How to break 158,962,555,217,826,360,000 codes ?
There are 158,962,555,217,826,360,000 possibilities of codes in the German Enigma Machine:
Flaw cracked by the genius Mathematician Alan Turing (Father of Artificial Intelligence) :
“A key can never be itself” — this is ‘the straw that breaks the camel’s back’, a critical clue to break the 158,962,555,217,826,360,000 possible codes !
These 3 jobs all require Math: Actuary, Statistician and Mathematician.
Others good to have Math : Software Engineer, Systems Analyst, Data Scientist.
I have written a guest post on https://chinesetuition88.wordpress.com on the very fascinating Chinese Remainder Theorem and its History (韩信点兵). Do check it out, you will be amazed at the genius of Chinese General Han Xin.
Students who are interested in Chinese Tuition may check out https://chinesetuition88.wordpress.com for more details.
淮安民间传说着一则故事——“韩信点兵”，其次有成语“韩信点兵，多多益善”。韩信带1500名兵士打仗，战死四五百人，站3人一排，多出2人；站5人一排，多出4人；站7人一排，多出6人。韩信马上说出人数：1049。
Translation:
In Ancient China, there was a General named Han Xin, who led an army of 1500 soldiers in a battle. An estimated 400-500 soldiers died in the battle. When the soldiers stood 3 in a row, there were 2 soldiers left over. When they lined up 5 in a row, there were 4 soldiers left over. When they lined up 7 in a row, there were 6 soldiers left over. Han Xin immediately said, “There are 1049 soldiers.”
Amazing! How did Han Xin do that?
Han Xin was not only a brilliant mathematician and general, he was also a very magnanimous guy full of wisdom.
Once, when he was suffering from hunger, he met a woman who provided him with food. He promised to repay her for her kindness after he had made great achievements in life, but it was rebuffed by her…
View original post 342 more words
Answer (scroll down)
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View original post 33 more words
For more logic puzzles, check out:
[Hint]: Think out of the box…
Answer below (scroll down)
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Answer:
1 + 13 +…
View original post 57 more words
We all know by now Singapore Math is not easy, but here is the viral Singapore Math problem that took the world by storm!
Question:
Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl marks 10 possible dates: May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, or August 17.
Then Cheryl tells Albert the month of her birthday, but not the day. She tells Bernard the day of her birthday, but not the month. Then she asked if they can figure it out.
Albert: I don’t know when Cheryl’s birthday is, but I know Bernard doesn’t know either.
Bernard: At first I didn’t know when Cheryl’s birthday is, but now I know.
Albert: If you know, then I know too!
When is Cheryl’s birthday?
Source: http://www.vox.com/2015/4/15/8420577/cheryls-birthday-singapore-math
There is a nice Numberphile video about it too.
Do give it a try! (The fun is in trying to solve the question)
Also, another fun part is sending this question to your friends!
Also see: Meet the mathematics lecturer behind ‘Cheryl’s birthday’ puzzle – See more at: http://www.straitstimes.com/news/singapore/more-singapore-stories/story/meet-the-mathematics-professor-behind-cheryls-birthday-p#sthash.qKZZtwpk.dpuf
To be honest, though Cheryl’s birthday puzzle is difficult, there are more challenging logic puzzles around. For a good challenge (and good practice), check out Puzzle Baron’s Logic Puzzles
Translation:
[The world’s most mysterious number is 142857.]
It is found in the ancient Egyptian Pyramids.
142857 x 1=142857
142857 x 2=285714
142857×3=428571
142857×4=571428
142857×5=714285
142857×6=857142
142857×7=999999
Amazing? Each multiple is a cyclic permutation of the original numer 142857.
You may read more about Egyptian mathematics in this wonderful book: Count Like an Egyptian: A Hands-on Introduction to Ancient Mathematics
Although in Singapore currently doctors and lawyers are the top jobs, the trend is changing, starting with the most technologically advanced country – America. 3 of the top jobs in America are about Math. As the world becomes more dependent on technology (and hence mathematics), Mathematics will play a more prominent role in the global scene. Eventually the change will come to Singapore too, as more and more jobs require mathematical skills.
According to our Law Minister Mr Shanmugam, Singapore is facing a glut (excessively abundant surplus) of lawyers, which means that Singapore may not have so many jobs for lawyers. “The study of law provides an excellent training of the mind, so I don’t want to be seen as discouraging people… but you have to have a realistic understanding of the market, the economy, the total structure,” said Mr Shanmugam – See more at: http://www.straitstimes.com/news/singapore/more-singapore-stories/story/singapore-facing-glut-lawyers-shanmugam-20140817#sthash.BojzeqhX.dpuf
Hence, young students may want to consider a new discipline that is Math related, like Actuary, Math, or Statistics. To read up more about what true Mathematics is (it is very different from high school mathematics, where students just practice differentiation and integration), check out this book How to Think Like a Mathematician: A Companion to Undergraduate Mathematics
Site: http://www.businessinsider.sg/best-jobs-of-2015-2015-4/#.VTEI7ZPoaKg
Perhaps if you had known that some of the best jobs of 2015 would require mathematical skills, you would’ve paid more attention in your high school algebra class.
Professions like actuary, mathematician, and statistician are three of the top jobs in America right now, according to CareerCast.com, a career guidance website that just released its 27th annual Jobs Rated report.
“Jobs in mathematics rank among the nation’s best jobs because they are financially lucrative, offer abundant opportunities for advancement, and provide the opportunity to do great work in a supportive environment,” says Tony Lee, publisher of CareerCast.com, in a press statement.
Here are the 10 best jobs of 2015:
| 2015 Rank | Job Title | Mid-level Income |
| 1 | Actuary | $94,209 |
| 2 | Audiologist | $71,133 |
| 3 | Mathematician | $102,182 |
| 4 | Statistician | $79,191 |
| 5 | Biomedical Engineer | $89,165 |
Read more at: http://www.businessinsider.sg/best-jobs-of-2015-2015-4/#.VTEI7ZPoaKg
Intriguing review (Chinese) of FLT by a non-mathematician. He aims to convey the beauty of Mathematics to students, who unfortunately treat Math as a tool to pass exams from PSLE, O and A level, university math course, then ditch Math upon graduation. Math is the beauty of the universe.
费马大定理 Fermat’s Last Theorem (FLT): 17世纪业余数学家法国大法官费马开的一个”玩笑”, 推动350年来现代数学突飞猛进。
FLT 数学长征英雄人物:
1. Fermat (费马 1601@ Toulouse, France)
2. Galois (伽罗瓦): Group Theory (群论)
3. Gauss (高斯)
4. Cauchy (柯西) Lamé (拉梅) Kummer (库马)
5. Solphie Germain
6. Euler (欧拉)
7. Taniyama (谷山丰), Shimura (志村五郎)
集大成:
8. Andrew Wiles (怀尔斯) (证明@1993 -1995)
1.Elliptic Curve (椭圆曲线)
2. Modular Form (模形式)
3. Fermat Last Theorem (费马大定理)
(1) = (2) = (3)
Music and math: The genius of Beethoven – Natalya St. Clair:
Music = Math
With humor and charm, mathematician Eduardo Sáenz de Cabezón answers a question that’s wracked the brains of bored students the world over: What is math for? He shows the beauty of math as the backbone of science — and shows that theorems, not diamonds, are forever. In Spanish, with English subtitles.
Yes, indeed, 1000 years from now, students will still be learning Pythagoras’ Theorem, while other fragments of human knowledge would have faded away.
Check out also this book: Arithmetic and Algebra Again: Leaving Math Anxiety Behind Forever
Check out this really funny video on a April Fools Prank during a Math Class!
The teacher played a trick on his math class for April Fool’s Day. In this one, he’s showing a “homework help” video that gets some trigonometry wrong.
Looking for more Math Jokes? Check out the book below!
Math Jokes 4 Mathy Folks
Strongly recommended free excellent MIT Math for high school, undergrads/grads and any self-study learners.
Thanks Prof. Gilbert Strang for the unselfish sharing.
http://ocw.mit.edu/faculty/gilbert-strang/
I find extremely pleasure when I discovered his brilliant lecture notes in “Generating Function” – a Discrete Math technique for computing sequencing using function, and the application in complex Combinatorics. Download here:
Example:
March’s Problem of the Month was a tough one on permutations. Only six people solved it! (Site: http://www.fen.bilkent.edu.tr/~cvmath/Problem/problem.htm)
The question goes as follows:
In each step one can choose two indices 
The official solution is beautiful and uses induction.
Personally, I used a more brute force technique to get the same answer using equivalence class theory which I learnt in my first year of undergraduate math! It is not so bad in this question, since n is only 100, but for higher values of n the approach in the official solution would be better.
If you are looking for recommended Math Olympiad books, check out this page. In particular, if you are looking for more Math Olympiad challenges, do check out this book Mathematical Olympiad Challenges
Recently, Professor Jo Boaler released her new book What’s Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success
The minute it came out, it became an instant best seller on Amazon. Currently, there are some issues on Math education in the United States, due to the very controversial syllabus called Common Core. Professor Jo Boaler attempts to address these controversies and give suggestions and advice to parents.
I totally agree with Professor Jo’s viewpoint that the first step to engage students in math learning is via practical means and showing them how mathematics is useful and relevant to their lives. Next is to always adopt a “growth mindset”, that no matter how weak or strong a child is in math, it is always possible to improve. Just having this mindset makes a huge difference. I took Prof. Jo Boaler’s online course on “How to Learn Math“, and what she said actually makes perfect sense. Hope a new generation appreciative of math will emerge due to new research on how to best learn Math, which Prof. Jo Boaler (PhD in Math Education) is an expert in.
Without further ado, I will link Prof. Jo Boaler’s introduction to her own book:
Hi Everyone,
I wanted you to be the first to know that my new book: What’s Math Got to Do With It:? How Teachers and Parents Can Transform Mathematics Learning and Inspire Success has just hit the bookstores and of course Amazon and other online outlets.
You can now get a copy here: What’s Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success
The changes from the original book include:
2 new chapters
A focus on mindset
Ideas for the Common Core
An infusion of new research through the book
Why not buy the book for your principal? Or your colleagues? your family? your students’ parents? or others who you think may need to understand the nature of good mathematics teaching? You may need people to know the research evidence behind what you are doing, as well as get some new ideas yourself.
For youcubers in the UK there will be a new edition of The Elephant in the Classroom coming out in the Autumn, we will let you know when, of course.
I also wanted you to know about some book signings that are planned:
Friday April 3 Portsmouth, New Hampshire. The Exeter High School Auditorium 7-8.30pm talk followed by book signing. See:
http://www.seacoastonline.com/article/20150330/NEWS/150339887/101019/NEWS
At NCSM:
Monday April 13th Boston, NCSM Following Jo’s keynote talk
At NCTM:
Thursday April 16th, 11.30 After Jo’s networking session.
We will also be arranging a book signing in the San Francisco bay area soon too.
I hope to see you at one of them. Below is our youcubed team reading the book yesterday 🙂
Viva La Revolution
Jo
Previously, the first YouTube video wasn’t working. I have added a new link to the interesting “Looking at Perfect Shuffles” video. 🙂
A magic trick based on the “Perfect Shuffle”. Featuring Professor Federico Ardila. I watched his videos on Hopf Algebras while learning the background material for my honours project on Quantum Groups.
Mathemagician Persi Diaconis discusses which is the best way to shuffle: Overhand shuffle, Riffle Shuffle, or “Smoosh” Shuffle? Watch the video to find out!
Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks
Official Website: http://www.nushigh.edu.sg/admission-n-outreach/admissions/eligibility-n-admissions-process
The official website, unfortunately, doesn’t tell much about how the NUS High Selection Test / DSA is like, in particular the format of the exam.
However, from online sources from students who took the test, we can have a glimpse of what the NUS High Selection Test (DSA) is like.
Disclaimer: I have not taken the NUS High Selection Test (DSA) before, and I am only listing down suggested format of the tests based on the online sources. I have taken the GEP Selection Test (both round 1 and round 2) though, at Primary 3.
Source 1: http://wwwdontmesswith6a.blogspot.sg/2011/06/nus-high-selection-test.html
This is a highly reliable blog post by the sister of Lim Jeck, a highly skilled Math Olympiad Participant who has achieved perfect score at IMO. From the blog post, we can tell that:
Read more at http://wwwdontmesswith6a.blogspot.sg/2011/06/nus-high-selection-test.html to get an idea of the original post and how the Science NUS High DSA (supposedly more difficult than the Math NUS High DSA) is like.
To deal with difficult NUS High DSA problems (last few questions of the Selection Test), most likely the student has to be trained in Math Olympiad. A book like The Art of Problem Solving Volume 1: The Basics AND Basics Solution Manual (2 Volume Set
If you are also interested in preparing for GEP (Primary 3 or Secondary 1 intake), do check out my most popular page on Recommended Books for GEP.
Other blogs with info on the NUS High DSA Selection Test:
Update (2016): Check out this Pattern Recognition (Visual Discrimination) book that is a guided tutorial for training for GEP / DSA Tests!
A magic trick based on the “Perfect Shuffle”. Featuring Professor Federico Ardila. I watched his videos on Hopf Algebras while learning the background material for my honours project on Quantum Groups.
Mathemagician Persi Diaconis discusses which is the best way to shuffle: Overhand shuffle, Riffle Shuffle, or “Smoosh” Shuffle? Watch the video to find out!
Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks
Recently, after taking the Coursera course on Cryptography, I had a better appreciation of mathematics and the role of cryptography in our modern society.
I was pleased to read this article Quantum compute this: Mathematicians build code to take on toughest of cyber attacks, and Washington State University mathematicians have designed an encryption code capable of fending off the phenomenal hacking power of a quantum computer.
The quantum computer, though not yet invented, is widely believed to be available soon in the next few years. In the hands of hackers, the quantum computer would be a formidable weapon as current cryptographic methods are extremely vulnerable to the quantum computer as it can factor numbers extremely quickly, leading to number theoretic codes being broken.
Quantum computers are near
Quantum computers operate on the subatomic level and theoretically provide processing power that is millions, if not billions of times faster than silicon-based computers. Several companies are in the race to develop quantum computers including Google.
Internet security is no match for a quantum computer, said Nathan Hamlin, instructor and director of the WSU Math Learning Center. That could spell future trouble for online transactions ranging from buying a book on Amazon to simply sending an email.
Hamlin said quantum computers would have no trouble breaking present security codes, which rely on public key encryption to protect the exchanges.
In a nutshell, public key code uses one public “key” for encryption and a second private “key” for decoding. The system is based on the factoring of impossibly large numbers and, so far, has done a good job keeping computers safe from hackers.
Quantum computers, however, can factor these large numbers very quickly, Hamlin said. But problems like the knapsack code slow them down.
Fortunately, many of the large data breaches in recent years are the result of employee carelessness or bribes and not of cracking the public key encryption code, he said.
Hence, when many people say mathematics is useless, they are actually extremely wrong, as mathematics permeates every aspect of life! Even though maths like calculus is not directly used in everyday life, it is part of our phone, computer, and every part of the modern lifestyle.
Kudos to the mathematicians who have averted a world disaster, before quantum computers are even invented!
If you are interested in what a quantum computer is, and what it can do (it is so powerful that whoever has one would hold the keys to the entire internet), check out this book Schrödinger’s Killer App: Race to Build the World’s First Quantum Computer
Written by a renowned quantum physicist closely involved in the U.S. government’s development of quantum information science, Schrödinger’s Killer App: Race to Build the World’s First Quantum Computer presents an inside look at the government’s quest to build a quantum computer capable of solving complex mathematical problems and hacking the public-key encryption codes used to secure the Internet. The “killer application” refers to Shor’s quantum factoring algorithm, which would unveil the encrypted communications of the entire Internet if a quantum computer could be built to run the algorithm. Schrödinger’s notion of quantum entanglement—and his infamous cat—is at the heart of it all.
This is an interesting question on vector subspaces (a topic from linear algebra):
Question:
If V and W are 2-dimensional subspaces of 
(A) 1 only
(B) 2 only
(C) 0 and 1 only
(D) 0, 1, and 2 only
(E) 0, 1, 2, 3, and 4
To begin this question, we would need this theorem on the dimension of sum and intersection of subspaces (for finite dimensional subspaces):
Note that this looks familiar to the Inclusion-Exclusion principle, which is indeed used in the proof.
Hence, we have 
Thus, 
Answer: Option D
If you are looking for a lighthearted introduction on linear algebra, do check out Linear Algebra For Dummies
Amazon just informed me of a new book which is the #1 New Release Math Book on Amazon!
The book is titled: Professor Stewarts Incredible Numbers
At its heart, mathematics is about numbers, our fundamental tools for understanding the world. In Professor Stewart’s Incredible Numbers, Ian Stewart offers a delightful introduction to the numbers that surround us, from the common (Pi and 2) to the uncommon but no less consequential (1.059463 and 43,252,003,274,489,856,000). Along the way, Stewart takes us through prime numbers, cubic equations, the concept of zero, the possible positions on the Rubik’s Cube, the role of numbers in human history, and beyond! An unfailingly genial guide, Stewart brings his characteristic wit and erudition to bear on these incredible numbers, offering an engaging primer on the principles and power of math.
Previously, I read Galois Theory, Third Edition (Chapman Hall/Crc Mathematics)
I read the Third Edition, featured above, but lately there is a newer and better fourth edition: Galois Theory, Fourth Edition
Most people don’t encounter Tensors (the higher level advanced version of Matrices) until they reach senior undergraduate, or even graduate level.
The best explanation I have ever seen, comes from this video by the author of A Student’s Guide to Vectors and Tensors
This YouTube video is watched over 200,000 times, a very commendable achievement for a math video!
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples of such relations include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of numerical values. The order (also degree) of a tensor is the dimensionality of the array needed to represent it, or equivalently, the number of indices needed to label a component of that array. For example, a linear map can be represented by a matrix (a 2-dimensional array) and therefore is a 2nd-order tensor. A vector can be represented as a 1-dimensional array and is a 1st-order tensor. Scalars are single numbers and are thus 0th-order tensors. The dimensionality of the array should not be confused with the dimension of the underlying vector space.
If you have some programming knowledge, you may view tensors as a type of multidimensional array. A more mathematical abstract way can be achieved by defining tensors in terms of elements of tensor products of vector spaces, which in turn are defined through a universal property.
Cool? The word “tensor” really strikes me as a word that is really sophisticated and complicated!
Given the equation of a plane: ax+by+cz=D, or in vector notation 
(When a question asks for the distance of a plane to the origin, by definition it means the shortest distance.)
One way to derive the formula is this:
Let X be the point on the plane nearest to the origin.
Furthermore, X lies on the plane, hence we have
Using the formula for dot product, we can get
Since 
Thus, we have 
The shortest distance from the point X to the origin is then
Ans: Shortest distance from point to plane is
If you are looking for a short summarized H2 Maths Notes, with Prelim Papers to practice, do check out our Highly Condensed H2 Maths Notes!
We will continue our series on GEP Questions. To learn more about Recommended Books for GEP, to practice GEP Questions, visit the link here.
Today, we will discuss the quintessential GEP Question: The Gauss Trick. This GEP question illustrates the fact that giftedness can be trained to a large extent.
Question:
Find the sum of 1+3+5+7+…+95+97+99.
Solution will be below after this text, scroll down after you are ready to see the answer!
If a 9 year child can solve this on his/her first attempt (i.e. see the question for the very first time), then the child is actually at the level of Carl Friedrich Gauss, the legendary mathematician! When Gauss was a young kid, his teacher set the class a difficult question, 1+2+3+…+99+100, hoping to keep the children quiet and busy while he could have some time to relax. Little did he expect Gauss to come up with the right answer minutes later.
Most 9 year old kids would not be able to solve this on their first try. I wouldn’t be able to solve it correctly even if given an hour when I was a kid! If the children have seen it before and practiced, that is a different story, as it becomes so easy even for a 7 year old kid. Hence, practicing GEP Questions actually leads to an indirect boost of IQ in this manner! The transition from ignorance to knowledge, leading to increased intelligence, can be accomplished by practice! To practice more of these GEP Questions, check out the Math Olympiad Recommended Books page. As a tutor, I know that this Gauss Trick is a “must-know” question for students aiming high for school Math / GEP Selection Test, since almost every kid knows this nowadays, and it is highly popular in tests.
In fact, this technique (Gauss Trick) commonly tested in GEP Questions can be used all the way to JC and beyond! In JC it is covered under the topic of AP/GP (Arithmetic Progression and Geometric Progression).
Solution:
Now, for the solution. This sum is the famous arithmetic progression, where each term differs from the next by a fixed constant. In this case, the constant is 2.
Some solutions use a number pair matching, which can be problematic to explain when the number of terms is odd, but still works nevertheless. We can use another method of writing the sum backwards.
First we note that there are 50 terms in this series. We can know that either by noting that they are the odd numbers from 1 to 100, and half of the numbers are odd, hence there are 100/2=50 terms. Alternatively, we can note that 1=2(1)-1, 3=2(2)-1, 5=2(3)-1, …, 99=2(50)-1, where the number in the brackets acting like a counter.
1+3+5+7+…+95+97+99
99+97+95+…+5+3+1
Note that each pair, 1+99, 3+97, 5+95 actually add up to the same thing, i.e. 100.
Adding up the two expressions above, we get 100×50=5000.
Dividing that by two (since we double counted), we will get 5000/2=2500.
Ans: 2500
Hope you enjoyed solving this question!
Check out this amazing book on Gauss: The Prince of Mathematics: Carl Friedrich Gauss
Gauss is a true Math genius, and you can read more about his life in this interesting biography! This historical narrative will inspire young readers and even curious adults with its touching story of personal achievement.
There is a great deal of confusion over how to calculate your JC Ranking Points. It would be calculated automatically after your exams, but it is good to learn how to calculate it by yourself!
New: JC Ranking Points Calculator App!
I have just written an automatic JavaScript App to calculate JC Ranking or UAS Score. To use, just make sure your JavaScript is enabled in your browser, and you are ready to calculate!
I also have written an article (one of my most popular posts) on how to calculate JC Ranking Points:
Basically, the JC Ranking Point system is a numerical system where students take their three best H2 subjects and assign a numerical value to their grade (A: 20, B: 17.5, C: 15 and so on in intervals of 2.5). Then, their next best H2 subject (or H1 subject) would be assigned half the points (A: 10, B: 8.75, C: 7.5, and so on). PW (Project Work) and GP (General Paper) will also be following the H1 point scheme. (A: 10, B: 8.75, C: 7.5, and so on)
If a student takes Mother Tongue (MTL), it will also follow the H1 point scheme, and the total points will be scaled to 90 points. (by dividing by 100 and then multiplying by 90).
This system gives a numerical way to distinguish between students for admission to university. E.g. just by grades, it is hard to tell who did better: a student who got ABC/A, or ABB/B, but ranking points may provide a distinguishing factor.
If you are still in JC or even doing O Levels, and checking this Rank Point System out, you still have time to change your destiny! Do check out the Highly Condensed Math Notes bundled with Exam Papers for practice!
You may also want to read some Motivational Books to motivate yourself to study. If you are in JC, you are one of the brightest in Singapore already, but however one still needs immense motivation to push through the two intensive years of JC and emerge victorious.
So it is official, scientists have discovered that a minimum of 1 hour of homework per day is needed to be better at Math!
Today I had a bad shock when my trusted TI-84 Plus Pocket SE Calculator could not be turned on. I just purchased it last year, and TI-84 Calculators are not exactly cheap as every JC student in Singapore would know.
On Amazon, the Texas Instruments TI-84 Plus Silver Edition Graphing Calculator, Silver
First, I did the common sense approach of changing all the AAA and even the button LR44 battery. No luck…
I noticed something strange happened to one of the batteries. A layer of yellow crust has coated the battery case. (I was using Ikea batteries)
I followed the 5 steps I found on the TI-84 troubleshooting site:
After trying all the 5 steps, I still couldn’t get the TI-84 to switch on.
I then used a piece of tissue to wipe of the battery contacts which were coated with the battery leak. Then, I tried step 5 again.
Thank God, the TI-84 sprung back to life again, and I can save $200 for now…
For those taking H2 Maths, do check out my H2 Maths Condensed Notes with Bundled Prelim Papers!
The Tap Question is another one of those questions that only involve fractions and whole numbers, and hence technically within the grasp of a 9 year old kid sitting for the GEP Screening Test.
However, looks are highly deceiving, and whoever tries the Tap Question for the very first time is highly likely to get stuck. (I was one of them years ago!) The Tap Question is highly popular as a challenging question, due to its psychological nature it is a hard question to grasp. This is a question you wouldn’t want to meet for the first time in the GEP Screening Test. However, if you know how to solve it, it is easy as a piece of cake, and you will be able to solve it during the GEP Screening Test no matter how they twist the question.
A fish tank is connected to three taps.
Tap A can fill the tank in 2 hours.
Tap B can fill the tank in 3 hours.
Tap C can drain the tank in 6 hours.
If all three taps are turned on at the same time, how long would it take to fill the empty fish tank?
Do try out the question before looking at the answer below!
There is a huge difference solving a question for the first time, and solving a question that one has seen before. To familiarize yourself with GEP Screening Test questions that can come out, do check out our Recommended GEP Books. Reading one of those books would increase your child’s repertoire of questions, and hence boost your child’s IQ indirectly. Also, do check out Math Olympiad books, as it is well known that GEP Screening Test Math questions do incorporate some Math Olympiad questions.
A book like The Original Collection of Math Contest Problems: Elementary and Middle School Math Contest problems
Solution:
The key insight is to find out what each tap can do in one hour.
Tap A can fill 1/2 of the tank in 1 hour.
Tap B can fill 1/3 of the tank in 1 hour.
Tap C can drain 1/6 of the tank in 1 hour.
Hence, if all of them are turned on simultaneously,
2/3 of the tank takes 1 hour to fill.
Multiplying this statement by 3,
2 tanks takes 3 hours to fill.
1 tank takes 3/2=1.5 hours to fill.
Ans: 1.5 hours
Hope you had fun solving this!
Also check out the following GEP Screening/Round 2 questions:
By the way, for my foreign readers who are curious what is GEP Screening Test, it is a test conducted in Singapore for entry to the GEP (Gifted Education Programme). The screening and selection tests are conducted at the end of Primary 3, equivalent to Grade 3 or age 9. The official website on GEP Screening Test is available at: http://www.moe.gov.sg/education/programmes/gifted-education-programme/faq/general/
The GEP Screening Math questions can be viewed as the epitome of the Singapore Math system, as it features highly challenging Math questions that are technically in syllabus but few students know how to solve!
After a long time, the Highly Condensed H2 Maths Notes is finally ready!
Are you looking for a short, summarized notes for H2 Vectors, H2 Complex Numbers, or even H2 Statistics? Do you remember the formula for the sum of a Geometric Progression?
Purchase this Highly Condensed H2 Maths Notes to quickly review and get ready for your exam!
Free Exam Papers included:
Numerous H2 Maths (Prelim) Free Exam Papers in PDF Format, with Solutions. Schools include ACJC, AJC, HCI, NJC, all the way to YJC, and more.
Note: The set of notes is 9 pages long (2 columns per page, total of 18 columns), i.e. highly condensed and summarized short notes.
Do check it out at our page of Math Resources for Sale!
Here is a type of a typical GEP Exam Question that can come out. Technically, this question is in syllabus since it only involves Whole Numbers. However, in practice, this is an extremely tough GEP Exam question for students who have not seen the Worker Question before.
GEP Test Question Sample (Worker Question):
6 men working 8 hours a day can paint a house in 2 days. In how many more days will 4 men, working 3 hours a day at the same rate, complete the same job?
Before you scroll down to check the answer, do give it a try! From personal experience as a tutor, even a 16 year old typical Secondary 4 student cannot solve this question if it is the first time they see it. However, once I explain the solution to them, it is extremely easy and students will get it immediately, even for primary school kids. Someone who has seen these types of questions before can solve it under a minute!
This shows the immense advantage one has if he/she has been exposed to certain types of questions. This is same for the GEP General Ability Test (GAT), a type of “IQ test”, which is basically pattern recognition. If a child has been exposed to books like Match Wits With Mensa: The Complete Quiz Book
Solution to GEP Sample Question (Worker Question):
There are many types of solutions to this question, but my favorite is using the man-days concept. Man-days is a unit for the amount of work that is needed for something. E.g. If building a house needs 10 man-days, it can be accomplished by either 1 man x 10 days = 10 man-days, or 5 men x 2 days = 10 man-days, etc.
For this question, we will use the unit of man-hours instead.
6x8x2=96 man-hours are needed to paint the entire house.
4 men working 3 hours a day would lead to 12 man-hours a day. Hence 96/12=8 days are needed.
Warning: This is where they trap the careless students! The question asks for how many more days. Hence, the answer is 8-2=6 more days.
Ans: 6 days
Do also check out the Chicken and Rabbit GEP Math Question, which is another type of popular GEP Selection Test and Screening Test question, and can be practiced beforehand as a GEP Mock Test.
Just to reblog this earlier post on a really challenging Probability O Level Question.
Also, do check out my other related posts on Probability:
Probability is becoming a really important branch of mathematics. One of the most famous Singaporean mathematicians is Professor Louis Chen Hsiao Yun who has a theorem named after him! (Stein-Chen method of Poisson approximation) Professor Chen researches on Probability.
To begin your journey in Probabilty, Introduction to Probability, 2nd Edition
is a good book to start learning from. An intuitive, yet precise introduction to probability theory, stochastic processes, and probabilistic models used in science, engineering, economics, and related fields. You may also wish to refer to our comprehensive list of Recommended Undergraduate Books.
(One of the best books to begin your journey in studying the mysterious topic of Probability)
A bag A contains 9 black balls, 6 white balls and 3 red balls. A bag B contains 6 black balls, 2 white balls and 4 green balls. Ali takes out 1 ball from each bag randomly. When Ali takes out 1 ball from one bag, he will put it into the other bag and then takes out one ball from that bag. Find the probability that
(a) the ball is black from bag A, followed by white from bag B,
(b) both the balls are white in colour,
(c) the ball is black or white from bag B, followed by red from bag A,
(d) both the balls are of different colours,
(e) both the balls are not black or white in colours.
Solution:
(a) $latex displaystylefrac{9}{18}timesfrac{2}{13}=frac{1}{13}$
(b) Probability of white ball from bag A, followed by white ball from bag B=$latex displaystyle=frac{1}{2}timesfrac{6}{18}timesfrac{3}{13}=frac{1}{26}$
Probability of white from B, followed…
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We sincerely wish our ex-Prime Minister Mr Lee Kuan Yew all the best, and may he have a good recovery.
Many people know that Lee Kuan Yew was a lawyer and politician, but few knew that he liked and was very good at Mathematics as a student. We can know these facts from the book The Singapore Story: Memoirs of Lee Kuan Yew, Vol. 1
Mathematical Excerpts from the book (Autobiography, so “I” means Lee Kuan Yew):
These are the quotes from the book that let us know more about the mathematical side of Lee Kuan Yew.
Hence, it is indeed apt and suitable that there is a prestigious award named “Lee Kuan Yew Award for Mathematics and Science” to honor students who have done well in Mathematics and Science subjects.
I hope you are inspired to learn more about Math, since even a famous person like LKY liked Math so much as to major in it. You can read more about Recommended Books for University Math where we list down all the best books for learning University Math.
Firstly, do check out our GEP Recommended Books if you are interested in books that will help boost your chances of entering GEP.
A typical sample GEP Exam Paper Question (Math) is the Chicken and Rabbit Question. This test is highly popular in the GEP Screening / GEP Selection Test, Round 1 or even extremely difficult Round 2.
A sample question would go like this:
A farmer has 36 chickens and rabbits in total.
He counted 124 legs altogether.
How many chickens and how many rabbits are there?
(By the way, this question is generated from my Chicken and Rabbit Question Generator!)
This is a highly typical GEP Exam paper question that may come out in the GEP test.
How do we solve it? One way is the trial and error or Guess and Check method. However, this method may not work for high numbers. What if the farmer had 10000 sheep?
The GEP Exam Paper has limited time, hence, we would need to solve it in an efficient way.
There is one method called the Assumption method, where students can remember the acronym ASSD!
Step 1 (A): Assume) Let’s assume all the 36 animals are chickens.
Then, there would be 36×2=72 legs in total.
Step 2 (S): Subtract) Clearly, 72 legs is too few.
In reality, there are 124-72=52 legs more.
Step 3 (S): Subtract) 4-2=2
Each rabbit has 2 more legs than a chicken.
Step 4 (D): Divide)
The extra 52 legs must be due to the rabbits, and each rabbit contributes 2 more legs.
Hence, there are 52/2=26 rabbits!
There must be 36-26=10 chickens then.
During the GEP Exam Paper, checking is essential to avoid careless mistakes. 26×4+10×2=124, which tallies! Hence, we are right!
To practice more Chicken and Rabbit questions, which is highly likely to come out in the GEP Exam Papers, check out our Chicken and Rabbit Worksheet Generator.
Lastly, do check out our GEP Recommended Books which may be the most useful books on the market (not found in Singapore since most major bookstores like Borders have closed down).
Scientists have proven that IQ can be increased, and hence reading a book like Match Wits With Mensa: The Complete Quiz Book
I have just completed the Coursera Cryptography I course by Dan Boneh successfully, and received the statement of accomplishment!
Difficulty: 4.9/5
This course is really difficult for those with no computer science background. Although there is a section on number theory, most of the sections are new to me as my background is mostly undergraduate mathematics. (Though I did take a course IT1002 (from NUS) called Introduction to Programming, which is mostly on Java Programming.)
Especially the programming exercises are very tough for people with limited programming knowledge! However, note that the programming assignments are entirely optional.
This course covers the theory and practice of cryptographic systems. Topics included symmetric encryption, data integrity, public-key encryption, and key exchange. The course emphasized the correct use of these primitives.
It is interesting to note how complex the field of cryptography is, and how smart hackers have become. It is possible to do a timing attack where even the time taken to respond to say a login, can be used by hackers to guess your password. Every logical operation in a computer takes time to execute, and the time can differ based on the input; with precise measurements of the time for each operation, an attacker can work backwards to the input. – Wikipedia
Needless to say, as our world becomes increasingly digital, cryptography becomes increasingly important.
I wrote two JavaScript applications to help solve some of the programming challenges in this course:
Sadly, WordPress doesn’t support JavaScript, so I have to write them on my sister blog: http://www.mathtuition88.blogspot.com
If you are interesting in programming, particularly app programming, why not check out this book Learning iOS Game Programming: A Hands-On Guide to Building Your First iPhone Game
Read this to be the next Flappy Bird creator! Michael Daley walks you through every step as you build a killer 2D game for the iPhone.
A while ago, I posted the HP A4 Paper Mysterious Question which goes like this:
Problem of the Week
Suppose 
Find the maximum of 
So far no one seems to have solved the question on the internet yet!
I have given it a try, and will post the solution below!
If you are interested in Math Olympiad, it is a good idea to invest in a good book to learn more tips and tricks about Math Olympiad. One excellent Math Olympiad author is Titu Andreescu, trainer of the USA IMO team. His book 104 Number Theory Problems: From the Training of the USA IMO Team
Now, to the solution of the Mysterious HP A4 Paper Question:
We will solve the problem in a few steps.
First, we will prove that 
Thus, we have proved that 
Next, we will prove a little lemma. Let 
Firstly, for the base case, 
Next, we will prove that if 
Since 
Continuing, we have
We will write 
where 
where 
Case 1: All the 
Thus, 
Case 2: Not all the 
Using Step 1, we have 
Ans: 10
Yao Sheng Tong (丘成桐) is the first Chinese (China – HK) who won the Fields Medal, he later also won the Wolf Prize after his PhD-thesis mentor SS. Chern (陈省身)。He is the first Chinese to head Harvard Math Faculty.
丘成桐 (ST Yao 1949~) Fields Medal @1982 [33岁] proved Calabi Conjecture
1. 读私立 培正中学, 高中遇 好数学老师. @香港中文大学, 发觉 Math Beauty, ‘豁然开朗’.
2. Best Math student not necessary Mathematician, only sufficient!
3. 一名数学科学家 都应对 文学,哲学 这类 学科有基本的涉猎. 好的数学 使你体验到庄子讲的
“天地与我并生, 万物与我为一” 的境界
4. 成功 = 要有数学热情.
Strategy:
a. 深入思考
b. 在心中或纸上仔细研究
c. Find clues from book, till get answer.
d. 出题目给自己
“庖丁解牛”数学方法
庄子讲庖丁(butcher)解牛有三个功夫階段：
1st Level: 看见一只全牛 (Whole Cow)
2nd Level: 三年后，不见全牛，只见牛的生理结構(Anatomy) ：骨骼，肌肉，筋腱。
3rd Level: 不以目视而是神视，＂与桑林之舞合拍，与经首之会同律。＂达到了”物我”两忘的境界。Intuition.
数学的方法也如此。
1st Level: Whole Math (Primary school to High School)
2nd Level: Component Structure – (Undergraduate Math)：
Macro- structure (Algebra : Group, Ring, Field, Vector Space… );
Micro-structure (Analysis : Calculus, Topology, etc)
3rd Level: 无处不数 Ubiquitous Math – (Graduate Math)
eg. Fermat’s Last Theorem used all Math theories available today to prove.
Singapore still follows the outdated UK Math pedagogy, using the old term “Advanced Calculus” (高等微积分) for the huge discipline of “Analysis” (分析) — the ‘Micro’ view of Math.
Mathematics is roughly divided into 2 categories:
‘Macro’ Math: Algebra
‘Micro’ Math: Analysis (or the outdated name Calculus)
Algebra has been transformed rapidly from 19th century after Galois’s invention of Group Theory, and expanded by David Hilbert and his students E. Noether, Artin, etc in Axiomatic Algebra, takes a very macro view of Mathematical structures in abstract thinking.
Analysis, also after 19th century Cauchy and Wierestrass’s invention of ‘epsilon-delta’ micro view of Calculus, transformed the Newton Calculus into rigourous Math.
The old school of division of Pure and Applied Math is no longer valid. Take for example, the Applied Math used in Google Search Algorithm uses abstract Vector Space of Matrices in Linear Algebra (Pure Math).
As every Singaporean should know by now, August 7 is going to be a public holiday this year, due to SG 50, i.e. Singapore’s 50th anniversary as a nation.
A quick check using the mental calculation of dates (Doomsday Algorithm), we can know that August 8 is Saturday, hence August 7 is a Friday. Thus, August 7 to August 10 is indeed a long weekend!
What are Singaporeans doing during the long weekend? Hint: It has something to do with studying. This is a really funny cartoon by Singaporean cartoonist “Chew on it”
Trial plan is free. Try it out to challenge with your friends!
On the cover of HP Printer A4 Paper, there is a mysterious Math Problem, which is strangely not solved by anyone on the internet yet! At least, when I gave it a search on Google, no one seems to have solved it…
Click on the images above to zoom in!
I will also type out the question in case the image is not clear:
Problem of the Week
Suppose
Find the maximum of
I will give it a try when I have free time this week.
Anyone who is interested to try please leave your answer in the comments below! I will help to verify if it is correct. If it is, you may be the first person on the internet to solve the HP A4 Paper Math Problem! 🙂
You may also forward this problem to your friends to try!
Update: Solution can be found here (Solution to HP A4 Printer Paper Mysterious Question)
Problems of Number Theory in Mathematical Competitions (Mathematical Olympiad Series)
Like many human endeavors, Math is one subject that requires motivation to excel.
There is this inspirational story that I found on https://schoolbag.sg/story/from-rock-bottom-to-top-of-the-class
With no interest in studying and thoroughly convinced he will never do well, M Thirukkumaran was on a downward spiral in secondary school and was almost retained in Secondary One and Two.
Fast forward nearly 10 years, Thiru, 23, is now studying Business Analytics at the National University of Singapore (NUS) under NEA and PUB’s National Environment and Water (NEW) Scholarship.
What turned Thiru a full 180 degrees around? It was the first taste of success, through the efforts of a teacher, Mr Tan Thiam Boon.
In Secondary Three at Monfort Secondary School, Thiru felt that he had hit rock bottom. During a rudimentary algebra test, he scored one mark out of a total score of 50. Instead of shaking his head in disbelief and despair, his mathematics teacher, Mr Tan, went out of his way to coach Thiru after school. But his efforts were in vain.
“Mentally, I had already accepted that I would not be able to do it,” said Thiru.
Undeterred, Mr Tan encouraged Thiru to pay full attention for the next topic, Trigonometry. Dejected and with little left to lose, Thiru came early to sit at the front of the class and followed the lesson attentively. About a week later, Mr Tan distributed the results of a test starting from the lowest to the highest scorers.
Thiru recalled the incident with great clarity.
“Naturally, I had expected my name to be the first to be called. But it was not and I was afraid my paper was lost. But instead, it was the last name called! I still remember the smile on Mr Tan’s face, and the confusion on everyone else’s that day. I sat dumbfounded. Mr Tan had managed to do what nobody else had. In one fell swoop, he eliminated the negative labels that society and I had placed on myself, and reinstated my confidence. More importantly, he showed me that I was capable, that I wasn’t a “delinquent” or a failure. It was akin to recovering from blindness.”
That initial spark ignited a passion, drive and desire to test his potential and accomplish what he did not even dare to imagine before. Thiru topped his class in mathematics and did well at the N-level and O-level examinations. At Tampines Junior College, his teachers gave him the opportunity to take H2 physics and mathematics, even though he did not take the prescribed secondary school subjects. Thiru’s hard work and determination paid off, often in the top 5% of the cohort for the regular examinations, and emerging as one of the college’s top students at the A-level examinations.
As a tutor, I know this is not easy. I have coached students from Fail to A grade. However, 1/50 is a really bad fail, and to achieve A in a matter of months requires extreme effort short of a miracle. For Thiru to overcome his challenges and extremely weak Math foundation to achieve his amazing accomplishments, required a motivational figure in the form of his Math teacher.
Such motivational teachers are rare, and I am glad that Thiru has found his mentor.
Helen Exley — ‘Books can be dangerous. The best ones should be labeled This could change your life.’ Books are another source of motivation. For those who have not yet found their motivational figure, do not wait as true motivational teachers are few and far between. To encounter one like Thiru requires luck and good fortune. However, good motivational books are there and available if you look for it. I have compiled a list of Motivational Books for the Student, which is available by clicking on the link.
In Math, you must always believe that you can solve the answer, in order to solve it. Just like Thiru, if you believe in yourself, you can do it! A nice book to read about Math and Motivation is Math Doesn’t Suck: How to Survive Middle School Math Without Losing Your Mind or Breaking a Nail
This is the book written by a truly gifted kid, the book of all gifted books. If you are looking for GEP books, this is the GEP book to rule all GEP books, written by the gifted kid himself.
The book is titled: We Can Do
I was reading the online news, and discovered this story about Moshe Kai Cavalin.
The one thing 14-year-old Moshe Kai Cavalin dislikes is being called a genius.
All he did, after all, was enroll in college at age 8 and earn his first of two Associate of Arts degrees from East Los Angeles Community College in 2009 at age 11, graduating with a perfect 4.0 grade point average.
Now, at 14, he’s poised to graduate from UCLA this year. He’s also just published an English edition of his first book, “We Can Do.”
Not only is he focused on academics (he researches on Wormhole Theory), he is also proficient in Chinese martial arts, scuba diving, and also writing books. He is also a math major!
Personally, I think the title “We Can Do” is a clever word play on Jeet Kune Do, a style of martial arts invented by Bruce Lee!
If you want to read the Chinese version, it is also available: （Go to College At 8-year-old!） 八歲進大學
Also, thanks to one of my readers who bought this book via my website: The Stanford Mathematics Problem Book: With Hints and Solutions (Dover Books on Mathematics)
This volume features a complete set of problems, hints, and solutions based on Stanford University’s well-known competitive examination in mathematics.
Video on Moshe Kai Cavalin, of half Jewish, half Chinese descent:
Hi everyone, please vote on this poll, and forward it to your friends! You may click on the buttons below this post to post to Facebook or Twitter!
Recently, I was reading this article online (Science and maths exams are harder than arts subjects, say researchers) by a reputed online magazine Guardian. In it, Durham researchers have found that, “At A-level, science, maths and technology subjects are not just more difficult than the non-sciences, they are without exception among the hardest of all A-levels. At GCSE, the sciences are a little more difficult than the non-sciences.“.
However, there are many rebuttals, for instance: Ian McNeilly, director of the National Association for the Teaching of English, said: “It seems scientists, on the one hand, decry the quality of their intake at universities and, on the other, say that their exams are so very hard. Any view of English as a ‘soft option’ is absolute nonsense. If the scientists tried to do it, they would find it wasn’t such a breeze.”
My view is that on one hand, yes Math is hard especially if the student doesn’t know what he/she is doing. It is very possible to get zero marks if the student makes a careless mistake in the very first step, or goes completely off tangent in approaching the question. However, in humanities, the only possible way to get zero marks is to hand up a blank script of paper. As long as you write something, most likely you will get some marks.
However, if the student knows what he/she is doing, and has sufficient practice, Math is actually easier. (We are not talking about the Riemann Hypothesis here, just high school and perhaps up to college mathematics.) In Math, we often hear of students scoring full marks, which they fully deserve if they know how to do every question. On the other hand, it is impossible to score full marks in English Literature, especially if there is an essay component. No essay is perfect!
Of course, there are ways to be good at both the Science and the Arts. For instance, From STEM to STEAM: Using Brain-Compatible Strategies to Integrate the Arts
Personally, I will not dismiss arts as “easy”, as arts do need time and effort to achieve mastery. Not everyone can write essays like Shakespeare, nor can everyone write calligraphy like the famous Wang Xizhi of China.
However, in the context of examinations, based on my personal experience, and readings on the internet, Math / Hard Science exams are indeed harder and more difficult than arts exams. This has a very negative effect of students dropping Math / Science subjects to major in “easier” subjects. Another point of concern is grade inflation in other subjects.
Most importantly, what do you think? Is Math easier or harder than other subjects?
Please vote, and forward this to your friends for voting! I will be really interested to know the results.
Source: https://sg.news.yahoo.com/groom-fails-math-test-indian-bride-walks-wedding-065433753.html
Here’s one more reason to learn math!
NEW DELHI (AP) — An Indian bride walked out of her wedding ceremony after the groom failed to solve a simple math problem, police said Friday.
The bride tested the groom on his math skills and when he got the sum wrong, she walked out.
The question she asked: How much is 15 plus six?
His reply: 17.
The news may seem like a joke, but on a more serious note, the usage of calculators in Primary 5 & Primary 6 has led to a drop in the mental arithmetic standards of children in Singapore! As a tutor, I have clearly observed the difference before and after the calculator usage was introduced. Many a times, students need a calculator to calculate what is 18+15, for example. Also, 8 times 8 (mental calculation) would pose a challenge nowadays to some students ages 9-12, whereas in the past students would know it is 64 in less than a second.
What is happening is that students are getting a bit too reliant on the calculator, and hence their human calculator (a.k.a brain) has lack of training in this aspect. How to remedy it is to practice and train in more multiplication questions.
is the book to refer to, to learn more mental calculation tips and tricks.
Tomorrow (March 14) is an important day for math lovers! It is the famous Pi Day! Pi is approximately 3.14 which corresponds to the date March 14.
Check out our post last year on Pi Day:
Here are some other previous posts on Pi:
is a very mysterious number. How mysterious? Take a look at the post to find out!If you are looking for a Pi Day T Shirt, Pi Day Once in a Century March 14, 2015 T-Shirt Large Black
Some interesting info on studying mathematics in university.
March 12th, 2015
[Note: This blog, written by Samantha Woodcock, was originally posted on http://www.topuniversities.com, and reposted here on Academic Exchange by permission from the author.]
Considering studying mathematics at university but not sure you fit the right mold? Think it’ll be too difficult, too nerdy, or won’t provide enough career options? Get ready to re-think your idea of the “typical” math student, and what’s involved in a mathematics degree…
1. Maths students are giant geeks.
Now there are the students out there that would remind you of the Sheldon Coopers of the world, but for the most part maths students are just normal people who have a passion for numbers. Not all of them wear glasses; they all don’t carry a calculator everywhere and they also don’t insist on wearing white shirts and plaid. Mathematics is also easy to combine with another subject, including art, all sciences…
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Very interesting Math Trivia
Mathtuition88 