Month: April 2015

Chinese Remainder Theorem History (韩信点兵)

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.

chinesetuition88's avatarChinese Tuition Singapore

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

Quiz: Can You Solve This Sum ?

For more logic puzzles, check out:

Puzzle Baron’s Logic Puzzles

tomcircle's avatarMath Online Tom Circle

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[Hint]: Think out of the box…

Answer below (scroll down)
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Answer:
1 + 13 +…

View original post 57 more words

Cheryl’s Birthday Problem

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. It is a very good practice for children gearing up for Math Olympiad since they love to test logic questions in Math Olympiad.

Egyptian Math Mystery

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.

egypt math

3 of the Top Jobs in America involve Math

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
mathematician
University of the Fraser Valley/flickr

Read more at: http://www.businessinsider.sg/best-jobs-of-2015-2015-4/#.VTEI7ZPoaKg

费马大定理 Fermat’s Last Theorem

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.

tomcircle's avatarMath Online Tom Circle

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)

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Math is Forever (Spanish)

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, suitable for students who really need some encouragement and motivation to overcome fear of math! Albert Einstein once said, “You never fail until you stop trying.” Hence, even if you have not done well in math for the past years, there is still hope, don’t give up!

April Fools Video Prank in Math Class

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

Excellent MITOpenCourseware

tomcircle's avatarMath Online Tom Circle

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:

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Example:
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Permutation Math Olympiad Question (Challenging)

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 1\leq k,l\leq 100 and transform the 100 tuple (a_1, \cdots, a_k, \cdots, a_l, \cdots, a_{100}) into the 100 tuple (a_1, \cdots, \frac{a_k}{2}, \cdots, a_l+\frac{a_k}{2}, \cdots, a_{100}) if a_k is an even number. We say that a permutation (a_1, \cdots, a_{100}) of (1, 2, \cdots, 100) is good if starting from (1,2,\cdots, 100) one can obtain it after finite number of steps. Find the total number of distinct good permutations of (1, 2, \cdots, 100).

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. In fact, any book by Titu Andreescu is highly recommended as he is the legendary IMO (International Math Olympiad) coach that led the USA team to a perfect score!

What’s Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success

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

The Math of Shuffling Cards

Previously, the first YouTube video wasn’t working. I have added a new link to the interesting “Looking at Perfect Shuffles” video. 🙂

mathtuition88's avatarMathtuition88

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 is an interesting book by Professor Diaconis, featuring Magic Tricks that have a mathematical background! This book is a great idea for a gift for students, teachers, or friends!

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NUS High Selection Test (DSA)

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:

  • The Math Paper is 1 and a half hours.
  • Math Paper is “ok” (easier than NMOS) Do take note that the blogger is very good at math, so “easy” is subjective.
  • Math Paper has 7 pages, inclusive of cover page and last page.
  • 23 Non-MCQ questions, where you have to shade the integer answer. (Do bring a pencil!)
  • “The first few Math questions are easy, like P6 Math questions. One of the easiest Math questions is, the average of 3 numbers is given, you add another 2 numbers and you get another given average, you have to find the sum of the 2 numbers added. There are varying marks for different questions. I think the harder questions carry 4 marks.”
    (Again, easy is subjective, what is easy for a Olympiad Gold Medalist may not be easy at all)
  • “Total marks for Maths and Science are 55 and 30 respectively.For Maths, max of answer shades is 4, so max answer may be 9999. Maths questions carry 1 mark, 2 marks, 3 marks and 4 marks. Think Q23 (last qn) is a 4-mark question.”
    (We can assume that due to the format of this test, all answers are integers!)

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 would be ideal in beginning the journey in Math Olympiad. Note that Math Olympiad is nothing like normal school math, and even a fresh university graduate in a math-related major say Engineering/Accounting would have great problems solving a Primary 6 Math Olympiad question, if he doesn’t have the necessary Math Olympiad background!

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!