely.

10. Up to 70% of Singapore students are probably one grade higher than their peers in the United States—for instance, a primary 2 student in a good neighborhood school in Singapore would have covered at least 60% of what a grade 3 student in the US had read, based on the textbooks’ contents from both countries.

9. Other than those few expensive-cheat math titles written by some lecturers or tuition centers' owners, most Singapore-published math textbooks and assessments are value-for-money titles vis-à-vis the expensive, thick, colorful—inch-deep, mile-wide—textbooks published in the US.

8. The power and beauty of the Singapore model (or bar) method is mostly appreciated by those outside Singapore, as compared to an unappreciative lot of local math teachers. Since the late eighties, they've been inundated with an unhealthy number of assessment (supplementary) books, aimed at promoting the visual heuristic—today, most local math teachers treat the model method as a hype or a bore.

7. Most elementary math teachers and graduates-parents have difficulty drawing a model when faced with a grade 5 0r 6 challenging word problem, preferring to use algebra instead to solve them—without peeking at the model-or bar-method solutions, most parents are unable to help their grades 5-6 children with their school homework.

6. Most math teachers feel uncomfortable or ill-prepared to coach their own students for math contests and competitions, leaving the task to trainers from private companies, or to coaches from mainland China.

Product Details
A decade-old
Singapore bestseller

5. Local mathletes are trained to answer questions that would defeat most secondary teachers, who are primarily drill-and-kill specialists employed to produce exam-smart students to outperform their peers from other Commonwealth countries.

4. Singapore-published textbooks and assessments are mostly written or ghostwritten by foreign-born authors, most of whom have never taught in primary or secondary schools, or by lecturers supervising trainee-teachers.

3. Most Singapore-published math titles are "edited" by non-Singapore citizens, who only have a smattering understanding of the local educational system.

2. An unhealthy number of school textbooks are rewritten or ghostwritten by editors for their PhD authors—many titles-conscious general editors or consultants are notoriously known to contribute quasi-zero input and to collect an undeserved royalty or lump sum payment.

1. Singapore is a haven for assessment math titles, but a hell-on-earth for mathophobics who are forced and terrorized by parents and tutors to go through hundreds of non-routine or challenging word problems, so that they'd remain ahead of the competition. The only consolation is Singapore's top ranking in TIMSS; more medals at contests and competitions, and more university places at top universities.

Indeed, Singapore firsts in math education comes with a high price and with much pain and suffering for students, teachers, tutors, parents, writers, editors, and publishers. Not to say, tens of thousands of students who feel shortchanged and alienated by the culture of mathematics challenge, resulting in poor self-esteem and a dislike for the subject!

© Yan Kow Cheong, November 2, 2011.

Tuesday, July 12, 2011

Mathematical Haikus for Kiasus

Some 17 odd hours ago, I posted "Mathematical Haikus for Goondus* and Suakus" on Facebook. I hypothesized that composing or formulating these 17-syllabled verses may help one to balance the left mathematical part with the often-atrophied right part of the brainHere's another lot of these 5-7-5-like crude non-seasonal poems.

Make Every Nanosecond Count

Three scores and ten
Roughly a billion heartbeats
Use your time wisely

How Many Misteaks Are There?

On the train platform
Train arriving in ''one mins''
That's non-SI time!

Issued by Japan in 1984 

Rightly Theirs
That kids can recall
Pythagorean Theorem
That proves their "math rights"

The Craze Is Back

Dare to sign up for
Rubik's Cube Competition?
That's raw math talent! 

A Modern-day Rubik Cube

Sudoku, again?
Think of something more worthy
To tickle your brain


Sam Loyd’s Alleged Invention

That cheap toy of yesteryear
Order it on eBay

A Math Competition for All

AMC* is best
The most popular contest
In the world today

*Australian Mathematics Competition

Where Cool Math Things Happen

MAA is cool!
The association to be
For math geeks and nerds

The World's Most Disliked Subject

Why you dislike math?
MATH is a four-letter word
A turn-off for kids

A Key to Unlocking the Universe’s Secrets

Math is a language
With notations and notions
To model the world

Some Like It With an “S”

Is it MATH or MATHS?
It all depends where you live!
Your location counts.

Product Details

The Most Quoted Verse

What's John 3:16?
God's numerical message
Of His Love for you

NUMBERS 1:1-36:13 

The Book of Numbers
Is not really a math book
But God's Almanac! 

Social Media MATH

To blog or to tweet?
It's hard to make up my mind
I choose to do both


Micromath 4 U
In 140 words
To share your sweet tweets


Five-minute math posts
On pop culture and gossip
For friends, fiends, and foes

The Chewing Gum Land

A math sanctuary
The ''fine city'' Singapore
Offers jail and cane

Triple Firsts in TIMSS

SINgapore’s success?
A haven of ''cheat/cheap'' books
© Yan Kow Cheong, 2011

To meet kiasus’ needs

Superstitiously Yours 

Friday the 13th 
An urban myth to promote 
Irrational fear

Apocalypse now

Faith in the Mayas
Dec 21, 2012
Where will you be then?

Singapore’s Papyrus

The model method
A mere fad or a cool tool?
To soothe the mind's eye

Faith or Fear in 1’s and 2’s

The Y2K scare
Next it's 12/21/12
Marketing faux fear?

© 2012 Yan Kow Cheong

Another Creative Math Title

Cre8tively yours
Who Took My Calculator?
Coming your way soon!

It's time to compose a few mathematical haikus to tickle the right part of your grey matter. And remember to share them with the rest of the mathematical brethren.

A 17-letter 
m-a-t-h-e-m-a-t-i-c-a-l H-A-I-K-U
Longs to be composed

* Suakus and Goondus are the Asian equivalents of Dummies, Idiots, Blockheads, and Morons.
# Kiasus are those who are afraid to lose out, displaying signs of self-centeredness and selfishness.

© Yan Kow Cheong, July 10, 2011