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Preschool Math Tuition Books and Toys

There seems to be an increasing demand for kindergarten Maths tuition for children of ages 2-6. (I have received many such requests lately.) Many parents are worried that their child may lose at the “starting line” which is Primary 1, hence are preparing beforehand during the ages of 2-6.

Possibly, at this age the best way to learn is through play, that is why Math Toys are very useful.

Skoolzy Rainbow Counting Bears with Matching Sorting Cups, Bear Counters and Dice Math Toddler Games 70pc Set

Skoolzy Rainbow Counting Bears Toy is suitable for age 2, 3, 4, 5 or 6 year old preschoolers to learn to add, count, sort & stack.

Goodnight, Numbers

Goodnight, Numbers is an award-winning bestselling bedtime story for kids, that help them familiarize with numbers.

Melissa & Doug Pattern Blocks and Boards – Classic Toy With 120 Solid Wood Shapes and 5 Double-Sided Panels

The Pattern Blocks and Boards toy is useful for developing pattern recognition and visualization skills; all very useful skills when it comes to the GEP Logic Section at age 9. PSLE has some very tough visualization questions on nets of cubes, only children with strong visualization skills will be able to solve them. It is notoriously hard to train for such questions (though there are some tips and techniques), it is almost like either you see it or you don’t.

Strong Visualization is needed to solve this type of questions, which appear in both GEP and PSLE. Source:

Times Tables the Fun Way Book for Kids: A Picture Method of Learning the Multiplication Facts

Times Tables would be the most challenging Math for children of the age group 3-6. Hence, early mastery of this subject will give your child a headstart to learn other more advanced math.

LeapFrog LeapStart Preschool 4-in-1 Activity Book Bundle with ABC, Shapes & Colors, Math, Animals

This 4-in-1 book bundle enables your child to learn about the alphabet, shapes, colors, math, as well as animals.

LEGO Education Set 45008 Math Train

This LEGO (Duplo) set teaches numbers, basic math skills, and includes fun activity cards. Very educational and fun toy.

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Cheap Digital Piano Singapore

Do you know that there is a decent digital piano selling for just $599 on Qoo10?

Qoo10 Link here (with other piano recommendations):

It is the Casio CDP-135 Digital Piano, with 88 Weighted Keys. It is an excellent choice for parents with children who are starting on piano. The child may show some initial interest in learning piano, but who knows, children may just give up after a few weeks. Buying a upright piano will cost at least S$3000, and if the child gives up it ends up as a “white elephant” collecting dust.

$599 is certainly affordable for most Singaporeans, and it will serve as a first piano for young children. If they have a sustained interest, after a few years an upgraded piano can be bought. An additional benefit is that digital pianos never go out of tune. With Singapore’s humid weather and young children hitting the keys, actual pianos may go out of tune frequently.

Video of the Casio CDP-130, which is an older version of Casio CDP-135. (That means the CDP-135 is even better than what is shown in the video below.) The pianist plays Fantaisie-Impromptu, which is a difficult classical piece (above Grade 8 level). Hence, playing grade 1-5 classical pieces is no problem on the Casio CDP-135.

If you are willing to fork out a little bit more cash, the Yamaha P-125 is a good catch (around $950). (Also found on my Qoo10 piano recommendations page.)

Basically, the Yamaha P-125 is the top tier of the P range, P for portable. For Yamaha, the ranking goes like this P<YDP<CLP (Clavinova). For beginners, P-125 will be more than enough. YDP is for intermediate players, and Clavinova is for musicians and for actual performance.

Finally, if your budget is around S$1800, the best digital piano is probably Korg G1 Air.

Also, read our previous post on Cheapest Digital Piano Singapore.

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Biggest Sheng Siong Outlet

Sheng Siong is a popular supermarket chain in Singapore. Many people like Sheng Siong because of its unique advantages of cheap price and wide variety of goods including exotic meat/vegetables not found in other supermarkets. Its pork section is exceptionally good and cheap compared to other supermarkets. This is apt as Sheng Siong’s CEO was formerly specializing in selling pork.

Sheng Siong outlets come in different sizes, from very big to very small. One of the biggest and largest Sheng Siong Outlet was the one at The Verge, also known as Tekka Mall, in Little India. Unfortunately, it has closed.

The biggest Sheng Siong seems to be now the one at Tampines Central, occupying 2 storeys.

How about in the West region, which is the biggest Sheng Siong? Please leave your comments below if you know.

According to Google Review,

544 Jurong West Street is “Not too bad but it’s kinda narrow and not much space to move around. Quite a number of variety considering it’s a much smaller store compared to the one at Teban Garden or West Coast”

Hence, we know that Teban Garden or West Coast Sheng Siong is bigger than 544 Jurong West Sheng Siong. I have been to Teban Garden Sheng Siong, it is not that big actually. Hence, probably the biggest Sheng Siong in the west is West Coast Sheng Siong?

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Nanyang Girls’ High School (NYGH) Cut off point

Nanyang Girls’ High School has the highest Cut-off point among all secondary schools in recent years.

For 2018 Secondary 1 admission, the cut-off was 264 (inclusive of bonus points for Higher Mother Tongue).

In the last 5 years, the cut-off point ranges from 261 to 264.


RGS Cut-off point

Its close competitor, Raffles Girls’ School (RGS), is slightly behind at 260.

The school’s cut–off point will only be made known by MOE after the results of the PSLE posting every year. We have yet to receive information from MOE on the cut-off point for our 2017 Secondary 1 students. RGS’ PSLE cut-off point last year was 260.

Source: RGS Website

Historical Cut off point of Nanyang Girls’ vs Raffles Girl’s School

The historical situation used to be reversed; RGS used to traditionally have a higher cut off point (COP) than NYGH, in the early 2000s. It is hard to find the historical data, but all students during that era will remember clearly that RGS cut off point was higher than NYGH by a few points.

Cut-off point reflects the popularity of a school, hence it can be said that NYGH is slightly more popular than RGS at the moment. Some benefits of NYGH is a strong Chinese traditional culture; while RGS is strong at leadership and all-roundedness.

Also check out the following posts related to Nanyang Girls’ High School:

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iPhone XR Price Reduction (Singapore)

Apple (US) has already started some form of price reduction for the iPhone XR, though only for trade-ins of iPhone 7 Plus. One can guess that their sales must be not as good as expected.

Though iPhone XR is indeed a good phone, the pricing of over S$1200 is absurd and unaffordable to the middle class. Note that the iPhone XR is supposed to be the cheapest model in the latest X-series. Even loyal fans of Apple are more likely to stick to older iPhone models like iPhone 7/8 than to upgrade. Other brands like Huawei and Samsung have much more affordable phones that are also decently good.

Basically, it is not sustainable for Apple to keep increasing the price significantly for every new model. Otherwise, the iPhone 20 will cost like S$3000? A corollary to Moore’s Law is that electronic devices are supposed to get better (but cheaper) as time passes. One of the first Apple Macintosh cost US$2495 in 1984, which is around US$5694 when adjusted for inflation.

When will iPhone XR start price reduction in Singapore?

This case will be good for protecting your iPhone XR if you get it after the price reduction:

Spigen Tough Armor Designed for Apple iPhone XR Case (2018) – Black

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2019 Calendar (Singapore) Printable

Singapore Calendar 2019

Quite a nice calendar template that I frequently use. From Calendar Labs.

For those looking for a more awesome and colorful calendar, do check out this “Hubble Space 2019 Calendar”. Possibly one of the most impressive calendars you can buy on the internet.

Goldistock -“Hubble Space Telescope” Eco-Friendly 2019 Large Wall Calendar – 12″ x 24″ (Open) – Thick & Sturdy Paper – Expand Your World & Your Mind

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What are the Most Scoring Topics of Mathematics in JEE Main?

With a few days left for the JEE Main Exam 2019, the stress of the candidates is at its peak. One of the cohesive and most scoring subject out of the three subjects in JEE Main is Mathematics. To prepare well for this section, you need to know the most scoring and important chapters of the subject to score good marks.


The subject which is based on calculations and formula and has the vast syllabus need lots of practice and smart strategies to score good marks. Having 16 chapters in the syllabus and everything is dependent on each other, the JEE Main Mathematics can be mastered yourself only by more practice. So, here we are discussing some of the important chapters of the subject Mathematics that you need to have at your fingertips to crack JEE Main 2019.


How to Score More Marks in JEE Main Mathematics?


After completing with your JEE Main syllabus 2019, you need to focus on the scoring topics in every section which demands little extra effort and time. Your revision days must include the clear idea of important and most scoring topics in JEE Main Mathematics subject. We have listed the scoring topics in Mathematics below for you:


The complete syllabus of Mathematics is divided into 4 sections which are:

  • Trigonometry
  • Calculus
  • Algebra
  • Coordinate Geometry


Now, before moving ahead with the important chapters, let’s study the complete syllabus of the maths. We believe that the subject Mathematics takes lesser time as compared to other subjects of JEE Main Physics and Chemistry. You can easily cover the major portion of the subject during the preparation of your boards. Check out the complete syllabus for the JEE Main 2019 for Maths Section:


Topics Units of Topics
Algebra Complex Numbers and Quadratic Equations

Matrix and Determinants

Permutations and Combinations

Mathematical Reasoning and Induction

Binomial Theorem

Sequences and Series

Statistics and Probability

Calculus Sets, Relations and Functions

Limit, Continuity and Differentiability

Applications of Derivations

Integral Calculus

Differential Equations

Coordinate Geometry Straight Lines

Circles and Conic Sections

Three-Dimensional Geometry

Vector Algebra

Trigonometry Trigonometric Identities and Equations

After the complete syllabus, we are providing here the important chapters which you need to put more focus on. The important topics which we have mentioned are based on the analysis of previous year question papers. The topics which we have started with are based on the weightage in the JEE Main Examination.


JEE Main Mathematics Topics

High Weightage Topics Low Weightage Topics
1.               3-Dimensional Geometry

2.               Vectors

3.               Straight Line, Circle

4.               Probability Distribution

5.               Determinant and Matrices

6.               Limits, Continuity and Differentiability

7.               Complex Numbers

8.               Quadratic Equation

9.               Binomial Theorem

10.   Conic Sections

1.               Trigonometric Equations

2.               Statistics

3.               Mathematical Logic

4.               Set and Relation

5.               Height and Distance

6.               Properties of Triangle

7.               Permutations and Combinations

8.               Indefinite Integral

9.               Trigonometric Ratio and Functions

Now, it’s the right time to peep into the scoring topics by checking out the number of questions asked from these topics in last years and the maximum marks these topics cover. Here is the tabulated form of the scoring topics of Mathematics in JEE Main 2019.


Also Checkout here – Top 5 jee main preparation apps


Topics Number of Questions Maximum Marks
Differential Calculus 7 28
Integral Calculus 4 16
Coordinate Geometry 3 12
Coordinate Geometry (3-D) 2 8
Matrix and Determinants 2 8
Probability 2 8
Trigonometry 2 8

Some Points to Remember for Mathematics in JEE Main 2019


  • Topics like Matrix and Determinants, Application of Derivations, Complex Numbers are simple and on the same hand scoring topics.
  • We recommend you focus on topics like Vectors and 3D, rather than Probability or Definite and Indefinite Integration. The reason being there is the marginal score for the examiner in these topics in the context of the questions.
  • Devote your time to Algebra as this chapter is scoring and easy on the same hand.
  • Pure Trigonometry questions are rarely asked in the JEE Main. This topic involves many formulas, once you mastered that you can score well.
  • The Circles chapter is the easiest chapter under the category of Analytical Geometry. If you have a good understanding of the concepts, you will be able to score well in this topic. As this chapter has no formula to mug up, only concept to clear, take no chance to skip this topic.
  • The chapter of Differential Calculus involves only one formula, so it is easy to master this topic with the sheer amount of practice.
  • If you ready to play brain game, go for the chapter Integral Calculus. Under this only one topic Definite Integrals has the minimal set of formulas, once mastered you can score excellently.
  • The chapter Vectors is the extension of the chapter Coordinate Geometry and so is easy and most scoring of all the topics. Chapters on Planes is easy and quick to solve.


Spend your time considerably in the reading theory of all the chapters and then move ahead with the practice. If you cover all the scoring topics which we have provided, we are sure you can score good marks in Mathematics in JEE Main 2019. Keep these points in your mind and start with your revision confidently.


Don’t forget to download JEE Main preparation app. It provides free study notes, quizzes, mock test, previous year papers, and all the updates related IIT JEE exams.


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Math Facebook Page

Do like and follow our Facebook page at:

It includes additional material not found on this website, including educational news and others.

On the PC version of, it is also accessible as a Facebook widget on the bottom right side of the webpage.

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Buy Enfagrow Milk (Official Store) at Discount

Everyone knows milk powder is very expensive in Singapore. Here is a good opportunity to buy Enfagrow Milk at discounted prices at the Official Enfagrow store:

Click here: Enfagrow SG Official Store Link

Buying from the Official Store is very important as it eliminates the risk of “fake milk powder”. It is well known that unscrupulous merchants (e.g. in China) sell “fake milk powder” with questionable ingredients packaged as branded milk powder.

Watch out for the dates (between 10 Dec and 15 Dec) where you can get the maximum discount.

Mechanics – Up to 14% Off* & get $30 Voucher^. T&Cs apply

Dates – 10 Dec (12pm) to 15 Dec (12pm)

Terms and conditions:

*Promotion is valid from 10-December (12pm) to 15-December (12pm)

^Voucher is valid from 16-December to 15-January 2019 and applicable only with a minimum spend of $200. Voucher will be emailed to you in your order confirmation email.

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Lion King Math

Recently the #3 trending video in YouTube is the Lion King Trailer. It would be quite amazing, to see the epic movie Lion King in “real life”.

Here is a very amusing video on Lion King and the “Math Song”:

If you want to watch the original Lion King movie, you may check it out on Amazon. Many people, including adults, rate Lion King as the best Disney movie ever produced.

The Lion King [Blu-ray]

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Singtel Voucher $100

Singtel Voucher Link: Click here to enter Official Singtel Voucher Website

As most people will know by now, Singtel is cutting down the recontract voucher. Nowadays it is hard to get Singtel recontract voucher, or even if you get one the amount is very little compared to last time.

However, now there is a new deal of a whopping $100 voucher for Singtel Combo plans. It is via referral codes and promo codes. Click on the link on the top of this post (or see the bottom of the post for more details).

Hi, heard you were looking for a new mobile phone? Here’s a Singtel mobile offer just for you. Sign up using my exclusive link below to enjoy an extra $100 OFF on selected phones when you switch to a new Singtel Combo plan.  When you’ve switched, I’ll be rewarded too!

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Tips and Tricks in Learning Math When You Know It’s Not Your Forte

Tips and Tricks in Learning Math When You Know It’s Not Your Forte

It’s a fact that not everyone is good at Math. It especially becomes a problem when you are about to take an important exam, such as an entrance exam for college or one of those daily quizzes your professor hurls at you during class. These are only some of the many reasons you should do your best in studying for your Math classes.

Here are tips and tricks in learning Math for you to get that sweet A+ on your report card:

  1. Study in a quiet place
    Math is a complex subject that needs absolute concentration. Avoid noisy places that can distract you or can hinder you from studying peacefully, although there are some ways in which you can study in a noisy environment where you cannot escape the noise. This subject needs your utmost concentration. An excellent study area is either your room or your school library, which lets you the focus in solving those tough equations.
  1. Check out online learning guides

Math has been around for a long time, and a lot of the topics in that subject has already been explored by experts before us. Luckily, we have the Internet now. You can scour the Internet for tips and tricks to learn new Math-related things. There are websites like that publish video guides to help you with your homework.

  1. Understand the processes
    Math is one of those subjects that you shouldn’t memorize. To excel in Math, you need to have a very firm understanding of mathematical concepts involved. It is recommended to understand everything, one topic at a time so you will a good grip on the subject.
  2. Study ahead
    In Math, it always pays off to read ahead of the scheduled lesson. Having a prior understanding of a certain topic can be an advantage to you. If you do advanced learning then in class you only need to clarify what you did not understand when you studied in advanced.
  3. Learn to use shortcuts
    Yes, you heard it. Math has shortcuts, too, and learning these shortcuts can help you a lot. There are a lot of simple things you can learn to speed up your calculations.


Here’s an example.
Add up large numbers in your head.
You are at a department store and you don’t have a calculator handy. You need to add up the cost of two items, for example, 384 + 297. Here’s a tip do this in your head quickly:

  • Round off the two numbers. You now have 380 instead of 384 and 297 becomes 300. If you add those up, the answer is 680.
  • Add up the numbers you set aside while you rounded off the two large numbers, +4 and -3. As a result, you have +1.
  • Sum this up with the 680 and you now have 681. You’re welcome!
  1. Enroll in peer-to-peer coaching
    Some schools initiate peer-to-peer coaching for specific subjects such as Trigonometry and Algebra. These programs allow students to review lessons, learn new things, and improve their understanding of a subject. This is beneficial to both sides as it is a form of a review for the mentor and a learning experience for the mentee.
  2. Have fun while learning!
    You have to have a mindset that Math is not a boring topic. If you have a computer, there are a lot of Math Games that you can find on the Internet. Playing Math games is a very productive way of learning while having fun. There are games made for both kids and adults.
  3. Practice, practice, and practice
    There is no quick way to learn something. Although some tips and tricks help make the journey easier. One good idea is to practice solving Math problems with a time limit. This way, your mind will be accustomed to the time pressure that you will feel during an exam. Always remember, nobody ever had it easy.Today, learning Math has never been easier for anyone with a classroom and a computer or even mobile device. You just have to start taking advantage of the avenues provided by the Internet today combined with the tricks mentioned prior. You can excel in Math with those tools, especially with the guidance of a good teacher.

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Chinese Primary School Math

Math Online Tom Circle

As shown in the picture below, find the height of the table?

(No Algebraic equation please for Primary School Math).

Answer: [hint = 130 cm]

Try yourself before scrolling down.

. .


The 2 sleeping cats and 2 sitting cats cancel out, left 260 (= 110+150) for the 2x Table Heights.

1 Table Height = 260/2 = 130 cm.

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How you pronounce German Mathematician’s Names ( and Physicists )

Math Online Tom Circle

Germany produced many Mathematicians and Physicists who succeeded the 18 CE Newtonian England, and the 19CE Napoleonic France before WW2, after which the Americans (mostly the Jewish German immigrants) take over till now.

German names are difficult to pronounce for foreigners.










For due respect, please learn to pronounce their German name correctly.

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Euler’s and Fermat’s last theorems, the Simpsons and CDC6600

Math Online Tom Circle

I am a fan of Fermat, not only because my university Alma Mater was in his hometown Toulouse (France) named after him “Lycée Pierre de Fermat (Classe Préparatoire Aux Grandes Ecoles) ” , but also the “Fermat’s Last Theorem” (FLT) has fascinated for 350 years all great Mathematicians including Euler, Gauss,… until 1993 finally proved by the Cambridge Professor Andrew Wiles. Another “Fermat’s Little Theorem” is applied in computer Cryptography .

Below is the explanation of (n = 4) case proved by Fermat and the latest proof by contradiction.

Euler Conjecture: extends FLT to 4 or more integers if FLT still holds? (a contradiction found).

Simpsons “Fool” Equality: Proof by contradiction (odd = even)

Proof of FLT by Andrew Wiles (1993):

The proof by Contradiction of FLT (n=4) is in Part 2 of the video after 20:30 mins (Warning: a bit heavy)

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Improper Integral vs. Infinite Series

Math Online Tom Circle

$latex displaystyle int_{2}^{infty}{frac{1}{x^{2} – 1}dx} =boxed{ ln {3}}$

$latex displaystyle sum_{n=2}^{infty}{frac{1}{n^{2} – 1}} = boxed{frac{3 } {2 }} $

To test if the infinite series
$latex displaystyle sum_{n=2}^{infty}{frac{1}{n^{2} – 1}} $ converges, provided it is

  • positive &
  • decreasing function

you can convert it to $latex displaystyle int_{2}^{infty}{frac{1}{x^{2} – 1}dx} $
(although they converge to different values.)

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Learning with Passion and Enthusiasm (Motivational Video)

The best way to learn is to learn with passion and enthusiasm. See the excellent motivational video below!

Without passion, learning becomes a chore like doing housework, and it becomes boring and tiring. Needless to say, such learning is not only burdensome but also ineffective and inefficient.

Sometimes kids who are smart don’t do well in school, because they don’t have the motivation or enthusiasm to learn. Check out our blog post on: Motivational Books for the Student (Educational).

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Michael De La De Maza Chess Book

Chess Books for Adults

This book is quite famous, or some may say “infamous”, in the circle of Chess books. The author is an adult player who managed to gain roughly 400 rating points in 400 days. To be precise, he gained 400 USCF rating points in the first year, and almost 300 USCF rating points in the second year, finally ending up with a rating of 2041. His original rating was 1321.

This is quite a meteoric rise for an adult. It is well documented that for many reasons, adults tend to improve slower than children when learning chess. Similarly, for music, languages, etc. Firstly, children’s brains are more “plastic” and flexible, and they have more time and less outside commitments to study chess.

The basic principles in his book seems to be to study tactics, which are “combo moves” in chess that can change the course of the game. Making a tactical mistake can spell the end of the game for either player.

Rapid Chess Improvement (Everyman Chess)

The opposite of tactics is strategy. Strategic masters are Anatoly Karpov, Tigran Petrosian, and the current World Champion Magnus Carlsen. They play in a calm and controlled manner and accumulate a small advantage. Strategic mistakes tend not to lose immediately, but may lead to a slow and steady deterioration of the position.

The best author on strategic chess (for beginning to intermediate players) is probably Jeremy Silman. His book “The Amateur’s Mind” is a classic book:

The Amateur’s Mind: Turning Chess Misconceptions into Chess Mastery

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Christmas Present for Gifted Kids

This microscope set will be ideal for gifted children who are interested in biology, or science in general. Learning about the microscope is the first step towards the study of life sciences/medicine/biological sciences.

This set is made and endorsed by National Geographic, a popular science magazine in the US.

National Geographic Dual LED Student Microscope – 50+ Pc Science Kit Includes Set of 10 Prepared Biological & 10 Blank Slides, Lab Shrimp Experiment, 10X-25X Optical Glass Lenses & More! (Silver)

See also: Recommended Educational Toys

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H2 Math Bell Curve

As mentioned in H2 Maths Distinction Rate (Percentage of As), the national distinction rate for H2 Math is usually around 50%. That is, 50% of all students will get ‘A’ grade for H2 Math. (Note that this statistic is heavily skewed by the top tier JCs where close to 80%-90% of all students get ‘A’ for H2 Math.)

According to numerous online sources, such as Reddit, this year’s (2018) H2 Math paper was significantly harder than previous years. It featured many “out-of-the-box” questions that are not found in the Ten-Year-Series. The “bug” question featuring a “bug” walking in a zig-zag fashion and falling into a “blackhole” stumped many students. Certain topics, such as PnC (Permutations and Combinations) are totally left out, prompting students to say that it is a case of “PnC —> “high investment Low returns “ More like “high investment NO returns”. (Update: The “bug” question is a disguised PnC question.)

Despite all these, there are still many students who honestly think that the paper was doable, or even easy. Probably these are the students who have practiced Prelim papers of top schools like Hwa Chong or Raffles, and hence are used to such tough questions.

The usual consensus is that a high 70s mark (>75) is sufficient to get A for H2 Math. Low 70s (70-74) is quite risky and may either be ‘B’ or ‘A’ grade depending on the difficulty of the paper.

Someone actually created a poll ( for the H2 Math 2018 scores. It is quite possible to predict your own score accurately since the answers will be circulating online soon after the exam. Based on the poll, ignoring the typo error that is “75-89” is meant to be “75-79”, we can see that the cut-off for ‘A’ grade for 2018 should be in the range 70-74. This is based on the hypothesis that the 50% distinction rate holds.

The bellcurve for A levels is apparently set by SEAB (Singapore Examinations and Assessment Board), while Cambridge does the marking. According to an “old examiner report“, Cambridge likened grading to swimming a lap. They said that a candidate who could ‘compete a lap’ would be given due credit regardless of the number of people who also completed the lap. After Cambridge finishes marking and gives the numerical score, the bell curving will be processed by SEAB to determine the final grade, that is, ‘A’, ‘B’, ‘C,’ and so on. This is probably necessary to ensure a “consistent” grade distribution so that university admissions can go smoothly, and that grades from different years are roughly comparable.

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Michael Jackson Math Songs

Quite a catchy song, and teaches an important concept. Check out the original song “Beat it” by Michael Jackson:

Also, see this multiplication table song set to the bass of “Billie Jean”:

Actual Billie Jean:

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Alexa Ranking: Reaching 1 million

Thanks to all readers, is reaching 1 million in global Alexa Rank. (Currently 1,093,528 global rank). Our next target will be to reach and maintain below 1 million global Alexa Rank! All our traffic is organic traffic, from search engines.

Alexa Traffic Rank is a measure of popularity of a website. The most popular website, in this case, will have a Alexa rank of 1. The second most popular,, will have a rank of 2, and so on.

Chrome users should install the free Alexa Chrome Extension, it is very good to check the traffic rank of your website. Also, when you visit your own website, it will be counted by Alexa, and your Alexa ranking will be improved. From my experience, it is quite fast, it does not slow down your internet browser at all.

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Boric Acid Singapore

In Singapore, cockroaches are impossible to completely eradicate due to the tropical climate which is ideal for cockroaches. In many other countries, a common method to kill cockroaches is using Boric Acid. Boric acid is considered a natural substance, and is effective at killing cockroaches. It works as a “stomach poison affecting the insects’ metabolism, and the dry powder is abrasive to the insects’ exoskeletons”. As such, it is impossible to develop a “resistance” to boric acid, unlike for insecticides where numerous cockroaches are now immune to insecticides.

Boric acid is also considered non-toxic, it is similar to table salt in its toxicity level. Boric acid has a “secondary kill” effect: cockroaches who die will be eaten by other cockroaches, leading to a chain reaction of cockroach deaths. Hence, it is more effective than sprays or traps which only kill the cockroach that is targeted.

However, when one starts to look for boric acid in Singapore, it is quite hard to find it in the pure powder form. Whether it is banned, or is simply not popular, it seems that no one on the internet knows.

I found a cockroach bait containing Boric Acid in Qoo10 below. It is a Malaysian brand. It is written on the packaging that boric acid is the active ingredient, and I also double confirmed with their Facebook customer service. Here it is:


[$2.60](▼43%)Pesso Eco Cockroach Bait


Cockroach from HDB Rubbish Chute

Another problem about old HDB estates is the interior rubbish chute. Though very convenient, it is also a direct pathway for cockroaches to enter your home. The solution is to install a new air-tight rubbish chute hopper such that the cockroaches cannot pass through.

[$175.00](▼42%)2018 Stainless Steel Rubbish Chute Hopper Installation 1 year warranty.


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My First Try On Nasi Lemak

Cooking For Fun

Nasi Lemak is a Malay dish. When I first came to Singapore, I tried it. But didn’t have any deep impressions on it. The specialty of it lies in rice, which is cooked with coconut milk. But when I had it before, rice was cold, so I couldn’t taste the fragrance of coconut milk.

The other day, I tried cooking rice with coconut milk, the rice turned out very nice. So I bought pandan leaves, lemon grass, sambal chili sauce and decided to cook my first nasi lemak.

Rice part:

1 cup rice ( I used the ones with some brown rice mixed in)

70 ml coconut milk

1 1/2 cup water

1 stalk pandan leave (make into a knot)

1 stalk lemon grass (pat lightly with knife)

1/2 tsp salt

Put all the ingredients above into rice cooker and press the cooking bottom. Trust me your kitchen will smell…

View original post 307 more words

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Checkmate Black King on 1st rank (Sicilian Defense, O’Kelly Variation)


I try to play Sicilian in the “Delayed Alapin” style with c3 followed by d4. The black player, though lower rated, is quite an experienced player with over 11,000 rapid games played on Lichess.

The turning point was 14. d5! I did not fully calculate all the lines, but I thought it was worth giving up the isolated d-pawn for a very active position, with my rook lined up against the black queen. The tactics worked out well in the end, in fact it turns out that white can always regain the pawn.

Lichess seems to have rating deflation: Many players (including myself) who were once rated over 2000 for rapid, are now dropping back to 1900 rating range. It seems due to an influx of good players into Lichess. Also see: Lichess to FIDE Elo Rating Conversion.

Final Position:

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Ohm Electricity Review

Do check out our Ohm Electricity Review at: Ohm $20 Promo Code: OHMREF25CC5A.

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Ohm Website “Flickering” or Refreshing?

Official Solution: Hi, the cause is most likely due to a cache and time-out issue. You could clear the cookies and cache, close off all the browsers and open that up again. Alternatively, you can also use a private “incognito browser” if you are using Chrome as your browser. Let us know if you still face issues and we will be happy to assist. Thank you, Team Ohm.

Ohm Energy: Free second electricity bill

Dear customer,

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You will also find more information on the Terms and Conditions of this Promotion (OhmNow and Save) on our website. If you have any questions or concerns, please contact our friendly Customer Care at 1800-835-6463 or for further assistance.


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Meat Ball With Running Egg

Cooking For Fun

When I made wantons in the morning, some filling was left. And I didn’t have any plan to make more wantons, so I used those left minced chicken filling to cook this dish. I know there is a Scotch Egg. But I kind of cook it in a different way, especially the meat part. So I just called it “Meat Ball With Running Egg”. 🙂

Original wanton filling is : 200g minced chicken, 5 chopped Shiitake mushroom, 4 chopped water chestnuts, 1 tablespoon light soy sauce, 2 tsp oyster sauce, 1 tsp salt, 1 tsp black pepper, 1/2 tsp minced ginger.

There was about 1/4 left. To add more flavor to the meat part, I also add in 2 tablespoon shredded mozzarella cheese, and 1 tablespoon chopped spring onion.

For the running egg. Boil some water ( doesn’t need to cover eggs, because eggs can be steamed too). When water…

View original post 210 more words

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What to do if fail O level Singapore

Recently, there is positive news that polytechnic students no longer need to “depend” on their O level results to enter university.

Most students will aim for JC (Junior College) or Poly (Polytechnic) admission after their O levels. Outright failure (F9) is quite rare for O levels, but students need to be careful as too many borderline passes like D7/C6 can lead to a score that is technically a pass, but not enough to enter JC or Poly. (See: Academic grading in Singapore: How many marks to get A in Maths for PSLE, O Levels, A Levels)

What to do if do badly for O level Singapore for key subjects

Method 1: Write Appeal Letters

Certain subjects (especially Mathematics) are more important to pass than others. According to this post on Quora, failing Mathematics can mean “auto rejection from all poly courses”.

Failing English is also quite serious as Poly seems to require D7 and above for English. This may be a problem for foreign students. There are indeed students who ace all subjects except English, they qualify for JC but not polytechnic! (See this post on Kiasuparents.)

The solution is: write appeal letters. Polytechnics do have some room for discretionary admission for borderline cases. So do try this option and do not give up.

“So off I went to most of the polytechnics to talk to their lecturers and submit my appeal. SP I did not go because I felt they were too good for me. Ngee Ann Poly I did not submit my appeal out of silly pride because the lecturer I talked to mocked & laughed at me. But for the rest, the lecturers I talked to were pretty kind even though they honestly tell me my chances are low. So I wrote my appeal letters talking about myself, my journey, what I did wrong, what I could have done & why I am suitable for the course I am applying.

RP was the first to accept me for their IT course and made me go through some programme first. Temasek come later wanting to interview me but I declined as I accepted RP offer already. NYP rejected me without an interview.”


Method 2: Alternative Route of ITE -> Poly -> Degree

This alternative route is also possible. The journey is not easy though, as this Quora post mentions:

“So I took my Higher Nitec in Information Technology in 2004. The only reason why I took Information Technology was because I was good with computers. I did HTML coding in secondary school and I was good at Photoshop. That was a great advantage. For me, Higher Nitec was much easier than O Levels. I easily scored a high GPA in 2006 and managed to get a place in Nanyang Polytechnic where I continued studying Information Technology.

To be honest, polytechnic was far more challenging but I was determined to prove my parents wrong. I continued my full-time studies while working part-time as a barista so that I can pay my school fees and daily expenses. It wasn’t easy and some days I did skip classes because I was burnt out from work and school. I did fairly good, scored a 2.6 GPA in 2009 and went on to serve my 2-year mandatory national service in the military.

In 2011, after finishing my national service, I was employed in an institution as an IT support. In 2012, I had plans to take up part-time degree. So I moved on in 2013, got a better paying job, took my part-time BSc in Cyber Forensics and Information Security Management and graduated. Working full-time and taking a part-time degree has got to be the most taxing feeling I have ever had. I had to juggle work with studies. There were school projects and essays. It was challenging but I made it.”


Method 3: O Level Private Candidate

Another possible route is retaking O levels as a private candidate. This route is also not easy as it involves discipline in studying by oneself. Possibly a better option is to enroll in private schools, or hire private tutors for those weaker subjects.

Method 4: Go Overseas

This method is unfortunately only for those who are well off (a.k.a. rich). The Singaporean O Level is the hardest version of the O Levels (compared to the UK or the Hongkong papers). Hence, chances are high that even a relatively academically weak Singaporean student can do quite well in overseas education systems like Australia, UK, US, etc.


Do read our blog on: Inspirational story: From EM3 and Normal (Technical) to PhD. Also very motivational is: From PSLE 124 to PhD A*Star Researcher. Nowadays, having a failure in the early section of education is not the end. There are different pathways to success, though some are longer than others. As Robert Frost wrote, “Two roads diverged in a wood, and I— I took the one less traveled by, And that has made all the difference.”

Also, read our highly popular posts on:

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Good news for Poly students: O Level “not counted” for Uni admissions

Should be good news for most Poly students. It is good to aim for a decent grade for key subjects like E maths and A maths though, as they may still be required for certain subjects in university. For example, a polytechnic student who wishes to apply to a computer science course at NUS would need at least a B3 in O-Level Additional Mathematics.

Source: Channel News Asia

SINGAPORE: Polytechnic graduates applying to the National University of Singapore (NUS) and the Nanyang Technological University (NTU) from 2020 will no longer have their O-Level grades be considered for admission, except for certain courses.

This comes with the Ministry of Education (MOE) removing an existing requirement for an applicant’s O-Level results to comprise 20 per cent of his University Admissions Score (UAS).

The UAS is a framework adopted by the six autonomous universities to benchmark and rank applicants’ academic abilities for admission to university.

Posted in a math tuition, nus cut off points | Tagged , , | 1 Comment

Sweden: From Safest to Most Dangerous Country in Europe

From young, through news and media my impression of Sweden is that it is a truly 1st world country on par with Switzerland, Norway, Denmark, etc. Also, my impression is that it used to be very safe, possibly one of the safest countries in the world.

But recently, Sweden seems to have risen to the rank of one of the most dangerous country in Europe, and some say the world! It is truly saddening and unbelievable. (I started to research on Sweden as I was searching for some jobs that had overseas travel requirements.)

From the way the online news media are portraying it, Sweden is very dangerous, almost on the level of Iraq/Iran/Afghanistan. I will definitely not dare to visit, let alone work there in the future.


“Mass immigration, mass raping mass crimes evil place to live was great now it’s a hell hole”

“STOCKHOLM — Sweden may be known for its popular music, IKEA and a generous welfare state. It is also increasingly associated with a rising number of Islamic State recruits, bombings and hand grenade attacks.”

The claim: Many young male migrants arrived in Sweden over the past few years, when the country accepted unprecedented numbers of refugees, and there has been a huge rise in sexual crime in Sweden especially in the southern port city of Malmo.”

“The number of explosions caused by hand grenades has increased in Sweden in recent years. There were fewer than five in 2014 but at least 20 in 2017, and a further 39 grenades were seized by police.”

YouTube video: Some people have started wearing bulletproof vests in Sweden.

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Motivational: Bear cub climbs ice mountain to reunite with mama bear

One of the most motivational nature videos of the year!

If you liked this video, you should also like this Iguana vs Snakes video:

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Useful Things to Know About Science Centre Singapore Before Visiting

Singapore has a reputation for not only being one of the most toured countries in Asia, but also for being a perfect place to incorporate a business. This is strongly attributed to its excellent tourist spots and also regulations that can allow even a foreigner to set up a business in Singapore. If you are a foreigner, you just need a Visa Express entrepass to start a new business or to relocate an existing one to Singapore. If you want to take a vacation in the next few months, then Singapore is the perfect place for you to go. If you are traveling with your children, you can rest assured that the kids will have an unforgettable experience from all the fun they will have. One of the places you can visit is Science Centre Singapore, situated at Jurong East, which is a museum full of exhibitions and learning opportunities for children. If you want to visit this place, here are some useful things you need to know.

1. Free entry for Singaporeans and permanent residents on specific days
When is the right time for you to visit Science Centre Singapore? If you are a Singaporean or a permanent resident, it is recommended that you visit during the off-peak days when you are not required to pay any entry fee. The off-peak days are usually normal weekdays during school time, excluding all public holidays. School holidays, public holidays, and weekends are usually the peak days, and visiting at these times means you will have to pay an entry fee. However, if you are a Singaporean or a permanent resident, you will still get a discounted entry fee during the peak days.

2. Extra admission charges
There are special exhibitions in Science Centre Singapore located at the Annex, which require extra fees if you want to view them. Science Centre Singapore also has areas like the Butterflies Up-Close Exhibition, which likewise demands additional entry fees.

3. Closed dates
Science Centre Singapore is open almost year-round, but it is closed on certain dates each month of the year. While most of these closure dates usually fall on Mondays (but not every Monday), it is recommended that you visit Science Centre Singapore’s website in order to get a full schedule outlining the closure dates. This is to make sure that you avoid making a trip there when it is closed.

4. Science shows
For the entire day, Science Centre Singapore conducts engaging science shows for its visitors. These shows include the dramatic and electrifying Tesla Oil Demonstration, as well as the Fire Tornado Demonstration. Just like with the closure dates, you can check out their website to find out the times for the shows.

5. Where to eat
You may be planning to spend a full day at Science Centre Singapore, and it is thus convenient to know where you can have your meals. The most convenient joints are the Coffee Bee, located at the Annex, or McDonald’s, situated at the main entrance. For additional dining options, you can go to the Omni Theatre Building, where you will find the Lijiang Restaurant and a kids’ cafe.

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How to Change UOB Card PIN Number

How to change UOB ATM PIN

It is recommended to change your UOB PIN number/password immediately upon receipt of the mailer. However, it is not very clear how exactly to change your UOB Debit Card PIN number.

The answer is that it is not possible to change the UOB PIN number online. (I called the UOB customer service at 1800 222 2121 to find this out.) Changing the UOB PIN Number is only possible at a UOB ATM machine.

How to change UOB Phone Banking PIN

Same as above, UOB Phone Banking PIN cannot be changed online. It can be changed at any UOB ATM machine.

The Phone Banking “Username”, or “Access Code” as it is called, is your NRIC number (numerical digits only, without “S” or the final letter).

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Factorize 2^n-1 (when n is composite)

How to Factorize 2^n-1

If n is composite, we may write n=ab where a,b>1.


\begin{aligned}    2^n-1&=2^{ab}-1\\    &=(2^a-1)(1+2^a+2^{2a}+2^{3a}+\dots+2^{(b-1)a}).    \end{aligned}

The key point is that both factors are greater than 1.


If a=2, b=5, we can see that 2^{10}-1=1023=(2^2-1)(1+2^2+2^4+2^6+2^8).

How to Remember this Result

The hard part is remembering this identity: \boxed{2^{ab}-1=(2^a-1)(1+2^a+2^{2a}+2^{3a}+\dots+2^{(b-1)a})}. It is essentially a “telescoping sum”, and you can prove it by expanding the right hand side and cancelling all like terms.

We can also view it as a Geometric Series as follows: \dfrac{1((2^a)^b-1)}{2^a-1} is the sum of a Geometric Progression with first term 1, and common ratio 2^a, for a total of b terms. That is,

\dfrac{1((2^a)^b-1)}{2^a-1}=1+2^a+2^{2a}+\dots+2^{(b-1)a}. Multiplying both sides by 2^a-1 gives the boxed result.


Primes of the form 2^n-1 are known as Mersenne Primes. Using the result above, we can show that if 2^n-1 is a prime, then n is a prime. (We prove the contrapositive: If n is composite, then we show that 2^n-1 is also composite.)

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Ohm $20 Promo Code: OHMREF25CC5A

*Ohm Promo Code: OHMREF25CC5A

After extensive research, I find that Ohm is one of the best alternative electricity suppliers. Choose either Fixed Ohm or Discount Ohm depending on your likings. I chose Discount Ohm as I have a feeling that the tariffs may drop. Fixed Ohm is also very good, and in fact better if tariffs rise.

Benefits of Ohm:

  1. Their discount off tariff rate is currently the best at 22%.
  2. Very smooth transition as they are partners with SP group. For instance, there is no need for any paper work nor additional payment of deposit.
  3. Very nicely done website. Signing up is immediate, just fill in some personal details online and you will get the Welcome Email in less than 15 minutes.
  4. Ohm absorbs the transmission loss. (Basically electrical energy is lost as heat while travelling through the cable, Ohm absorbs this and charges consumer based on metered usage. This is worth approximately 3.4% savings.)
  5. 3-month trial period. If within 3 months you decide to change plans, there is no penalty. Hence, you are not locked in to a fixed price which may turn out to be undesirable.

Use my Ohm Referral Code to get $20 off: OHMREF25CC5A

Fixed Rate vs Discount off Tariff

Currently (1st November 2018), Ohm fixed rate is 18.14 cents/kWh (incl. GST) while the Discount off Tariff (22%) is effectively 20.14 cents/kWh (incl. GST).

However, it is not impossible for the SP Tariff to drop (see graph below). If the SP Tariff drops, say to 20.20 (excl. GST) which is the Jan 2017 price, then 22% Discount off Tariff would be effectively 16.86 cents/kWh (incl. GST) which beats the fixed rate! (Working: 20.20 x 1.07 x 0.78=16.86)

Nobody knows for sure whether the SP Tariff will drop as it depends on a lot of complex factors including oil price, etc. However, currently with 12 electricity providers in a small country like Singapore, there is a great deal of competition, and SP is likely to lower its price to remain competitive.

Historical SP Tariff (excluding GST). Source: SP Group

Conclusion: Choose Ohm

Everyone should change a supplier in the open electricity market for savings. There are no drawbacks of doing so. In particular, there is no such worry of “blackouts” or “unreliable electricity supply” as SP group is still operating the national power grid.

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Scientists Use Math To Discover Why BTS Jin Is The Most Handsome

Recently, the hottest topic in Singapore is “BTS concert”. BTS is a Korean band that is extremely popular among the young students in Singapore. The BTS tickets in Singapore are completely sold out in less than 4 hours. (Possibly there are many people “scalping” tickets for resale, also known as “Yellow Cow Tickets” 黄牛票.)

Fans of BTS are known as “The Army”.

According to this website, Korean scientists have “mathematically” proven why Jin, a member of BTS, is the most handsome of all. His nickname is “worldwide handsome”.

Essentially, the idea is using the Golden Ratio, a special number that appears frequently in nature. See also our previous posts:

Source: Koreaboo

A plastic surgeon has reported that he studied 269 Asian male faces and concluded that BTS’s Jin is scientifically the most proportional and handsome Asian face.

The doctor mentioned 1:1.618 as the “golden ratio” for having a perfectly proportional face. Most often, when applied to facial symmetry, this ratio uses the measures of the width from one ear to the other (1) and for the hairline to the chin (1.618)… And Jin’s face exactly measures up.

Many of BTS videos have hundreds of millions of views. This is one of them:

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The Hardest H3 Math Question (Combinatorics)

I think this may be one of the hardest H3 Math Questions in history. It is taken from RI H3 Prelim 2018. It seems that even in top schools like RI, there are less than 50 people taking H3 Maths in any given year. Part (d) is extremely hard to get the formula for general r. In fact during the exam it is probably wise to skip such questions or give partial answers (e.g. the formula for r=3) as it is not worth the time for 3 marks.

See also our related blog posts:

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Test your Circuit Breaker today; Circuit breaker can save you from electrocution

Do forward this to your family and friends. Especially for older homes, the circuit breaker could have broken down silently without anyone noticing.


Pressing a button at home regularly could save your life.

This is the “test” button, typically denoted by a “T” on the circuit breaker, a safety device found in all households and buildings.

If you press that button and all electricity cuts off, that means it is functioning. A working circuit breaker will cut off electricity in the event of a current leakage, like when someone touches a live wire, minimising the chances of fatal electrocution.

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New York Professor: Blockchain is a lie, and the least useful technology

The most famous application of Blockchain is “Bitcoin”. Whoever bought just $1000 worth of Bitcoin around year 2010 would be a millionaire now. However, other than “Bitcoin” and cryptocurrencies, blockchain technology do have other uses. So is blockchain useful or not? Only time will tell.

See also our previous posts:

Source: Channel News Asia

Blockchain has been heralded as a potential panacea for everything from poverty and famine to cancer. In fact, it is the most overhyped – and least useful – technology in human history.

In practice, blockchain is nothing more than a glorified spreadsheet. But it has also become the byword for a libertarian ideology that treats all governments, central banks, traditional financial institutions, and real-world currencies as evil concentrations of power that must be destroyed.

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Integration by parts, DI method

This DI method is American way, better than the UK Cambridge O level technique.

Math Online Tom Circle

(Traditional Method learnt at O-Level) Integration by Parts:

NEW Technique taught by this UC-Berkley Chinese Mathematician: “DI” Method

First Stop: “D column at…

2nd Stop:

3rd Stop:

$latex displaystyle int {e^{x}}sin {x} dx$

View original post 1 more word

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The Best Movies For Mathematicians

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Analysis by Timothy Gowers

Math Online Tom Circle

Why easy analysis problems are easy
by Timothy Gowers (UK, Fields Medal 1998)

Timothy Gowers is teaching in Cambridge, he wrote the thick volume of “Princeton Math Encyclopedia.”

He is a very good mathematician, who likes to explain simple fundamental Math questions (like why 2+2=4, multiplication is commutative,…), in the process making abstract math simple to understand.

If you have recently met epsilons and deltas for the first time, then you may find the problems you are asked to solve on examples sheets very hard. On the other hand, you will notice that your lecturers, supervisors etc. do not find them hard at all. Why is this? ” Read on …

Below is my attempt to rewrite the Example 1 with Latex epsilon-delta notation for easy reading.

Example 1.

I wish to prove that the sequence (1,0,1,0,1,0,…) does not converge.

$latex text{Let me set the sequence }

View original post 174 more words

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Book of Wisdom

1 Wis 7:7-11 (Book of Wisdom or “Wisdom of Solomon”)

I prayed, and prudence was given me; I pleaded, and the spirit of wisdom came to me.

I preferred her to scepter and throne, and deemed riches nothing in comparison with her, nor did I liken any priceless gem to her; because all gold, in view of her, is a little sand, and before her, silver is to be accounted mire.

mire – deep mud

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No Exams Singapore

The latest policy update is that there will be reduced or no exams for certain levels such as lower primary and lower secondary.

Do check out our Poll writeup for more details: Poll: Opinion on Fewer exams for Primary and Lower Secondary Students?

It seems that it is quite a dilemma; parents find exams stressful, yet exams are viewed as necessary by most parents to measure how their child is learning. According to Today Online, tuition centres have great success offering their own “in-house exams” to fill in the gap. At the end of the day, the “big exams” like PSLE, O-Levels, A-Levels are still present, and that is the major feature in the education landscape.

It is quite like “lump sum payment” (single exam at end of the year) versus “payment by installments” (multiple exams throughout the year). In the end, the full sum still has to be paid (students still have to study the material to pass the final exam).

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Grabfood $8 Promo Code

It seems Grabfood is aggressively fighting for market share, giving very generous $8 Promo Code currently: 8TASTY.

New users of Grabfood can use the Promo Code 8TASTY twice. The current delivery charges are $3-$5 depending on surge. That means you can get a $3-$5 free meal delivered for free to your home. E.g. Bubble Tea or some McDonald’s food.

There is also another Grabfood Promo Code (also $8): CHILLOUT

The “CHILLOUT” Promo code is limited to a one-time use per eater and is only valid from 13 to 14 October 2018.

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Inequalities : RMS, AM, GM, HM

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Integral of x^x from 0 to 1

Math Online Tom Circle

$latex boxed {displaystyleint_{0}^{1}{x^{x}dx }= frac {1}{1^1} – frac {1}{2^2} +frac {1}{3^3} – frac {1}{4^4} + frac {1}{5^5} +…}$

$latex boxed {displaystyleint_{0}^{1}{x^{-x}dx }= frac {1}{1^1} +frac {1}{2^2} +frac {1}{3^3} + frac {1}{4^4} + frac {1}{5^5} + …}$

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10 cents Mr Bean Soya Milk (with Masterpass @ Qoo10)

A small reward for yourself/ your children this October? Coupled with the ongoing Masterpass promotion ($1 off) at Qoo10, and this Mr Bean promotion, the soya milk ends up costing only 10 cents!

This is probably the cheapest drink in Singapore currently! (other than NEWater 🙂 )


[$1.10](▼48%)[Mr Bean][ 10.10 Special ] Grass Jelly Soya Milk or Classic Soya Milk – $1.10 only!


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H3 Mathematics Resource Page

H3 Mathematics is the pinnacle of the Junior College Mathematics syllabus in Singapore. It contains a glimpse of actual Math that Mathematicians do, and it requires true mathematical understanding and technique to do well. (H1/H2 math requires a lot of practice, but not true understanding. It is quite common for students to “apply the method” and get the correct answer without having any idea of what they are actually doing.)

Topics in H3 Mathematics include Functions, Sequence and Series, Combinatorics, and even Number Theory. Certain schools also include topics like Linear Algebra and Differential Equations. Certainly, the H3 Math questions have a Math Olympiad style to them.

Here are some practice questions for H3 Math (more will be added in the future), with some hints. Questions are adapted from actual H3 prelim papers.


Q1) The function f is such that f(x+2)=af(x+1)-f(x), for all real x and some constant a.

(i) In the case that f is a linear function, find all possibilities for f and a.

(ii) In the case that f(0)=0 and |f(1)|=1, use mathematical induction to prove that f(n-1)f(n+1)+1=[f(n)]^2 for all positive integers n.

(iii) In the case that a=2, sketch one possibility for f which is not linear.


(i) Write f(x)=mx+c and substitute it into the question. You should reach two cases, a=2 or m=0. For the case a=2, there is no other restriction. For the case m=0, either a=2 or c=0.

(iii) Try a step function.

Sequence and Series


Q1) Mr H uses a software to generate distinct codes of the form \overline{x_1x_2x_3x_4x_5}, where x_i\in\{0,1,2,3,4,5\}. What is the least number of codes Mr H should generate such that there are at least two distinct codes that satisfy: \boxed{x_1+x_3=x_2+x_4}. [3 marks]

Hint: No, the answer is not 12. Draw a possibility diagram for x_1+x_3, essentially a table with 6×6=36 entries detailing what combinations are there for x_1+x_3. Then list cases (a popular technique for combinatorics), based on each value of x_1+x_3. For example, the case x_1+x_3=0 only has 1 option namely x_1=x_3=0. Then x_2+x_4 can have 35 other options such that x_2+x_4 is different from x_1+x_3. The last digit x_5 has no restrictions so there are 6 options. Hence the total number of ways for this case is 1x35x6=210. Finally, after adding up all the cases, use pigeonhole principle (add one) to conclude the answer, which is 6901.

Q2) See The Hardest H3 Math Question (Combinatorics)

Number Theory

Q1) Let m, n, M and N be positive integers. Given that gcd(M,N)=1 and mN^2=nM^2, use Mathematical Induction to show that M^2\mid m and N^2\mid n. [5 marks]

Hint: First prove for the trivial case M=1 or N=1. Then write M=p_1p_2\dots p_k and N=q_1q_2\dots q_l, where p_i, q_j are primes. Use Euclid’s Lemma twice to prove that p_1^2\mid m. By induction, show that M^2\mid m. I think the key point is that Generalized Euclid’s Lemma is probably not allowed otherwise it is very easy.

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My favorite Fermat Little Theorem with Pascal Triangle

Math Online Tom Circle

Fermat Little Theorem: For any prime integer p, any integer m

$latex boxed {m^{p} equiv m mod p} &s=3$

When m = 2,

$latex boxed{2^{p} equiv 2 mod p}&fg=aa0000$

Note: 九章算数 Fermat Little Theorem (m=2)

Pascal Triangle (1653 AD France )= (杨辉三角 1238 AD – 1298 AD)

$latex 1 : 1 implies sum = 2 = 2^1 equiv 2 mod 1$

$latex 1: 2 :1implies sum = 4 = 2^2 equiv 2 mod 2 ;(equiv 0 mod 2)$

$latex 1 :3 :3 :1 implies sum = 8= 2^3 equiv 2 mod 3$

1 4 6 4 1 => sum = 16= 2^4 (4 is non-prime)

$latex 1 :5 :10: 10: 5: 1 implies sum = 32= 2^5 equiv 2 mod 5$


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