Linear System of Differential Equations, Solutions, Phase Portrait Sketching

Solutions of Homogeneous Linear System of DE
\displaystyle \mathbf{y}'=\mathbf{A}\mathbf{y}
\displaystyle \mathbf{y}(t)=\mathbf{v}e^{rt}
where r and \mathbf{v} are eigenvalue and eigenvector for \mathbf{A} respectively.

Superposition Principle
If \mathbf{x_1}(t) and \mathbf{x_2}(t) are two solutions to a homogeneous SDE \mathbf{y'}=\mathbf{Ay}, then \displaystyle \mathbf{y}=c_1\mathbf{x_1}(t)+c_2\mathbf{x_2}(t) is also a solution for any scalars c_1, c_2.

Euler’s formula
\displaystyle e^{i\theta}=\cos\theta+i\sin\theta

General Solutions (Complex Eigenvalues)

1) Let r_1=a+bi be an eigenvalue corresponding to eigenvector \mathbf{v_1}. (The eigenvectors are complex conjugates: \mathbf{v_1,v_2}=\mathbf{p}\pm \mathbf{q} i.)
2) Construct
\displaystyle \mathbf{x}_\text{Re}(t)=e^{at}(\mathbf{p}\cos bt-\mathbf{q}\sin bt)
\displaystyle \mathbf{x}_\text{Im}(t)=e^{at}(\mathbf{p}\sin bt+\mathbf{q}\cos bt)
3) The general solution is \displaystyle \mathbf{y}=c_1\mathbf{x}_\text{Re}(t)+c_2\mathbf{x}_\text{Im}(t).

How to Sketch Phase Portrait

Probably the best video on how to sketch Phase Portrait:

The Self-Driven Child (i.e. How to make your child motivated to study)

I think this book is a excellent read for Singaporean parents. Have your child ever seem to be too laid back and resist improvement, although he or she has great potential?

Often children in Singapore have to be “forced” to go to tuition to improve their grades, “forced” to study, “forced” to practice piano etc. Wouldn’t it be great if the child has self-motivation to willingly go for tuition / self-study? This change of mindset is explored in this book: “The Self-Driven Child”.

“Forcing” children to study works in the short term, producing temporary improved academic results, but in the long term it leads to psychological problems and even mental breakdowns in severe cases. Not to mention there is no joy of learning experienced by the child. The alternative solution is described in “The Self-Driven Child”.

“It is not an overstatement to say that this is one of the most radical and important books on raising healthy, resilient, purpose-driven kids.” – Madeline Levine, author of The Price of Privilege

“An invaluable resource for the thinking parent.” – Lisa Damour, author

The Self-Driven Child: The Science and Sense of Giving Your Kids More Control Over Their Lives

Check out other recommended Motivational Books here.

Characteristic Polynomial, Eigenvalues, Eigenvectors

Characteristic Polynomial, \det(\lambda I-A)
\begin{aligned}  \lambda\ \text{is an eigenvalue of }A&\iff\det(\lambda I-A)=0\\  &\iff \lambda\ \text{is a root of the characteristic polynomial}.  \end{aligned}

The solution space of (\lambda I-A)\mathbf{x}=0 is called the eigenspace of A associated with the eigenvalue \lambda. The eigenspace is denoted by E_\lambda.

Sum/Product of Eigenvalues
– The sum of all eigenvalues of A (including repeated eigenvalues) is the same as Tr(A) (trace of A, i.e. the sum of diagonal elements of A)
– The product of all eigenvalues of A (including repeated eigenvalues) is the same as \det(A).

Finding Least Squares Solution Review and Others

Rotation Matrix

The rotation matrix
\displaystyle  R=\begin{pmatrix}  \cos\theta & -\sin\theta\\  \sin\theta & \cos\theta  \end{pmatrix}
rotates points in the xy-plane counterclockwise through an angle \theta about the origin.

For example rotating the vector (1,0) 45 degrees counterclockwise gives us:
\displaystyle  \begin{pmatrix}  \cos 45^\circ & -\sin 45^\circ\\  \sin 45^\circ & \cos 45^\circ  \end{pmatrix}  \begin{pmatrix}  1\\  0  \end{pmatrix}  =  \begin{pmatrix}  \frac{\sqrt{2}}{2}\\  \frac{\sqrt{2}}{2}  \end{pmatrix}.

Finding Least Squares Solution

Given Ax=b (inconsistent system), solve
\displaystyle A^TAx=A^Tb instead to get a least squares solution of the original equation.


If we know a least squares solution \mathbf{u} of A\mathbf{x}=\mathbf{b}, we can find the projection \mathbf{p} of \mathbf{b} onto the column space of A by \displaystyle \mathbf{p}=A\mathbf{u}.

Dimension Theorem for Matrices (Also known as Rank-Nullity Theorem)

If A is a matrix with n columns, then \displaystyle rank(A)+nullity(A)=n.

(rank(A)=number of pivot columns,

nullity(A)=number of non-pivot columns.)

Linear Independence and the Wronskian
A set of vector functions \vec{f_1}(x), \dots, \vec{f_n}(x) from \mathbb{R} to \mathbb{R}^n is linearly independent in the interval (\alpha,\beta) if \displaystyle W[\vec{f_1}(x),\dots,\vec{f_n}(x)]\neq 0 for at least one value of x in the interval (\alpha,\beta).

Oven-roasted Scampi with Butter & Garlic

If you are an avid cook or food lover, do check out this cooking blog. 100% free recipes with high definition photographs, featuring Chinese, Western and other cuisines like Thai, Malaysian, Indian etc.

All recipes are suitable for beginner, amateur chefs. Not much equipment required other than the usual kitchen appliances like stove, oven.

Free Newsletter Subscription:

Some exotic recipes:

  1. Ge Da Soup
  2. King Crab
  3. Green Curry Chicken
  4. Pork Chop with Roasted Apple
  5. Homemade Prawn Ribs Noodle Soup

Cooking For Fun

When my husband and I were in Bangkok, we had a very giant Prawn in a restaurant. And several days ago, we found such a giant Prawn in supermarket. So we bought two.

Preheat oven to 200 degree C.

1. Clean Prawns and cut them into halves. I didn’t cut them through just left a big cut on their belly.

2. Sprinkle salt and black pepper over Prawns.

3. In a small bowl, add in 10 g butter. Then use a garlic press to press 4 cloves garlic. Add in 1/2 tsp black pepper. Mix well.

4. Press butter garlic mixture into the cut on belly. Brush shell with a little honey.

5. Into the oven and roast for 15-20 mins.

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Latest Secondary School Cut off Point (COP)

Something interesting in the latest Secondary School (based on PSLE results) Cut off Points (COP) is that MGS Cut-off points has risen sky high to 261, on par with RI and above HCI and RGS.

Nanyang Girls’ High School IP SAP (girls) – 264
Methodist Girls’ School (Secondary) IB (girls) – 261
Raffles Institution IP (boys) – 261
Hwa Chong Institution IP SAP (boys) – 260
Raffles Girls’ School (Secondary) IP (girls) – 260
CHIJ St. Nicholas Girls’ School IP SAP (girls) – 258
Dunman High School IP SAP (co-ed) – 258
National Junior College IP (co-ed) – 258
Anglo-Chinese School (Independent) IB (boys) – 256
Cedar Girls’ Secondary School IP (girls) – 254
Victoria School IP (boys) – 254
Catholic High School IP SAP (boys) – 253
River Valley High School IP SAP (co-ed) – 253
Singapore Chinese Girls’ School IP (girls) – 253
St. Joseph’s Institution IB (boys) – 253
Temasek Junior College IP (co-ed) – 253
CHIJ St. Nicholas Girls’ School O-levels SAP (girls) – 253
Methodist Girls’ School (Secondary) O-levels (girls) – 253
Singapore Chinese Girls’ School O-levels (girls) – 252
Catholic High School O-levels SAP (boys) – 251
Anglo-Chinese School (Independent) O-levels (boys) – 250
Cedar Girls’ Secondary School – O-levels (girls) – 250
Victoria School – O-levels (boys) – 248
Anderson Secondary School (co-ed) – 247
Bukit Panjang Govt. High School (co-ed) – 247
CHIJ Secondary (Toa Payoh) (girls) – 247
Paya Lebar Methodist Girls’ School (Secondary) (girls) – 247
St. Joseph’s Institution O-levels (boys) – 246
Nan Hua High School SAP (co-ed) – 245
St. Andrew’s Secondary School (boys) – 245
St. Margaret’s Secondary School (girls) – 244
Crescent Girls’ School (girls) – 243
Fairfield Methodist School (Secondary) (co-ed) – 243
Nan Chiau High School SAP (co-ed) – 243
Swiss Cottage Secondary School (co-ed) – 241
Anglican High School SAP (co-ed) – 240
Anglo-Chinese School (Barker Road) (boys) – 239
Chung Cheng High School (Main) SAP (co-ed) – 239
Commonwealth Secondary School (co-ed) – 239
Ngee Ann Secondary School (co-ed) – 238
Chung Cheng High School (Yishun) (co-ed) – 237
Yishun Town Secondary School (co-ed) – 237
Tanjong Katong Girls’ School (girls) – 236
Xinmin Secondary School (co-ed) – 236
CHIJ St. Theresa’s Convent (girls) – 235
Maris Stella High School SAP (boys) – 235
Fuhua Secondary School (co-ed) – 234
Zhonghua Secondary School (co-ed) – 234
Clementi Town Secondary School (co-ed) – 233
Presbyterian High School (co-ed) – 233
Tanjong Katong Secondary School (co-ed) – 233
Kranji Secondary School (co-ed) – 232
Kuo Chuan Presbyterian Secondary School (co-ed) – 232
Riverside Secondary School (co-ed) – 232
St. Patrick’s School (boys) – 232
CHIJ St. Joseph’s Convent (girls) – 231
Dunman Secondary School (co-ed) – 231
Holy Innocents’ High School (co-ed) – 229
St. Hilda’s Secondary School (co-ed) – 229
Ang Mo Kio Secondary School (co-ed) – 227
Bowen Secondary School (co-ed) – 225
Evergreen Secondary School (co-ed) – 225
Gan Eng Seng School (co-ed) – 225
Jurong Secondary School (co-ed) – 225
Edgefield Secondary School (co-ed) – 224
St. Gabriel’s Secondary School (boys) – 224
Geylang Methodist School (Secondary) (co-ed) – 223
Bukit Batok Secondary School (co-ed) – 222
CHIJ Katong Convent (girls) – 222
Hua Yi Secondary School (co-ed) – 222
Hai Sing Catholic School (co-ed) – 221
Pasir Ris Secondary School (co-ed) – 221
West Spring Secondary School (co-ed) – 221
Mayflower Secondary School (co-ed) – 220
St. Anthony’s Canossian Secondary School (girls) – 219
Ahmad Ibrahim Secondary School (co-ed) – 218
Pasir Ris Crest Secondary School (co-ed) – 216
Pei Hwa Secondary School (co-ed) – 216
Queensway Secondary School (co-ed) – 216
Woodlands Ring Secondary School (co-ed) – 215
Beatty Secondary School (co-ed) – 214
Bedok View Secondary School (co-ed) – 214
Chua Chu Kang Secondary School (co-ed) – 214
Peirce Secondary School (co-ed) – 214
Deyi Secondary School (co-ed) – 210
Yuan Ching Secondary School (co-ed) – 210
Zhenghua Secondary School (co-ed) – 210
Compassvale Secondary School (co-ed) – 209
Hillgrove Secondary School (co-ed) – 208
North Vista Secondary School (co-ed) – 207
Bedok South Secondary School (co-ed) – 206
Orchid Park Secondary School (co-ed) – 204
Woodgrove Secondary School (co-ed) – 204
Bukit View Secondary School (co-ed) – 203
Coral Secondary School (co-ed) – 203
Greendale Secondary School (co-ed) – 203
New Town Secondary School (co-ed) – 202
Yishun Secondary School (co-ed) – 201
Chong Boon Secondary School (co-ed) – 200
Jurongville Secondary School (co-ed) – 200
Kent Ridge Secondary School (co-ed) – 200
Westwood Secondary School (co-ed) – 200
Christ Church Secondary School (co-ed) – 199
Guangyang Secondary School (co-ed) – 197
Admiralty Secondary School (co-ed) – 196
Manjusri Secondary School (co-ed) – 196
Tampines Secondary School (co-ed) – 196
Greenridge Secondary School (co-ed) – 195
Greenview Secondary School (co-ed) – 195
Canberra Secondary School (co-ed) – 194
Punggol Secondary School (co-ed) – 194
Bishan Park Secondary School (co-ed) – 193
Jurong West Secondary School (co-ed) – 192
Juying Secondary School (co-ed) – 192
Seng Kang Secondary School (co-ed) – 191
Damai Secondary School (co-ed) – 190
Junyuan Secondary School (co-ed) – 190
Ping Yi Secondary School (co-ed) – 190
Bedok Green Secondary School (co-ed) – 189
Montfort Secondary School (boys) – 189
Naval Base Secondary School (co-ed) – 189
Northland Secondary School (co-ed) – 189
Queenstown Secondary School (co-ed) – 189
Regent Secondary School (co-ed) – 189
Yuhua Secondary School (co-ed) – 189
Assumption English School (co-ed) – 188
Bartley Secondary School (co-ed) – 188
Bendemeer Secondary School (co-ed) – 188
Boon Lay Secondary School (co-ed) – 188
Broadrick Secondary School (co-ed) – 188
Bukit Merah Secondary School (co-ed) – 188
Changkat Changi Secondary School (co-ed) – 188
Dunearn Secondary School (co-ed) – 188
East Spring Secondary School (co-ed) – 188
East View Secondary School (co-ed) – 188
Fajar Secondary School (co-ed) – 188
Fuchun Secondary School (co-ed) – 188
Hong Kah Secondary School (co-ed) – 188
Hougang Secondary School (co-ed) – 188
Loyang Secondary School (co-ed) – 188
Marsiling Secondary School (co-ed) – 188
Northbrooks Secondary School (co-ed) – 188
Outram Secondary School (co-ed) – 188
Peicai Secondary School (co-ed) – 188
Sembawang Secondary School (co-ed) – 188
Serangoon Garden Secondary School (co-ed) – 188
Serangoon Secondary School (co-ed) – 188
Shuqun Secondary School (co-ed) – 188
Springfield Secondary School (co-ed) – 188
Tanglin Secondary School (co-ed) – 188
Whitley Secondary School (co-ed) – 188
Woodlands Secondary School (co-ed) – 188
Yio Chu Kang Secondary School (co-ed) – 188
Yusof Ishak Secondary School (co-ed) – 188
Yuying Secondary School (co-ed) – 188


(Dated 2017)

Do double check from the official MOE PSLE Cutoff point list:—english.pdf

Also, do read this article on how to choose secondary schools based on the latest PSLE Cut off Points (COP) and other factors: More to secondary schools than cut-off point: Minister Heng.

5 Ways To Make Math More Fun And Meaningful For Kids

5 Ways To Make Math More Fun And Meaningful For Kids

Fun and meaningful – these are two words that children rarely use to describe math. There are several reasons why many kids dislike math, but according to kids and learning experts, the top reasons always include:

  • They always have to memorize mathematical formulas and concepts
  • They often have to make numerous complex and lengthy calculations (such as finding the surface area of cuboid or cylinder)
  • They always feel pressure to get perfect quiz or test scores
  • They have a hard time finding practical applications for the advanced mathematical formulas and concepts they’re learning

Because of these reasons (and more), parents always struggle to get kids to like math and excel in this subject.

How to Help Kids Change Their Attitude towards Math

According to a study published on the website of Stanford University’s Graduate School of Education, kids who outgrow their dislike and fear of of this subject will find it easier to do better on this subject.

If you are a parent or teacher, you can help children change their attitude towards math by making it more fun and meaningful for them. You can do this through the following ways:

1.    Enable your kids to realize the importance of math

When children understand that math is not all about theories and principles, they will start viewing the study of math as a valuable learning opportunity and thus become more interested in it. As such, you need to constantly show them how useful math is in real life.

For instance:

  • Teach your kids about basic finance whenever you go shopping
  • Train younger kids to sort coins and bills and how to use them when buying individual or small amounts of items
  • Allow your older children to help find the best prices for the items on your shopping list. Ask them to tally simple sums while grocery shopping.

Other activities that will help kids understand the relevance of math in real life include using measurements and basic operations when cooking and baking, telling time, checking temperatures, etc.

Although these activities seem simple, they are still effective ways of teaching kids the importance of knowing the right concepts and applications of certain mathematical operations.

2.    Take math outdoors

If you’re an educator, when you take math learning outside the classroom, you provide kids excellent ways of realizing that math can be found and used everywhere. This will also allow you to transfer lessons outside the classroom, and vice versa.

Below are examples of fun activities that will enable you to take math outdoors:

  • Treasure or scavenger hunt
  • Multiplication hopscotch
  • Leaf logic
  • Counting maze (for preschoolers)

3.    Enroll your kids in an after-school tutoring program

Sometimes, children need outside help to discover that math is interesting and meaningful. If you and your kids decide to get help from a tutor, find a tutoring center that specializes in teaching kids math.

The right math tutoring center will follow a unitary method that will help their students make sense of all the theories and concepts they are learning. They will assess the needs of the students and design a personalized learning program that will address their specific requirements.

Most tutoring centers today do not simply provide additional explanations and activities for kids to learn a particular concept. Tutors tailor their teaching techniques to ensure the students learn by heart and apply their knowledge.

As such, they also employ fun and creative methods to teach their students. They also check progress along the way to make sure kids truly understand, apply, and retain the concepts they learned.

4.    Incorporate math in games

Bring out your board games, a pack of cards, a puzzle, or even or old blocks and turn the game into a family competition. Activities and games that incorporate or focus on math are great in reinforcing the right mathematical skills and concepts.

Regardless of the activity, you can reward even small accomplishments and help your kids know that they just completed a fun math-related task. Children will love the recognition and prize, especially if they can compete with their siblings. They will also realize that knowing mathematical operations can be fun and applying them can be rewarding.

5.    Be supportive

Lastly, although you may want to empathize with your kids, saying things like “I was also never good at math” won’t do anything good for them. It is best to encourage your children to embrace challenges and see the fun in learning even if they are having a hard time with some mathematical concepts.

Be as involved as you can be in your children’s schoolwork and show enthusiasm. When you help your kids learn to associate math with fun, pleasure, parental love and attention, they will be excited about the subject throughout their learning years.

As a parent or educator, your support and willingness to think outside the box will go a long way in helping your kids think differently about math and eventually excel in the subject.


Maloy Burman is the Chief Executive Officer and Managing Director of Premier Genie FZ LLC. He is responsible for driving Premier Genie into a leadership position in STEM (Science, Technology, Engineering and Mathematics) Education space in Asia, Middle East and Africa and building a solid brand value. Premier Genie is currently running 5 centers in Dubai and 5 centers in India with a goal to multiply that over the next 5 years.

Positive attitude toward math predicts math achievement in kids

Science has proven that attitude is really important. Positive attitude toward math has great impact on math achievement in children.

One way to instill positive attitude toward math is via interesting math books. Another way is via parental example, it is important for parents themselves to have a positive attitude to math as it can influence their child.

Source: Science Daily

For the first time, scientists have identified the brain pathway that links a positive attitude toward math to achievement in the subject. In a study of elementary school students, researchers found that having a positive attitude about math was connected to better function of the hippocampus, an important memory center in the brain, during performance of arithmetic problems.

In a study of elementary school students, researchers at the Stanford University School of Medicine found that having a positive attitude about math was connected to better function of the hippocampus, an important memory center in the brain, during performance of arithmetic problems.

The findings will be published online Jan. 24 in Psychological Science.

Educators have long observed higher math scores in children who show more interest in math and perceive themselves as being better at it. But it has not been clear if this attitude simply reflects other capacities, such as higher intelligence.

The new study found that, even once IQ and other confounding factors were accounted for, a positive attitude toward math still predicted which students had stronger math performance.

‘Attitude is really important’

“Attitude is really important,” said Lang Chen, PhD, the study’s lead author and a postdoctoral scholar in psychiatry and behavioral sciences. “Based on our data, the unique contribution of positive attitude to math achievement is as large as the contribution from IQ.”

The scientists had not expected the contribution of attitude to be so large, Chen said. The mechanism underlying its link to cognitive performance was also unexpected.

“It was really surprising to see that the link works through a very classical learning and memory system in the brain,” said the study’s senior author, Vinod Menon, PhD, professor of psychiatry and behavioral sciences. Researchers had previously hypothesized that the brain’s reward centers might drive the link between attitude and achievement — perhaps children with better attitudes were better at math because they found it more rewarding or motivating. “Instead, we saw that if you have a strong interest and self-perceived ability in math, it results in enhanced memory and more efficient engagement of the brain’s problem-solving capacities,” Menon said.

The researchers administered standard questionnaires to 240 children ages 7 to 10, assessing demographics, IQ, reading ability and working-memory capacity. The children’s level of math achievement was measured with tests of their knowledge of arithmetic facts and ability to solve math word problems. Parents or guardians answered surveys about the children’s behavioral and emotional characteristics, as well as their anxiety about math and general anxiety. Children also answered a survey that assessed their attitude toward math, including questions about interest in math and self-perceived math ability, as well as their attitude toward academics in general.

Forty-seven children from the group also participated in MRI brain scans while performing arithmetic problems. Tests were conducted outside the MRI scanner to discern which problem-solving strategies they used. An independent group of 28 children also was given MRI scans and other assessments in an attempt to replicate the findings from the cohort previously given brain scans.

Lenovo Miix 320 Review

The Lenovo Miix 320 is quite a good budget computer actually, much better than what the reviews on Amazon state.

Some special features is that it is able to detach into a tablet, has touchscreen, and is compatible with a screen pen called the Lenovo Active Pen. Overall, it is the best laptop in its price range. Be sure to buy the 4 GB RAM version, as the 2 GB version will be too slow.

Size wise, its screen is basically the same size as an iPad. Hence, the Lenovo Miix 320 is suitable for people who want an “iPad with a keyboard”. If the screen font is too small, just setting the system font to 125% will solve the problem.

Conclusion: A very decent budget laptop that is suitable for basic usage such as surfing the web, email and word processing. Value for money (probably the best in its price range).

Lenovo Miix 320, 10.1-Inch Windows Laptop, 2 in 1 Laptop, (Intel Atom X5-Z8350, 1.44 GHz, 4 GB DDR3L, 64 GB eMMC, Windows 10 Home), Platinum, 80XF00DRUS

Alternatives for people with higher budget

The most classic 2 in 1 Laptop is probably the Microsoft Surface. Hence, if you have a higher budget than the Lenovo Miix, probably you may want to consider the Surface instead.

Microsoft Surface Book (128 GB, 8 GB RAM, Intel Core i5)

Dot Product and Span Summary

Dot Product

\text{span}\{\mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k}\}=\{c_1\mathbf{u_1}+c_2\mathbf{u_2}+\dots+c_k\mathbf{u_k}\mid c_1,c_2,\dots,c_k\in\mathbb{R}\}=\text{set of all linear combinations of } \{\mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k}\}.

V\subseteq\mathbb{R}^n is a subspace of \mathbb{R}^n if
1) V=\text{span}\{\mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k}\} for some vectors \mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k}.
2) V satisfies the closure properties:

(i) for all \mathbf{u},\mathbf{v}\in V, we must have \mathbf{u}+\mathbf{v}\in V.

(ii) for all \mathbf{u}\in V and c\in\mathbb{R}, we must have c\mathbf{u}\in V.

3) V is the solution set of a homogeneous system.

(Sufficient to check either one of Condition 1, 2, 3.)

For V to be a subspace, zero vector \mathbf{0} must be in V. (Since for \mathbf{u}\in V, 0\in\mathbb{R}, we have 0\mathbf{u}\in V.)

Linear Independence and Dependence
\mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k} are linearly independent if the system \displaystyle c_1\mathbf{u_1}+c_2\mathbf{u_2}+\dots+c_k\mathbf{u_k}=0 has only the trivial solution, i.e. c_1=c_2=\dots=c_k=0.

If the system has non-trivial solutions, i.e. at least one c_i not zero, then \mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k} are linearly dependent.

Lichess 2000 rating

Finally reached Lichess rating of 2000! (Rapid rating)

That had been my goal for some time but I only managed to achieve it after 1000 games or so. (See my Lichess profile.) I have no coach so I mostly learn from YouTube videos such as from GingerGM and the Saint Louis Chess Club. Those have helped a lot.

Note: Lichess rating is inflated, so 2000 Lichess is nowhere near 2000 FIDE unfortunately.

Check out my previous post on how to convert Lichess ratings.

Tip: Also World Champion Garry Kasparov’s Masterclass is incredibly helpful for beginning to intermediate players to improve their game. In fact, my ideal opening is Kasparov’s Queen’s Gambit variation with Bd3 and Ne2, aiming for a f3 and e4 expanding in the center. This opening is incredibly powerful and commonly played by Kasparov and his teacher Botvinnik.

Gaussian Elimination Summary

Row echelon form (REF)
For each non-zero row, the leading entry is to the right of the leading entry of the row above.

E.g. \begin{pmatrix}  0 & \mathbf{1} & 7 & 2\\  0 & 0 & \mathbf{9} & 3\\  0 & 0 & 0 & 0  \end{pmatrix}

Note that the leading entry 9 of the second row is to the right of the leading entry 1 of the first row.

Reduced row echelon form (RREF)
A row echelon form is said to be reduced, if in each of its pivot columns, the leading entry is 1 and all other entries are 0.

E.g. \begin{pmatrix}  1 & 0 & 0 & 2\\  0 & 1 & 0 & 3\\  0 & 0 & 1 & 4  \end{pmatrix}

Elementary Row Operations
1) cR_i — multiply the ith row by the constant c
2) R_i \leftrightarrow R_j — swap the ith and the jth row
3) R_i+cR_j — add c times of the jth row to the ith row.

Gaussian Elimination Summary
Gaussian Elimination is essentially using the elementary row operations (in any order) to make the matrix to row echelon form.

Gauss-Jordan Elimination
After reaching row echelon form, continue to use elementary row operations to make the matrix to reduced row echelon form.

Butterfly Pea Health Benefits (Research Papers)

Butterfly Pea Tea is a blue colored tea from the Butterfly Pea flower. It is apparently not very well known outside Thailand (and some other countries). Even in Singapore, which is a South East Asian country, few people know it.

(I bought the dried flowers from Qoo10. Quite good quality from this seller, sourced from Thailand, and nice air-tight packaging.)

Here are some health benefits of Butterfly Pea Tea (from this website).

We only choose sources from published journal articles as they are more reliable.

  1. Protection of Clitoria ternatea flower petal extract against free radical-induced hemolysis and oxidative damage in canine erythrocytes.
  2. Inhibitory effect of Clitoria ternatea flower petal extract on fructose-induced protein glycation and oxidation-dependent damages to albumin in vitro.
  3. Anti Inflammatory, Analgesic Properties

Basically, from what I understand, the benefits of Butterfly Pea (Clitoria ternatea) is that it is an anti-oxidant, potentially useful to prevent diabetes, and also anti-inflammatory and may have some mild painkilling properties (analgesic).

Quite interesting, I may decide to plant it in the future! Another interesting property is that if you add acid (e.g. lemon juice or lime juice), the color changes from blue to purple!

Purchase link from Qoo10:

“Yu Gong” of India: Mountain Man Motivational Story

Most Chinese would have heard of the story Yu Gong Yi Shan, which is a fable of how a man moved a mountain bit by bit using his perseverance.

Today I learn that there is a true story of Dashrath Manjhi from India who literally cut through a mountain with just a simple tool over 20 years! Incredibly motivational story.

Dashrath Manjhi (c. 1934[1] – 17 August 2007[2]), also known as Mountain Man,[3] was a poor labourer in Gehlaur village, near Gaya in Bihar, India, who carved a path 110 m long (360 ft), 9.1 m (30 ft) wide and 7.6 m (25 ft) deep through a hillock using only a hammer and chisel.[1][4][5] After 22 years of work, Dashrath shortened travel between the Atri and Wazirganj blocks of Gaya town from 55 km to 15 km.[6]

Dashrath Manjhi ran away from his home at a young age and worked at Dhanbad‘s coal mines. He returned to his village and married Falguni Devi. While crossing Gehlour hills to bring him lunch, she slipped continuously and seriously injured herself, which eventually led to her death. Manjhi was deeply disturbed and that very night decided to carve a path through the Gehlour hills so that his village could have easier access to medical attention.[1] He carved a path 110 m long, 7.7 m deep in places and 9.1 m wide to form a road through the rocks in Gehlour hill.[a] He said, “When I started hammering the hill, people called me a lunatic but that steeled my resolve.”

The Fields Medal fallacy: Why this math prize should return to its roots

Source: Science Daily

The Fields Medal, whose origins date back to the 1930s, will be issued again this year in August to up to four of the world’s most accomplished mathematicians under the age of 40. In a commentary for Nature, Michael Barany, a Society of Fellows post-doctoral fellow in history at Dartmouth, proposes that the Fields Medal return to its roots as a tool intended to shape the future of mathematics, rather than recognizing those who have already found the spotlight.

Issued every four years at the International Congress of Mathematicians, the Fields Medal has been likened to a Nobel Prize in mathematics. To date, 55 of the 56 recipients of this prize have been male and come from a narrow selection of institutional, disciplinary and geographic backgrounds compared with the discipline as a whole

“The current approach to the Fields Medal reinforces biases both within the field of mathematics and in broader public attitudes on what makes for a brilliant mathematician and typically overlooks how mathematics can be used to advance public good. Medalists today must achieve notable milestones in their research and careers at a young age — something that is a lot easier for those with the most social and institutional privilege,” explained Barany, who presented this research Friday at the Joint Mathematics Meetings of the American Mathematical Society and Mathematical Association of America.

Through research in Harvard University’s archives, Barany draws on forgotten records behind the deliberations from two early Fields Medal committees, which show, in a way not possible with existing published materials, how the medal’s purpose was interpreted following its origins in interwar international conflict.

In 1932, Canadian mathematician John Charles Fields called for the creation of an “International Medal for Outstanding Discoveries in Mathematics” that should be “in recognition of work already done” and “an encouragement for further achievement.” According to Barany, early committees avoided rewarding the best-regarded mathematicians and focused instead on identifying those who could benefit from the boost of an international prize to make a difference in the discipline.

However, in 1966, the medal’s meaning radically changed. The committee established a strict age limit of 40-years old, which opened the prize up to those who become famous early on in their careers. In addition, the prestige of the Fields Medal was elevated following a controversy relating to one of the medalists, whose colleagues defended him by comparing the Fields Medal to the much higher-profile Nobel prizes.

As Barany explains, shifting recognition to more established mathematicians embraced the kinds of judgement and rivalry that Fields aimed to avoid in its founding, and was counter to the Fields Medal’s heritage. He says that future Fields Medal committees can learn from their predecessors’ insistence on using the award to make a difference for the future, and address ongoing challenges facing mathematics and its place in the world.

Does Reboiled Water Cause Cancer?

Just to share with all. Hard to imagine that a simple act like reboiling water is actually causing harm to health.

The logic makes sense though: reboiling water evaporates some of the water which is lost as water vapor. The remaining salts (including arsenic / nitrates) become more concentrated, and hence more harmful.

Do share this with your loved ones!

Source: ThoughtCo

There is a concern that reboiled water may lead a person to develop cancer. This concern is not unfounded. While the boiled water is fine, increasing the concentration of toxic substances may put you at risk for certain illnesses, including cancer. For example, excessive intake of nitrates has been linked to methemoglobinemia and certain types of cancer. Arsenic exposure may produce symptoms of arsenic toxicity, plus it has been associated with some forms of cancer. Even “healthy” minerals may become concentrated to dangerous levels. For example, excessive intake of calcium salt, commonly found in drinking water and mineral water, can lead to kidney stones, hardening of the arteries, arthritis, and gallstones.

Star Wars “Math” Song

The following is a very original and interesting “song” using a pencil to mark out the rhythm of the famous “Cantina Theme” in Star Wars. Very creative indeed!

Technically, the equation is mathematically valid too, provided A is viewed as a function of x (and f, v). Or is the v a square root? Who knows?

Viewers need to have heard the original Cantina Theme to appreciate the pencil scribbling:

Also read my other related Star Wars math posts below:

Latest JC Cut-off points COP

JC Cut off Points for 2018:

Something amazing has happened in this latest JC COP. Nanyang Junior College (NYJC) has officially “surpassed” National Junior College (NJC) in terms of COP. (Traditionally NJC was one of the top three JCs, together with RI and HCI.) Coupled with a fantastic location (Lorong Chuan) and a caring principal Mr Kwek who has instilled a culture of care towards students, NYJC has deservingly propelled itself to the top tiers of the JC Cut of Points List.

Junior Colleges 2018 COP Arts Science / IB
Raffles Institution, RI  5 5
Hwa Chong Institution, HCI  5 5
Victoria Junior College, VJC  8 6
Anglo-Chinese School (Independent), ACSi NA 6
National Junior College, NJC  9 8
Nanyang Junior College, NYJC  8 7
Anglo-Chinese Junior College, ACJC  10 9
St. Joseph’s Institution, SJI NA 8
Eunoia Junior College, EJC  11 10
Temasek Junior College, TJC  11 10
St. Andrew’s Junior College, SAJC  12 11
Anderson Junior College, AJC  12  12
Catholic Junior College, CJC  14 15
Meridian Junior College, MJC  13  14
Pioneer Junior College, PJC  17  14
Yishun Junior College, YJC  20  20

JC Cut off Points for 2017:

Something interesting this year is that Nanyang JC (NYJC) has reached the same level (Cut off Points) as National JC (NJC). This is unimaginable say 10 years ago. Congratulations to NYJC for the great improvement! The great location of NYJC (near Lorong Chuan MRT) must have made a great factor too, it is so convenient for many students.

Also check out my post on Which JC is good?. Cut off points is not the sole indicator of how “good” a JC is. Do consider other factors and choose your JC wisely.

JC Arts Science
RI 4 4
HCI 4 4
VJC 7 5
ACS(I) 5
NJC 7 6
NYJC 7 6
ACJC 8 7
EJC 9 9
TJC 9 9
SAJC 10 9
AJC 11 9
CJC 10 11
MJC 11 11
SRJC 13 12
PJC 13 12
TPJC 13 14
JJC 15 15
YJC 17 16
IJC 19 20

JC Cut off Points for 2016:

Arts Science
RI 4 3
HCI 4 4
ACS(I) 5
VJC 6 5
NJC 6 5
NYJC 7 6
ACJC 7 7
AJC 9 8
TJC 9 9
SAJC 9 9
CJC 10 10
MJC 10 10
SRJC 11 11
PJC 13 12
TJC 13 13
JJC 15 15
YJC 17 16
IJC 20 20

It seems majority of the schools are under 10 points for the cut off, though there are JCs that cater to up to 20 pointers.

Citibank SMRT Credit Card Review ($120 Cashback Signup Bonus)

Sign up via this link to get $120 Cashback (Official Citibank site)

The Citibank SMRT card is one of the best credit cards in Singapore for your savings. It is very suitable for heartlander Singaporeans who spend mostly on groceries, transport and bills.

  • 5% savings* on groceries (7.3% for FairPrice Xtra Kallang Wave)
  • 5% savings* on fast food, movies and coffee
  • 3% savings* on online shopping
  • Redeem for cash rebate, vouchers or SMRT rides
  • No convenience fee charge for EZ-Reload Auto Top-up transactions

To compare with another credit card (POSB Everyday), here are the benefits of Citibank SMRT over POSB Everyday. Note that you can have both cards to reap the maximum benefits too!

Citibank SMRT vs POSB Everyday Card

  • Citibank SMRT has 5% savings for Fairprice, Giant AND Sheng Siong. POSB everyday 5% rebate is only for Sheng Siong.
  • Citibank SMRT card has EZ-Reload Auto Top-up 2% savings. No other credit card in Singapore has this feature. (Please correct me if I am wrong)
  • 1% savings on insurance bills for ACE Insurance and Prudential. (POSB Everyday has recently excluded rebates for insurance, i.e. 0% rebate for insurance and in fact most payments other than Starhub and SP)
  • Citibank SMRT card has 5% savings at Selected Town Councils – Service and Conservancy Charges. (As mentioned, POSB Everyday has excluded most bill payments from earning rebates).
  • Citibank SMRT has 1% saving* on telecommunications bills (M1, Singtel, Starhub). POSB Everyday only works for Starhub.
  • Citibank SMRT has 2% savings on both Guardian’s and Watsons. To be fair, POSB Everyday has 3% savings on Watsons (only). So we should use Citibank SMRT for Guardian and POSB Everyday for Watsons.
  • Many other savings benefits (e.g. movies, Starbucks) for Citibank SMRT

(Information valid as of Jan 2018. Do check out the official sites: Citibank SMRT and POSB Everyday for updated official information.)

$120 Cashback for Signing up Citibank SMRT Card

Once again, to get $120 cashback for signing up for Citibank SMRT, sign up via this link here.


More schools to merge in 2019, JCs included

Seems like the declining population is quite serious indeed. Not too long ago there was another merger: 8 JCs to merge (i.e. 4 JCs to close down). “Merge” is just a nice way to say that the affected schools are closing down.

I think at this rate, quite a few jobs may be affected, like paediatrician (children’s doctor), childcare, and even tuition. I think a previous Today article mentioned the impact of declining population on the tuition industry.

The following Primary/Secondary schools will be merged in 2019:

Bendemeer Primary – from merger of Balestier Hill Primary and Bendemeer Primary

Casuarina Primary – from merger of Loyang Primary and Casuarina Primary

Cedar Primary – from merger of MacPherson Primary and Cedar Primary

White Sands Primary – from merger of Coral Primary and White Sands Primary

Damai Primary – from merger of East Coast Primary and Damai Primary

Jing Shan Primary – from merger of Da Qiao Primary and Jing Shan Primary

Junyuan Primary – from merger of East View Primary and Junyuan Primary

The merged secondary schools will be:

East Spring Secondary – from merger of East View Secondary and East Spring Secondary

Jurongville Secondary – from merger of Hong Kah Secondary and Jurongville Secondary

Yuhua Secondary – from merger of Shuqun Secondary and Yuhua Secondary

The merged schools will be located at the schools whose name has been chosen for the combined entities. For example, students from Loyang Primary will thus have to go to the existing Casuarina Primary School from 2019.


Star Wars: The Last Jedi Review

I find the plot quite disappointing. While the CGI graphics are good, and the acting is decent, the storyline is what I find lacking. Ever since George Lucas sold the Star Wars franchise to Disney, I find that the essence of Star Wars has changed.

Not just The Last Jedi, the previous Force Awakens plot was formulaic. There seems to be a fixed pattern to it:

  1. Some pilots must try to sneak in to disable/destroy the large enemy ship, be it the Death Star or the “Dreadnaught”.
  2. Go to a new planet with some fancy new animals/lifeforms
  3. The battle between the Sith and the Jedi. (This was lacking in the Last Jedi though. There were close to zero lightsaber battles.)

There are some loopholes that don’t fit the style of the previous Star Wars movies. Rey defeating Kylo Ren without any prior training, and Luke Skywalker doing some kind of “Astral Projection”. As a fan who have watched all prior Star Wars movies, the style and plot of the new movies (post-Disney) are strikingly different in a bad way. In fact, it is so bad that I don’t think I will watch any Star Wars movies in movie theatres anymore.

Highly Motivational Math Video (in Chinese)

This is actually one of the best motivational videos on Math I have seen. Unfortunately there is no English translation. It covers how useful Math is, and also some history of Math in ancient China. (It is rarely known, but China discovered negative numbers and calculated pi to high accuracy much earlier than in Western civilization.)

However, (according to the video), Math in ancient China went downhill in the Ming dynasty after it was scrapped from the imperial examination. Seems like removing Math from the examination syllabus is always a bad idea!

Finally, the video ends off with a note not to discourage budding mathematicians. Many budding mathematicians, will face strange looks from well-intentioned friends and society. Will learning math be useful or can it make money? Such thoughts can discourage people from learning mathematics (like the speaker himself).

By the way, at the start of the video, the speaker tells a humorous story of how he used Math to propose to his crush in England. This is related to my earlier post on Valentine’s Day Math on how to draw a heart using math.

Uni Grads: Choosing the hawker life over the tried and tested route

Very nice to see their passion in cooking, and daring to choose the road less travelled. Do support them at their stall, Prawn Village, which currently at 20 Ghim Moh Road, #01-62, 199583.

SINGAPORE: They had not yet graduated from university, but like many of their peers, Joanne Heng and Chan Kheng Yee were already out and about looking for a full-time job. A posting on online portal Gumtree caught their attention, and they responded to the call for “interns” the very next day.

Like any job application, the two friends had to go through a job interview filled with questions about their sincerity, motivation and commitment to the position.

But this job required very different skill sets from your typical office job: Anson Loo, the person who made the post on Gumtree, was looking for young interns to help him run his hawker stall.

“I realised that there are young people who want to become hawkers, but face a lot of financial constraints,” said Anson, who sells prawn mee at Ghim Moh food centre. “So I thought, why not target young people with no experience, so I can give them the training from scratch?”


Of the seven who applied, Anson said Joanne and Kheng Yee fit his requirements the best. They were young, passionate and completely new to the hawker trade. And indeed, the two, who met while they were studying in the polytechnic, recalled bonding over a shared interest in F&B even as students.


8 of 10 Self-Made Millionaires Were Not ‘A’ Students. Instead, They Share 1 Trait

I think this is the latest best selling motivational book (out in Jan 2018). It is quite true, many successful people aren’t necessarily ‘A’ students. They do have some common traits that everyone can learn from.

The book is quite applicable to students. Most often, for students who aren’t performing well in school, the main reason is motivation. They are either not motivated, or their motivation is misplaced (e.g. motivated in computer games). Most parents will actually notice that their child is quite intelligent, but underperforming in school. The problem is psychological (motivation) rather than anything else.

The Motivation Myth: How High Achievers Really Set Themselves Up to Win

Source: Jeff Haden Linkedin

Listen to most teachers — and most parents — and it’s easy to assume that getting good grades in school is a requirement for professional success.


Tom Corley, an accountant and financial planner, surveyed a number of high net-worth individuals. Many of them are self-made millionaires. (Not that you have to be a millionaire to be successful, of course.) He found most of the people surveyed did not earn high GPAs in school.

In fact, only 21% of the self-made millionaires were “A” students. 41% reported they were “B” students, and 29% were “C” students.

That’s right: More of the self-made millionaires were C students than were A students.

And if you’re wondering if family background played a part, 59% of the self-made millionaires came from middle-class households and 41% came from poor households — proving where you start does not dictate where you finish.

As Corley writes:

“…success in life does not come easy. It is fraught with pitfalls, obstacles, failure, and mistakes. Success requires persistence, mental toughness and emotional toughness in overcoming these pitfalls. Its pursuit pushes you to the edge emotionally and physically. You must grow a thick skin and become accustomed to struggle if you hope to succeed.

“Individuals who struggle academically may be more accustomed to dealing with struggle and making it a daily habit to overcome pitfalls.”

In short, they become mentally tough, which creates a foundation for long-term success.

Express vs Normal Academic for borderline PSLE score

For students scoring in the borderline of around 190- 200 PSLE score, there is a dilemma of going to Express stream in a neighborhood school, or Normal Academic N(A) in a more established school. Note that even good schools like Anderson, ACS (Barker) do have N(A) streams.

Here are some advices from people who have experience:

Source: Hardwarezone

First opinion says that N(A) in a good school is better:

“NA in good school

i myself is living example

in the end my parent chose express at neighborhood school

in the end that school is a s**** school. i cannot keep up with studies.

by secondary 2 i drop to NA

then stuck with the lousy neighborhood school and my whole life got ruin until now

if i can go back time i sure chose NA at good school”

This post (from Hardwarezone) made a good argument of why express stream is better:

“I think you have been grossly misinformed about the life of a NA student. You say your son is addicted to computer games but it seems like he is doing relatively ok to be able to qualify for the express stream.

Yes he will be learning at a slower pace but over the next 4 years your son will have to be resilient to constant peer pressure and have a good amount of perseverance as many if not most students from NA are known to be more boisterous and mischievous causing them to deviate from their studies. Not to mention the stigma associated with being in NA. During my time in one of your “preferred NA schools”, I’ve seen many good students fall out of their studies because of the influence of their peers. The “just because I’m in NA I cmi” mentality will sadly creep up to many.

In my batch, iirc out of 3 classes of NA students, only 1 class of about 30 people was able/ motivated enough to take their O Levels. If your son decides not to take his O’s in the future and goes to ITE instead, whatever good name the school has will not have any effect on his resume.

Moreover, it is not like “school reputation” has any standing in the hiring process. Unless you’re from a top school like RI/Hwa Chong, no one would give two hoots. Recruiters mostly look at your highest education obtained or post secondary education to see if there is any relation to the job scope you are applying for.

In my opinion, you are better off applying to an express neighbourhood school and see if he is able to cope rather to a “prestigious” school in the NA stream. He can always drop out of the express stream if he’s not suitable for it.”

This parent from Kiasuparents gives very compelling reasons in favor of Express:

“Every year, there are parents asking the same question: Exp or NA?

Every year, I would encourage parents to choose Express for their children, for the simple reason that O level syllabus is very rigorous. Children are expected to work hard right from the start, unlike NA.

Even though the school you get will not be a ‘good school’ (if you choose Express), the classmates your child mixes with are likely to be of similar academic profile, with some that only have ‘Exp’ as their option. Similarly, if your child chooses NA, the children he or she mixes with will also have some that only have ‘NA’ as their option.

There are children who transfer from NA to Exp, but how many are there? From what I heard, very few manage to do that. You may want to check with the schools you are interested in choosing NA for for the probability that your child could transfer to Exp though.

I just want to let you know that if it happens to my own child, I would choose Exp. If the child is willing to work hard, he will make it anywhere. And if he can make it anywhere, then why wouldn’t I choose a better stream for him so that he can get used to the rigorous syllabus earlier?”

This parent from Kiasuparents recommended Anderson and Presbyterian High as two good schools with Normal Academic:

“I would choose one where the school discipline is good, and the teachers are supportive of students etc, with good academic and non – academic programs to expose / stretch the child. If going for NA, you can consider Anderson Sec, Presbyterian High.. these are generally pretty good schools.”

Amos Yee Math Talent

Amos Yee (“famous” for posting controversial videos) actually has good talent and aptitude in Mathematics. His mother is a Math secondary teacher with decades of experience. Many people know him for his infamous videos and his “American English” pronunciation, but few know that he was actually one of the top students in his secondary school in terms of Maths. His English results were good too, and so was his Pure Chemistry. Perhaps even more impressive (and rare), is Amos Yee has a Grade 8 guitar certification (ABRSM merit) (Source: Amos Yee “Happy Teachers’ Day” YouTube video). In March 2011, Yee also won awards for Best Short Film and Best Actor at The New Paper’s First Film Fest (FFF) for his film Jan. He was also an actor in Jack Neo’s movie We Not Naughty.

Unfortunately, Yee seemed to have not used his talents well to benefit society, but instead got himself into a lot of trouble. Who knows, if he turns over a new leaf it could be still possible to have a bright future.

Quote from Amos Yee Facebook:

“OK my fellow friends, sorry it’s been so late, I shall announce my O level results.

Apparently I did better than I expected, for all the wrong subjects, so if you truly want to see the innate comedy of my results, you should check out the results which I’d predicted before in my previous post before reading this, and then I think you’ll laugh as much as I did.

E Maths (Dogs truly can get A1 for E Maths)

English: A1 (Well this was surprising, I’d finally gained the coveted A1 for English that I had always hoped for in my Secondary School, and mastered the art of English Comprehension.

A Maths: A2 (So apparently if you leave out 4 entire questions that are 7 marks each, you can still get an A2. So I guess I did really ****ing well for the questions that I did. My mother, being a highly coveted A maths and E maths teacher for 3 decades, threw herself out of the building when she heard her darling son did not attain an A1 for A maths. If you look out of the window, you can still hear the faint cries of ‘****! MY SON IS SUCH A DISAPPOINTMENT’. Well, you threatened to disown me when I became an Atheist, and in the end you didn’t, so I think you’ll do just fine.)


Raw Aggregate Score: 7(-2 for CCA, -2 for MSP)

Best school I can go to: Nanyang JC”

Source: Amos Yee Facebook, January 14, 2015

Amos also claimed that he “had only studied extensively for the first E maths and A maths papers during the O levels period”. He didn’t really study during the month before the O levels, rather he was “abandoning studying just for that last month, and instead using that month to Complete 4 seasons of Daria, play Spirit Tracks and smash brothers on the DS, create a tuition namecard I now rarely use, and listening to all the best albums of the Beatles”. His Prelim results were also great:


“For reference and comparison, this was my mark for prelim 2, the final exam I took in school that isn’t O levels, I think by Prelim 2, the tedium of studying and the uselessness of it was already bearing down on me, and I studied for all the papers, 2 days before and tried to do the best I can with the retained knowledge I had for CA1 (Which I also got an L1R5 of 12 but a slightly higher % of 72 % compared to CA2’s 70%, due largely in part to 93% and 88% for E maths and A maths respectively)
E Maths: A1 (82%)
A maths: A1 (82%)
English: A2 (72%)
Chemistry: A2 (71%)
Chinese: B3 (68%/)
Literature: B3 (65%)
Combined Humans: B3 (65%)
Malay: B4 (64%)
L1R5: 12

Source: Amos Yee Facebook, January 11, 2015

His prediction of his O Level results are also quite accurate:


“So here it is, my predicted O level results that are coming out tomorrow:

E maths: A1 (Dogs can get A1 for E maths)

English: A2 (Honestly, I might get an A1 in lieu of the bellcurve,but I never got A1 for English in any full paper before, neither would I feel proud if I did, why would I feel proud about mastering the art of the language of robots.)

Chemistry A2(SPA will help me probably, and though I didn’t study, retention from previous exams was surprisingly good when I did the paper)

A Maths: B3 (Though this was my best subject ever in previous exams(I got 100 plus bonus mark for an A maths paper for God’s sakes), apparently the few weeks I didn’t study was significant enough to make me forget my concepts, to the point that I had skipped an entire 2-3 questions with about 7 marks each, and I think with my inclination to be careless and forget units, it’s going to be a deprovement that will send shock waves)”

Source: Amos Yee Facebook, January 11, 2015

Finally, Amos Yee was also a top student in Secondary 3, 4 (Zhonghua Secondary School) and nominated for a Humanities scholarship. He was consistently getting As for E maths, A maths, Chemistry and English. (Source: Amos Yee WordPress blog, January 25, 2015).

PSLE 181 to NUS Medicine (Miracle Story)

Quite an amazing story. The road from PSLE 181 to NUS Medicine is a long and arduous journey. Read how Mr Tan Jun Xiang, 22, managed to overcome the odds to enter the prestigious NUS Medicine faculty.

NUS Medicine is very hard to get in (even perfect scorers can get rejected). His overjoyed father “booked two tables at a restaurant and invited (his) relatives to celebrate.”


SINGAPORE – Mr Tan Jun Xiang, 22, is not your typical medical student who aced all his school examinations.

In fact, he scored only 181 points in the Primary School Leaving Examination (PSLE) and had to go into the longer five-year Normal stream in secondary school.

The polytechnic graduate, who made it to the prestigious medicine faculty at the National University of Singapore (NUS), is among the rare few who do not fit the mold.

When he was younger, he never thought he would go to university – much less the highly competitive Yong Loo Lin School of Medicine at NUS, where only about one in seven applicants get in.

So what sparked his stunning academic turnaround?

A few things: seeing how disappointed his parents were with his results, getting into the secondary school of his choice after an appeal and discovering that he could indeed do well if he put his mind to it.

Read more at Straits Times

Math Books for Christmas

Wishing all readers a joyous Christmas ahead! Here are some ideas for a mathematical Christmas gift for your loved ones who are math lovers:


This Christmas-themed Math book is the perfect gift for your child. According to Amazon, it is rated 4.5/5, and one reviewer even remarked that his 7 year old daughter loved reading it:

“I don’t write reviews normally but I was sitting in bed reading it when my 7 year old daughter snuggled up next to me to read it too – she would not let me turn the pages till she finished which was cute even though I had to wait.” (Amazon)

The Indisputable Existence of Santa Claus: The Mathematics of Christmas


This book is rated very highly on Amazon; it is one of the best sellers in the Math category. It is ideal for homeschoolers, and for Singaporean primary school students who want to learn in advance, during the school holidays. (American Middle School syllabus should be accessible to upper primary Singaporean students) It is written in a very interesting manner as well.

Everything You Need to Ace Math in One Big Fat Notebook: The Complete Middle School Study Guide (Big Fat Notebooks)


This book is extremely popular in the United States. It is a #1 New York Times bestseller, as well as based on true history. “The phenomenal true story of the black female mathematicians at NASA whose calculations helped fuel some of America’s greatest achievements in space. Soon to be a major motion picture starring Taraji P. Henson, Octavia Spencer, Janelle Monae, Kirsten Dunst, and Kevin Costner.”

Hidden Figures: The American Dream and the Untold Story of the Black Women Mathematicians Who Helped Win the Space Race

What are some useful, but little-known, features of the tools used in professional mathematics?

What's new

A few days ago, I was talking with Ed Dunne, who is currently the Executive Editor of Mathematical Reviews (and in particular with its online incarnation at MathSciNet).  At the time, I was mentioning how laborious it was for me to create a BibTeX file for dozens of references by using MathSciNet to locate each reference separately, and to export each one to BibTeX format.  He then informed me that underneath to every MathSciNet reference there was a little link to add the reference to a Clipboard, and then one could export the entire Clipboard at once to whatever format one wished.  In retrospect, this was a functionality of the site that had always been visible, but I had never bothered to explore it, and now I can populate a BibTeX file much more quickly.

This made me realise that perhaps there are many other useful features of…

View original post 695 more words


We show how to solve “Clock” olympiad questions, which appear often in APMOPS/ SMOPS Olympiad questions.

Question: (SMOPS 2001, Q9)

Between 12 o’clock and 1 o’clock, at what time will the hour hand and minute hand make an angle of 110 degrees?

Full Solution:

We first analyse the hour hand:

It takes 60 min for the hour hand to move 360/12=30 degrees.

30 deg — 60 min

1 deg — 60/30=2 min

x deg — 2x min

(We measure the degree from the 12 o’clock vertical position.)

Next we analyse the minute hand:

It takes 5 min for the minute hand to move 30 degrees.

30 deg — 5 min

1 deg — 5/30=1/6 min

(x+110) deg — (x+110)/6 min

Now, we want the hour hand to be at x deg, and the minute hand to be at (x+110) deg simultaneously:





Since x deg — 2x min, hence the answer is

2x=20 min after 12 o’clock

Ans: 12.20

Tessellations of Pentagons

Tessellation is a cool topic in primary level to PSLE math. Most students will enjoy it even if they hate other types of Math. It is a natural human instinct to be amazed at how different shapes can fit together perfectly to tile the plane.

Apparently, tessellation is going to be removed from the entire PSLE syllabus soon (see That is certainly quite sad for many reasons.

Triangles and quadrilaterals (even irregularly shaped ones) can be easily tessellated. However for pentagons, it is less clear and some pentagons (including the regular pentagon) cannot be tessellated!

For more information read the article here at:

50 Life Lessons from an 80 Year Old

Quite interesting list. Wonder who is the 80 year old?


We absolutely love these pieces of advice from an 80 year old man. 

  1. Have a firm handshake.
  2. Look people in the eye.
  3. Sing in the shower.
  4. Own a great stereo system.
  5. If in a fight, hit first and hit hard.
  6. Keep secrets.
  7. Never give up on anybody. Miracles happen everyday.
  8. Always accept an outstretched hand.
  9. Be brave. Even if you’re not, pretend to be. No one can tell the difference.
  10. Whistle.
  11. Avoid sarcastic remarks.
  12. Choose your life’s mate carefully. From this one decision will come 90 per cent of all your happiness or misery.
  13. Make it a habit to do nice things for people who will never find out.
  14. Lend only those books you never care to see again.
  15. Never deprive someone of hope; it might be all that they have.
  16. When playing games with children, let them win.
  17. Give people a second chance, but not a third.
  18. Be romantic.
  19. Become the most positive and enthusiastic person you know.
  20. Loosen up. Relax. Except for rare life-and-death matters, nothing is as important as it first seems.
  21. Don’t allow the phone to interrupt important moments. It’s there for our convenience, not the caller’s.
  22. Be a good loser..
  23. Be a good winner.
  24. Think twice before burdening a friend with a secret.
  25. When someone hugs you, let them be the first to let go.
  26. Be modest. A lot was accomplished before you were born.
  27. Keep it simple.
  28. Beware of the person who has nothing to lose.
  29. Don’t burn bridges. You’ll be surprised how many times you have to cross the same river.
  30. Live your life so that your epitaph could read, No Regrets
  31. Be bold and courageous. When you look back on life, you’ll regret the things you didn’t do more than the ones you did.
  32. Never waste an opportunity to tell someone you love them.
  33. Remember no one makes it alone. Have a grateful heart and be quick to acknowledge those who helped you.
  34. Take charge of your attitude. Don’t let someone else choose it for you.
  35. Visit friends and relatives when they are in hospital; you need only stay a few minutes.
  36. Begin each day with some of your favourite music.
  37. Once in a while, take the scenic route.
  38. Send a lot of Valentine cards. Sign them, ‘Someone who thinks you’re terrific.’
  39. Answer the phone with enthusiasm and energy in your voice.
  40. Keep a note pad and pencil on your bed-side table. Million-dollar ideas sometimes strike at 3 .
  41. Show respect for everyone who works for a living, regardless of how trivial their job.
  42. Send your loved ones flowers. Think of a reason later.
  43. Make someone’s day by paying the toll for the person in the car behind you.
  44. Become someone’s hero.
  45. Marry only for love.
  46. Count your blessings.
  47. Compliment the meal when you’re a guest in someone’s home.
  48. Wave at the children on a school bus.
  49. Remember that 80 per cent of the success in any job is based on your ability to deal with people.
  50. Don’t expect life to be fair.

Biological evidence that Jesus actually was born in December

The type of sheep in Israel is the “Awassi” sheep, a type of desert sheep. In Israel, the principal lambing season is December through January. It makes sense, since the winter in Israel is around 8 degrees Celsius, which is not too freezing cold. Something interesting to know!

Source: Aleteia

Long ago, I accepted the idea that December 25 was probably not the actual date of Christ’s birth, that the real date was unknown but probably in the spring. Knowing the exact date doesn’t really impact the liturgical celebration, after all. It was just one more sad thing about being an adult, one more little bit of wonder gone from life.

Since then, I’ve become well acquainted with the historical evidence in favor of a date of December 25. The date can be derived historically from the dating of Zechariah’s entry into the temple to burn incense. It can also be derived theologically from the ancient tradition that a great prophet entered and left the world on the same calendar day. Thus, the Annunciation was determined to have occurred on the same day as the crucifixion, March 25. December 25 naturally follows nine months later. They are good arguments, held to strict standards of historical research and logic, within their own fields.

But neither ever quite satisfied my desire for something really concrete. One continual objection was that the shepherds in the fields at night were presumed to be attending to the dropping of lambs. And lambs didn’t drop in December. Lambs dropped in the spring, not the winter.

So, when yet another person asked “Why do we celebrate Christmas in December if lambs are born in the spring?” instead of explaining the significance of March 25, I suddenly wondered: ARE lambs actually born in the spring in Israel? Can I find out?

The Awassi sheep is a desert sheep, a fat-tailed breed that has existed in the Middle East for an estimated 5,000 years. It is the only indigenous breed of sheep in Israel. They are raised for wool, meat, and milk. Awassi sheep breed in the summer and drop lambs in the winter, when there is sufficient pasture for the ewes in milk. In Israel, the principal lambing season is December through January.

This is practical, I thought. This is fact. This is biology.

$1 (Bid) R2-D2 Droid for kids interested in Robotics

For $1, you can bid on this R2-D2 Droid inventor kit at Hachi.Tech ( This is a very reliable online site that is owned by Challenger.

The Droid is perfect for those kids who want to learn more about robotics during their holidays. You may also purchase it from Amazon:

Star Wars Droid Inventor Kit

America’s Lost Einsteins


Millions of children from poor families who excel in math and science rarely live up to their potential—and that hurts everyone.

Consider two American children, one rich and one poor, both brilliant. The rich one is much more likely to become an inventor, creating products that help improve America’s quality of life. The poor child probably will not.

That’s the conclusion of a new study by the Equality of Opportunity project, a team of researchers led by the Stanford economist Raj Chetty. Chetty and his team look at who becomes inventors in the United States, a career path that can contribute to vast improvements in Americans’ standard of living. They find that children from families in the the top 1 percent of income distribution are 10 times as likely to have filed for a patent as those from below-median-income families, and that white children are three times as likely to have filed a patent as black children. This means, they say, that there could be millions of “lost Einsteins”—individuals who might have become inventors and changed the course of American life, had they grown up in different neighborhoods. “There are very large gaps in innovation by income, race, and gender,” Chetty told me. “These gaps don’t seem to be about differences in ability to innovate—they seem directly related to environment.”

Appeal from RGS to NYGH (Success)

Just read that appealing to transfer from RGS (Raffles Girls School) to NYGH (Nanyang Girls High) is possible:

dd moved out of rgs to nygh. she got 262+2. she appealed to nygh and was granted interview on Thursday. was given the good news after her interview.

nygh her first choice. she probably missed by decimal points. thus tried to appeal. was telling her both schools are equally good thus if not successful for nygh in her appeal, just move on 🙂

NYGH cut off point has been higher than RGS for the recent past years. It is the opposite situation of their boy school counterparts: RI cut off point is usually higher than Hwa Chong (Chinese High).

From Medical Doctor to Math Professor

Just read about this rather amazing biography: From a medical doctor, Hau-tieng Wu pursued a Ph.D. in math, and is now a math professor at Duke. Quite an interesting transition, that is quite rare, possibly less than 100 such cases in the world. Most mathematicians know little about medicine, and most medical doctors know little about math. It is rare to have someone know both fields.

Listen to your heartbeat with a stethoscope and you’ll hear a rhythmic lub-dub, lub-dub that repeats roughly 60 to 100 times a minute, 100,000 times a day.

But the normal rhythm of a healthy heart isn’t as steady as you might think, says Hau-tieng Wu, M.D., Ph.D., an associate professor of mathematics and statistical science who joined the Duke University faculty this year.

Rather than beating like a metronome, heart rhythm varies depending on whether you’re asleep or awake, sitting or jogging, calm or driving in rush hour. Breathing rate, brain activity and other physiological signals vary in much the same way, Wu says.

He should know. Before becoming a professor, Wu trained as a medical doctor in Taiwan. In his fifth year of medical school he was doing clinical rotations in the hospital when he was struck by the complex fluctuations in heart rhythm during anesthesia and surgery.

Where some saw noisy patterns — such as the spikes and dips on an electrocardiogram, or ECG — Wu saw hidden information and mathematical problems. “I realized there are so many interesting medical data that aren’t fully analyzed,” Wu said.

When a patient is in the hospital, sensors continuously monitor their heart rate and rhythm, breathing, oxygen saturation, blood pressure, brain activity and other vital signs.

The signals are sent to computers, which analyze and display the results and sound an alarm if anything veers outside normal ranges.

An ECG, for example, translates the heart’s electrical activity into a squiggly line of peaks and valleys whose frequency, size and shape can change from one moment to the next.

Wu is using techniques from differential geometry and harmonic analysis to detect patterns hidden in these oscillating signals and quantify how they change over time.

His methods have been applied to issues in cardiology, obstetrics, anesthesiology, sleep research and intensive care.

Why do Americans have such trouble with fractions—and what can be done?

Fractions indeed are challenging for kids from ages 7-12. Probably the mental picture of a pie/pizza helps. Also, making fractions to the same denominator is a technique that once mastered will make addition/subtraction of fractions much easier.

Source: Scientific American

Many children never master fractions. When asked whether 12/13 + 7/8 was closest to 1, 2, 19, or 21, only 24% of a nationally representative sample of more than 20,000 US 8th graders answered correctly. This test was given almost 40 years ago, which gave Hugo Lortie-Forgues and me hope that the work of innumerable teachers, mathematics coaches, researchers, and government commissions had made a positive difference. Our hopes were dashed by the data, though; we found that in all of those years, accuracy on the same problem improved only from 24% to 27% correct.

Such difficulties are not limited to fraction estimation problems nor do they end in 8th grade. On standard fraction addition, subtraction, multiplication, and division problems with equal denominators (e.g., 3/5+4/5) and unequal denominators (e.g., 3/5+2/3), 6th and 8th graders tend to answer correctly only about 50% of items. Studies of community college students have revealed similarly poor fraction arithmetic performance. Children in the US do much worse on such problems than their peers in European countries, such as Belgium and Germany, and in Asian countries such as China and Korea.

Kids struggling in math? Try this “magic” method from Japan (VIDEO)

URL: Aleteia

It’s an Asian-style mathematics similar to Common Core that’s actually fun to do.

Confession time: I’m terrible at math. I don’t just mean like “struggled with calculus” bad, I mean like “had to watch YouTube videos to relearn long division in order to help my 4th grader with her homework” bad. I don’t know my times tables, except the easy ones. I can’t do fractions or percentages. I count on my fingers. 

It’s sad and shameful, and I was determined that my children would not share my fate. So when my oldest daughter was 5, I bought the insanely expensive starter package from Right Start Math and set about teaching her how to do math the right way.

It did not go well, nor did it last long. I found even the very simple activities baffling because I couldn’t grasp the intention. It was like trying to teach my daughter a foreign language I didn’t know.

However, my abject failure to understand it did not diminish my enthusiasm for the Asian method of mathematics. One of the reasons I like Common Core math is because there are lots of similarities. If you’ve never been exposed to the wonder of Asian-style mathematics, allow me to remedy that for you:

Check out the video on the page, it is quite amazing. (Japanese method of multiplying with lines).

URL: Aleteia

Asus VivoBook E12 Review (Singapore)

[$369.00](▼27%)[ASUS]ASUS VivoBook E12 E203NA / lightweight 11.6-inch laptop /1 Years International warranty


I think currently this is the best budget laptop below $400. Size wise it is about the size of an iPad (around A4 size). The 4GB RAM makes it better and faster than those budget laptops with only 2GB RAM.

From the reviews on Qoo10, it is good for everyday uses such as YouTube, Microsoft Word etc.

Buying from Qoo10 has free shipping, Free Xiaomi Smart Scale and also “Free MS Office 365 Personal 1 Year” Pre-Installed Worth $98. (as of Nov 2017, check the updated terms and conditions to see if it is still valid)


[$369.00](▼27%)[ASUS]ASUS VivoBook E12 E203NA / lightweight 11.6-inch laptop /1 Years International warranty



US students aren’t bad at math—they’re just not motivated

It turns out that US students aren’t that bad at math, they just have no motivation to do the PISA test properly. (The PISA test is an external test that has no bearing on their school academic results.)


It’s no secret that young Americans perform poorly on math and science tests, especially compared to their peers in countries like Singapore, Korea and China, where math scores are among the highest in the world. Now, a working paper surfaces a fundamental reason for that weak performance: American students are simply not trying hard enough.

In the latest results of the Programme for International Student Assessment (PISA), US students ranked roughly average among the 75 participating countries. The PISA tests, administered by the OECD every three years, assess 15-year-olds around the world on math, science and reading. Governments and policy makers point to the outcomes when making the case for education reform.

The researchers also ran a simulation, and found that if the 15-year-olds in the US had been given the same cash bonus in 2012 when taking the assessment, America would have ranked 19th in the PISA math test instead of 36th among 65 nations.

Hokkaido University Photo Gallery and Trip Guide

These photos are from a trip to Hokkaido University in August 2017. Hokkaido University is just walking distance away from Sapporo station, and is worth spending an afternoon there. Admission is free. Even in the hottest summer, Hokkaido has cool weather, ranging from around 17 degrees to 25 degrees.

Some places to visit are the Poplar Avenue, and also the Hokkaido University Museum (check the time, it can close as early as 5pm.) Also, the statue of one of the founders, Dr. William Smith Clark, is also a good place to visit. There is also a monument of the school motto: “Be ambitious!” (少年よ、大志を抱け )

Raven/crows can be spotted all around the campus of Hokkaido University. They are the largest raven I have ever seen, about the size of a small eagle.

A random willow tree in Hokkaido University.
Students can be found reading or just sitting in the field. I even spotted one student practicing violin.
The famous Poplar Avenue of Hokkaido University.

A small stream running through the middle of Hokkaido University.
Hokkaido University Museum: A Town Plan of Hokkaido University.
Poster in Hokkaido University Museum.
Chemical Structure in Hokkaido University Museum.
The Hokkaido University Museum is 3 storeys. It is quite big actually, there should be enough things to see for around one hour or more.

The below are some photos from inside the Hokkaido University Museum.


A signboard in a garden.


A large raven in Hokkaido University. It is much larger than it looks in the photo.
The famous motto of Hokkaido University: “Be ambitious!” (original words: “boys, be ambitious!”)

8 Facts About Infinity That Will Blow Your Mind

Nice article on infinity. Also little known is the fact that the symbol of infinity was introduced by clergyman and mathematician John Wallis, hundreds of years ago in 1655. Although not well-known, John Wallis was a talented individual as can be deduced from his biography. His works include integral calculus, analytic geometry, and collision of bodies. He was the one who coined the term “momentum”.

Source: ThoughtCo

Infinity has its own special symbol: ∞. The symbol, sometimes called the lemniscate, was introduced by clergyman and mathematician John Wallis in 1655. The word “lemniscate” comes from the Latin word lemniscus, which means “ribbon,” while the word “infinity” comes from the Latin word infinitas, which means “boundless.”

Wallis may have based the symbol on the Roman numeral for 1000, which the Romans used to indicate “countless” in addition to the number. It’s also possible the symbol is based on omega (Ω or ω), the last letter in the Greek alphabet.

Fave (previously Groupon) FREE Promo Code

FREE Promo Code discount:

Fave (previously known as Groupon) is a fantastic deals site that offers many discounts. It includes discounts to many restaurants/buffets in Singapore.

Quite many famous restaurants are offering discounts there: e.g. Dancing crab, Mouth Restaurant, Ah Yat Abalone, JUMBO Seafood, Charcoal Thai and more.

Here is a free Promo Code, for first-time users who sign up on Fave: KJMDZ

You will get a further first-timer’s discount upon sign-up.

Click here to sign up with the Promo Code:

Increased Tax for US PhD Students

This is very bad news. Graduate students are already poor (stipend is already low to begin with and fixed for the duration of study). A very bad policy for grad students.

The tax overhaul passed Thursday by House Republicans could cost graduate students thousands of dollars, prompting a backlash from students and university leaders who say the proposal could make graduate degrees unaffordable, especially for low-income students.

A provision tucked away in the House’s bill would count the tuition discounts given to many grad students as income, meaning that students would pay taxes on tens of thousands of dollars that they never see. The bill would double or even triple many students’ taxes.

Source: Buzzfeed

Source: PhD Comics