## A Maths List of Formula to Remember

Are you looking for a list of Additional Mathematics (A Maths) Formulas to remember?

Check it out at: https://mathtuition88.com/math-notes-worksheets-sale/

Updated to include: Supplementary Angles, Complementary Angles, and Half Angle Formulas for Trigonometry

Remember, memorizing the formula is not enough. We need to know how to apply and use the formula! (The next level is to know how to derive the formulas, but that will not be tested in the exams. 🙂 )

Do you really really hate Math? Is it your most dreaded subject?

Why not learn to love Math as it is pretty much a compulsory subject until high school? Read this book, it may change your mindset about Math. From a well-known actress, math genius and popular contestant on “Dancing With The Stars”—a groundbreaking guide to mathematics for middle school girls, their parents, and educators

## Can you read this Math Clock?

If you can read this clock, you are without a doubt a geek. Each hour is marked by a simple math problem. Solve it and solve the riddle of time. Matte black powder coated metal. Requires 1 AA battery (not included). 11-1/2″ Diameter.

Who knew there was such an interesting clock? 😛

Also interesting is this mug below. If you can read it, it shows you are well versed in Mathematics!

I Ate Some Pie and It Was Delicious Mug– Perfect For Any Math Nerd!!– Funny High Quality Coffee Mug!! (11oz, Black I Ate Sum Pi)

## Very Inspirational Math Video

Just to share a video here:

Very Inspirational Math Video

It is a video of a girl who once did a math quiz and totally blanked out for the whole quiz. However, it turned out that her teacher did not actually ask for the quiz back, and gave her as much time as she wanted to complete the quiz. Under the relaxed circumstances, she completed the quiz and got a ‘C’. (big improvement from totally blank).

Then, she went to UCLA (very good school in US), and became a mathematics major, and wrote the book that is listed below the video!

Truly inspiring. For some kids, too much pressure may result in Math anxiety and totally blank out, while for other kids a little bit of pressure is needed to ensure that they do take studies seriously. Need to find the perfect balance for each child.

**The book the girl above wrote is: Math Doesn’t Suck: How to Survive Middle School Math Without Losing Your Mind or Breaking a Nail

## Why is e irrational?

Anyone who has taken high school math is familiar with the constant $\boxed{e=2.718281828\cdots}$.

Today we are going to prove that e is in fact irrational! We will go through Joseph Fourier‘s famous proof by contradiction. The maths background we need is to know the power series expansion: $\displaystyle \boxed{e=\sum_{n=0}^{\infty}\frac{1}{n!}}$. The proof is slightly tricky so stay focussed!

Suppose to the contrary that e is a rational number, so $\displaystyle e=\frac{a}{b}$.

Using the power series formula mentioned above, we have $\displaystyle\sum_{n=0}^\infty \frac{1}{n!}=\frac{a}{b}$

Multiply both sides by $b!$, $\displaystyle \sum_{n=0}^{\infty}\frac{b!}{n!}=\frac{ab!}{b}=a(b-1)!$

Now, we split the sum into two parts:

$\displaystyle \sum_{n=0}^b \frac{b!}{n!}+\sum_{n=b+1}^\infty \frac{b!}{n!}=a(b-1)!$

Rearranging,

$\displaystyle \sum_{n=b+1}^\infty \frac{b!}{n!}=a(b-1)!-\sum_{n=0}^b \frac{b!}{n!}$

Now, denote $\displaystyle x=\sum_{n=b+1}^\infty \frac{b!}{n!}>0$. $x$ is an integer since both $\displaystyle a(b-1)!$ and $\displaystyle\sum_{n=0}^b \frac{b!}{n!}$ are integers and their difference (which is x) will be an integer.

We now prove that $x<1$. For all terms with $n\geq b+1$ we have the upper estimate

$\displaystyle\begin{array}{rcl} \frac{b!}{n!}&=&\frac{1\times 2\times \cdots \times b}{1\times 2\times \cdots \times b \times (b+1) \times \cdots \times n}\\ &=&\frac{1}{(b+1)(b+2)\cdots (b+(n-b))}\\ &\leq& \frac{1}{(b+1)^{n-b}} \end{array}$

This inequality is strict for every $n\geq b+2$. Changing the index of summation to $k=n-b$ and using the formula for the infinite geometric progression $S_\infty = \frac{a}{1-r}$, we obtain:

$\displaystyle x=\sum_{n=b+1}^\infty \frac{b!}{n!} < \sum_{n=b+1}^\infty \frac{1}{(b+1)^{n-b}}=\sum_{k=1}^\infty \frac{1}{(b+1)^k}=\frac{\frac{1}{b+1}}{1-\frac{1}{b+1}}=\frac{1}{b}\leq 1$

We have that $x$ is an integer but $0. This is a contradiction (since there is no integer strictly between 0 and 1), and so $e$ must be irrational. (QED)

Interesting? 🙂

Did you know the constant e is sometimes called Euler’s number?

Learn more about Euler in this wonderful book. Rated 4.9/5 stars, it is one of the highest rated books on the whole of Amazon.

Leonhard Euler was one of the most prolific mathematicians that have ever lived. This book examines the huge scope of mathematical areas explored and developed by Euler, which includes number theory, combinatorics, geometry, complex variables and many more. The information known to Euler over 300 years ago is discussed, and many of his advances are reconstructed. Readers will be left in no doubt about the brilliance and pervasive influence of Euler’s work.

Watch this video for another proof that e is irrational!

## Math Handheld Computer Game

Featured Item:

Educational Insights Math Whiz

Is your child disinterested in Math? Looking for some fun and educational Math games?

Math Whiz plays like a video game and teaches like electronic flash cards. This portable ELA quizzes kids on addition, subtraction, multiplication and division, AND works as a full-function calculator at the press of a button. Problems are displayed on the LCD screen. Features eight skill levels, as well as lights and sounds for instant feedback. Two AAA batteries required (not included).

## NUS Math Ranked among Top in Asia

In the latest Quacquarelli Symonds (QS) World University Rankings by Subject (2014), NUS Math is ranked among the best mathematics departments in Asia.

Chinese students typically outperform U.S. students on international comparisons of mathematics competency. Paradoxically, Chinese teachers receive far less education than U.S. teachers–11 to 12 years of schooling versus 16 to 18 years of schooling.

Studies of U.S. teacher knowledge often document insufficient subject matter knowledge in mathematics. But, they give few examples of the knowledge teachers need to support teaching, particularly the kind of teaching demanded by recent reforms in mathematics education.

This book describes the nature and development of the “profound understanding of fundamental mathematics” that elementary teachers need to become accomplished mathematics teachers, and suggests why such teaching knowledge is much more common in China than the United States, despite the fact that Chinese teachers have less formal education than their U.S. counterparts.

# Coordinate Geometry Notes

Check out these Formulas for:

• Distance between 2 points
• Midpoint between 2 points
• and more

Featured book: The Ultimate book of Formulas (2000+ Formulas!)

Schaum’s Outline of Mathematical Handbook of Formulas and Tables, 4th Edition: 2,400 Formulas + Tables (Schaum’s Outline Series)

This Schaum’s Outline gives you

• More than 2,400 formulas and tables
• Covers elementary to advanced math topics
• Arranged by topics for easy reference

## News: Singapore Education Ranked Third in World

 Singapore takes third spot in global education rankings Straits Times Teacher Anthony Tan conducting an English lesson with a class of Primary 6 pupils at Woodlands Primary School. Singapore’s education system has … Singapore offers Saudi Arabia help in education Arab News PROPOSAL: Singapore Senior Minister of State Lee Yi Shyan with Mazen Batterjee, vice chairman of the JCCI, on Wednesday. (AN photo by Irfan … In Singapore, Training Teachers for the ‘Classroom of the Future’ Education Week News Welcome to the Classroom of the Future—a mock-up housed by Singapore’s National Institute of Education (NIE) to demonstrate what learning might … Singapore Polytechnic Assists CDIO Implementation At Malaysia’s Polytechnic Bernama PUTRAJAYA, May 6 (Bernama) — Singapore Polytechnic is assisting Malaysia on the implemention of innovative engineering education framework … Lift education standards: Linfox boss The Australian “Most of our graduates are now coming out of Thailand, Vietnam, Singapore and China because they are just so well educated,” he said. “I can get … In search of education The News International Unless we start investing massively in education, science, technology and innovation, as was done by Singapore, Korea, Malaysia, China and others, … Sultanate, Singapore and the Indian Ocean Oman Daily Observer These are thoughtful words from your education minister (Heng Swee Keat), … a pragmatism which incidentally I believe we share with Singapore. Direct School Admission not meant to lower academic standards TODAYonline In Singapore, there is no compromising a good education. Having a talent does not give a student the licence not to pursue academic excellence. NAFA inspires The Hindu The safe and comfortable cosmopolitan environment Nanyang Academy of Fine Arts, Singapore makes it the perfect destination for education abroad. Japan’s Education Minister visits SMU Perspectives@SMU Singapore Management University (SMU) received a special guest on its campus on 3 May 2014 – Japan’s Minister of Education, Culture, Sports, …

## Monster Group

Check out this Youtube video on the Monster Group (related to Group Theory, a branch in Mathematics)

In the mathematical field of group theory, the monster group M or F1 (also known as the FischerGriess monster, or the Friendly Giant) is a group of finite order. (See Wikipedia: http://en.wikipedia.org/wiki/Monster_group)

Featured book:

The Symmetries of Things

This book is written by John Conway, one of the mathematicians who worked on the Monster Group. Rated highly on Amazon.

Start with a single shape. Repeat it in some way—translation, reflection over a line, rotation around a point—and you have created symmetry.

Symmetry is a fundamental phenomenon in art, science, and nature that has been captured, described, and analyzed using mathematical concepts for a long time. Inspired by the geometric intuition of Bill Thurston and empowered by his own analytical skills, John Conway, with his coauthors, has developed a comprehensive mathematical theory of symmetry that allows the description and classification of symmetries in numerous geometric environments.

This richly and compellingly illustrated book addresses the phenomenological, analytical, and mathematical aspects of symmetry on three levels that build on one another and will speak to interested lay people, artists, working mathematicians, and researchers.

## How to prove square root of 2 is irrational?

A rational number is a number that can be expressed in a fraction with integers as numerators and denominators.

Some examples of rational numbers are 1/3, 0, -1/2, etc. Now, we know that $\sqrt{2}\approx 1.41421\cdots$.

Is the square root of 2 rational? Or is it irrational (the opposite of rational)? How do we prove it? It turns out we can prove that the square root of two is irrational using a technique called proof by contradiction. (One of the earlier posts on this blog also used proof by contradiction to show that there are infinitely many prime numbers.)

First, we suppose that $\displaystyle\sqrt{2}=\frac{p}{q}$, where $\displaystyle\frac{p}{q}$ is a fraction in its lowest terms.

Next, we square both sides to get $\displaystyle 2=\frac{p^2}{q^2}$.

Hence, $2q^2=p^2$. We can conclude that $p^2$ is even since it is a multiple of 2. Thus, $p$ itself is also even. (the square of an odd number is odd).

Thus, we can write $p=2k$ for some integer k. Substituting this back into $2q^2=p^2$, we get $2q^2=4k^2$, which can be simplified to $q^2=2k^2$.

Hence, $q^2$ is also even, and hence $q$ is also even!

But if both $p$ and $q$ are even, then $\displaystyle\frac{p}{q}$ is not in the lowest terms! (we could divide them by two). This contradicts our initial hypothesis!

Thus, the only possible conclusion is that the square root of two is not a rational number to begin with!

Featured book:

Math Jokes 4 Mathy Folks

Who says math can’t be funny? In Math Jokes 4 Mathy Folks, Patrick Vennebush dispels the myth of the humorless mathematician. His quick wit comes through in this incredible compilation of jokes and stories. Intended for all math types, Math Jokes 4 Mathy Folks provides a comprehensive collection of math humor, containing over 400 jokes.

– Highly rated on Amazon.com

## Studies and Studying: How do top students study?

Check out this post by MIT almost perfect-scorer, on how to study. His secret is to study the material in advance, before the lessons even start! This is really a useful strategy, if implemented correctly. Imagine being in Primary 3 and already knowing the Primary 4 syllabus! Primary 3 Math will be a breeze then. This is one of the reasons why China students are so good at Math – they have already studied it back in China, where the Math syllabus is more advanced!

Do try out this strategy if you are really motivated to improve in your studies. The prime time to do this is during the June and December holidays – take some time to read ahead what is going to be learnt during the next semester.

This is an excerpt of the thread:

I graduated from MIT with a GPA of 4.8 (out of 5.0) in mathematics. I had two non-As, both of which were non-math classes.

That doesn’t imply that I have good study methods, but anyway, here’s how I studied at MIT. My main study method as an undergraduate, for math classes, was knowing a sizable chunk of the material in advance.

This isn’t a method that will work for everybody. I did a lot of mathematics outside of the classroom both in high school and at MIT, and I often saw a substantial portion of the material in a given class before I took it. I can’t emphasize enough how much easier this makes a class, and not just for the reasons you might expect: one of the most valuable things you get out of knowing a lot of the material already is just not being intimidated by it. (And you can get this benefit even if you’ve only seen some of the material before and possibly forgotten some of it too.) You’re much more relaxed, and that makes it easier to process the part of the material that you don’t know.

What that translates to in terms of practical advice is this:

• cultivate a sense of curiosity,
• don’t restrict your learning to the classroom,
• only take classes that actually seem really interesting to you, and
• try to learn something related to those classes the semester before.

None of this is advice for studying for a class you’re taking now, but it’s advice for reducing the extent to which you will need to study for classes you’ll take in the future.

– Qiaochu Yuan

## Math News: Math student from Nanyang Technological University detects OAuth, OpenID security vulnerability

Is it safe to log in through well known sites such as Facebook and Google? Think again, for Wang Jing, a PhD student in mathematics at the Nanyang Technological University in Singapore, has detected critical security vulnerabilities in the OAuth, OpenID security protocols. (Source: http://phys.org/news/2014-05-math-student-oauth-openid-vulnerability.html) [Second article in the list below]

Forward this information to your friends via the Tweet button below to warn them of the potential danger!

 Unique Math Learning Center Opens in Lake Forest: Local After-School Program to Provide … Chicago Tribune Mathnasium – The Math Learning Center opened its doors in Lake Forest in March to students looking for math help and math enrichment. The new … Math student detects OAuth, OpenID security vulnerability Phys.Org (Phys.org) —To get right to the point, a doctoral candidate in math has discovered two holes in OAuth and OpenID that could leak data and redirect … A math lesson for city: Teachers’ contract likely to cost billions SILive.com Mayor Bill de Blasio, who visited Staten Island Wednesday evening to speak at an SIEDC cocktail reception at the Hilton Garden Inn, has cleared his … Math error halts Bank of America’s stock buy back and dividend increase UPI.com WILMINGTON, Del., April 28 (UPI) — Bank of America has had to halt its proposed stock buy back and dividend increase because of a math error in its … Philadelphia girl makes math a game – and excels Philly.com Josephine Nyugen, a sixth-grader at St. Cecelia, right, with her father, Joseph, left, plays the math game ‘Into the Vortex’ on the First in Math website. Math for public works borrowing bill proves tough St. Cloud Times Lawmakers from the Minnesota House Capital Investment Committee look around the prison yard near the loading dock inside the Minnesota … Houghton Mifflin Harcourt takes Go Math! Academy into the home market Boston Globe Houghton Mifflin Harcourt, a Boston-based education company, developed the program for students in kindergarten to sixth grade to learn basic math … Steve Ballmer’s math on Apple innovation doesn’t add up ZDNet His math of Apple innovation appears lacking to this longtime Mac user. Let me add a few more of his “tricks” to the list: The Apple II platform. Ballmer … Math class helps special needs student try to win a wheelchair van KOB.com There’s a lot more going on than just addition and subtraction in Mr. Green’s math class. “We just want to help out this family,” Said Cody Green. Budget math takes a U-turn: Christie blames federal tax law that brought windfall last year NorthJersey.com The same federal tax policy that Governor Christie is now blaming for New Jersey’s $807 million budget shortfall helped save his budget last year … ## Education News Update  The Straits Times holds its first Education Forum on Sunday Straits Times The Straits Times’ first Education Forum on May 4, 2014, held at the Singapore Management University’s Mochtar Riady Auditorium. — ST PHOTO: … All 300 places at The Straits Times’ first education forum this Sunday taken up Straits Times Mr David Hoe, an undergraduate at the National University of Singapore (NUS), is one of the speakers at the inaugural The Straits Times Education … Many turn up at E Plus International Education fair The Hindu The aspirants evinced keen interest in countries like Holland, Singapore, New … Official boards of all the countries presented seminars on education … Tuition and divorce The Independent Singapore News In September 2013, The Independent Singapore reported on Senior Minister of State for Education Ms Indranee Rajah’s observation on the perceived … NS committee may propose changes to IPPT management TODAYonline SINGAPORE — Suggestions to improve the management of the Individual … Veterans’ League, which was founded to promote National Education. Should India Embrace Socialism, Singapore Style? Businessinsider India This is because the Singapore government only borrows to develop a … What offers a ray of hope to Indian educators is that Singapore’s education … How does one of the top-performing countries in the world think about technology? The Hechinger Report SINGAPORE—Forty students in bright yellow shirts hunched over their … Investments in education technology have been a key part of Singapore’s … Why Indonesian education is in crisis Jakarta Post Does anyone seriously believe “education” in Indonesia is on par with the west, or even Asian countries like Japan, Korea or Singapore? Ask the … Are you getting a little crazy in your classroom? T.H.E. Journal We have asked Dr. Zachary Walker, an assistant professor at the National Institute of Education, Singapore, an American who is traveling the world … GEMS Education eyes expansion in the region Business Times (subscription) GEMS Education, the world’s largest operator of private schools, aims to … from kindergarten to pre-university, will open in Singapore later this year. ## Riemann Hypothesis Proof # Latest News: Riemann Hypothesis Proved? Recently, I saw on Arxiv (an online Math journal) that a professor from South-China Normal University, Mingchun Xu, has proved the notoriously difficult Riemann Hypothesis. Quote: “By using a theorem of Hurwitz for the analytic functions and a theorem due to T.J.Stieltjes and I. Schur, the Riemann Hypothesis has been proved considering the alternating Riemann zeta function. “ His paper can be downloaded here: http://arxiv.org/pdf/1402.5952v2 More verification is needed to check if it is indeed a proof. What is the Riemann Hypothesis about? Watch this Youtube Video: In 1859, Bernhard Riemann, a little-known thirty-two year old mathematician, made a hypothesis while presenting a paper to the Berlin Academy titled “On the Number of Prime Numbers Less Than a Given Quantity.” Today, after 150 years of careful research and exhaustive study, the Riemann Hyphothesis remains unsolved, with a one-million-dollar prize earmarked for the first person to conquer it. Rated: 4.5 stars on Amazon ## Maclaurin Series Informal Proof Most students will encounter the Maclaurin Series (also known as the Taylor’s Series centered at zero) when they are studying JC H2 or College Maths. The formula looks pretty intimidating at the start: $\displaystyle \boxed{f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+\cdots+\frac{x^n}{n!}f^{(n)}(0)+\cdots}$ How on earth does one come up with that formula? However, it turns out it is not that hard to prove the Maclaurin Series informally, or at least to derive the above formula. (The hard part is related to rigorous proof of convergence, etc.) The idea is to approximate a function by a power series (a kind of infinite polynomial) and then find out what are the coefficients. So, we assume we can write the function as such: $\boxed{f(x)=f_0+xf_1+x^2f_2+x^3f_3+\cdots+x^nf_n+\cdots\;(\dagger)}$, where $f_i$ are the coefficients of the polynomial (to be determined). We also assume that the above equation holds for all $x$. Then, letting $x=0$, we get $f(0)=f_0$. We have just found the first coefficient! Next, we differentiate the equation $(\dagger)$ to get: $f'(x)=f_1+2xf_2+3x^2f_3+\cdots+nx^{n-1}f_n+\cdots$ Letting $x=0$ again, we get: $f'(0)=f_1$. Now, differentiating the above equation one more time gives us: $f''(x)=2f_2+6xf_3+\cdots+n(n-1)x^{n-2}f_n+\cdots$ Letting $x=0$ gives $\displaystyle f_2=\frac{f''(0)}{2}$. Keep on differentiating, and we will see that $\displaystyle f_n=\frac{f^{(n)}(0)}{n!}$. This is how we get the Maclaurin Series: 🙂 $\displaystyle \boxed{f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+\cdots+\frac{x^n}{n!}f^{(n)}(0)+\cdots}$ Excellent Video by Khan Academy Interested to learn more about Calculus? After reading this thick book about Calculus, you will probably know more Calculus than Isaac Newton himself! ## SG Education News: Even Saudis are learning Singapore way of Teaching  Saudis learning the Singapore way of teaching Straits Times Since last October, the National Institute of Education (NIE) has taken leadership trainers from the kingdom under its wing, training them in curriculum … All 300 places at The Straits Times’ first education forum this Sunday taken up Straits Times Mr David Hoe, an undergraduate at the National University of Singapore (NUS), is one of the speakers at the inaugural The Straits Times Education … Singapore Plows Ahead of US With Tech in Schools NBCNews.com In the late 1990s, the Singapore Ministry of Education unveiled its master plan for technology. The first phase was spent building up infrastructure and … Govt mulls more recognition for NSmen in housing, health, education Channel News Asia SINGAPORE: More recognition could be given to National Servicemen (NSmen) in areas such as housing, healthcare and education. Defence … Singapore to beef up nuclear technology expertise Channel News Asia Singapore is beefing up its nuclear technology expertise with a newly-announced programme. The 10-year Nuclear Safety Research and Education … Many turn up at E Plus International Education fair The Hindu The aspirants evinced keen interest in countries like Holland, Singapore, New … Official boards of all the countries presented seminars on education … AWARE’s pushback on more benefits for NSmen ignites debate TODAYonline SINGAPORE — The Government’s plan to enhance housing, healthcare and education benefits for operationally ready national servicemen has … ## Math News April 28 # Can Monkeys do advanced Math? Read the below news for the answer and more!  Math wiz monkeys providing researchers with insights into human brain activity Fox News Monkeys trained to solve math problems are providing researchers with new insights into understanding a human learning disability in which children … Math and Science Pay, But High Schoolers Care Less Wall Street Journal (blog) Math and science are the peas and carrots of the jobs market: great for a career future, but resolutely unpopular with the young. Even amid a relatively … Math wrath in Pincher Creek? Pincher Creek Echo Protesters gather during a rally to support a petition calling for math curriculum reform at the Alberta Legislature Building in Edmonton, Alta., … Monkey Math and Other Number-Crunching Critters Discovery News Rhesus monkeys are able to perform math at an advanced level, reports a study this week from Harvard Medical Medical school. The monkeys were … Math department wins national award for exemplary program The Williams record According to Susan Loepp, professor of mathematics, the College’s department is unique in several regards. “Everyone likes math even if they don’t … From math failure to savant: How a mugging made a numbers whiz CTV News Padgett describes the brutal bar attack and his subsequent transformation into a math savant in his new book, Struck by Genius: How a Brain Injury … ## Fibonacci Numbers and the Mysterious Golden Ratio ## What are Fibonacci Numbers? Fibonacci Numbers, named after Leonardo Fibonacci, is a sequence of numbers: $F_0=0, F_1=1, F_2=1, F_3=2, F_4=3, F_5=5$, with a recurrence relation $F_n=F_{n-1}+F_{n-2}$. ## Relation to Golden Ratio Fibonacci Numbers are linked to the mysterious Golden Ratio, $\displaystyle \phi=\frac{1+\sqrt{5}}{2}\approx 1.61803$ In fact, the ratio of successive Fibonacci numbers converges to the Golden Ratio! The first person to observe this is Johannes Kepler. How do we prove it? Recall the recurrence relation: $F_n=F_{n-1}+F_{n-2}$ Dividing throughout by $F_{n-1}$, we get $\displaystyle \frac{F_n}{F_{n-1}}=1+\frac{F_{n-2}}{F_{n-1}}$ (We will first assume $\displaystyle\lim_{n\to\infty}\frac{F_n}{F_{n-1}}$ exists for the time being, and prove it later) Taking limits, we get $\displaystyle\lim_{n\to\infty}\frac{F_n}{F_{n-1}}=1+\lim_{n\to\infty}\frac{F_{n-2}}{F_{n-1}}$. Denoting $\displaystyle\lim_{n\to\infty}\frac{F_n}{F_{n-1}}$ as $\phi$, we get: $\displaystyle \phi=1+\frac{1}{\phi}$ Multiplying by $\phi$, we get $\phi^2=\phi +1$ $\phi^2-\phi-1=0$ This is a quadratic equation, solving using the quadratic equation, we get: $\displaystyle \phi=\frac{1\pm\sqrt{1^2-4(1)(-1)}}{2}=\frac{1\pm\sqrt{5}}{2}$ Since $\phi$ is clearly positive, we have $\displaystyle \phi=\frac{1+\sqrt{5}}{2}$ which is the Golden Ratio! For a complete proof, actually we will need to prove that $\displaystyle\frac{F_n}{F_{n-1}}$ converges. This is a bit tricky and requires some algebra. Interested readers can refer to the excellent website at: http://pages.pacificcoast.net/~cazelais/222/fib-limit.pdf for more details. Interesting video on Fibonacci numbers! Fibonacci numbers and the Golden Ratio can also be used for trading stocks. ## Best Fibonacci Number Videos on Youtube Math is logical, functional and just … awesome. Mathemagician Arthur Benjamin explores hidden properties of that weird and wonderful set of numbers, the Fibonacci series. (And reminds you that mathematics can be inspiring, too!) Dr James Grime on the Pisano Period – a seemingly strange property of the Fibonacci Sequence. Fibonacci Fun: Fascinating Activities With Intriguing Numbers From “Raising Rabbits” to “Prickly Pinecones”, 24 easy-to-use, reproducible activities and projects introduce students to Fibonacci numbers and the golden ratio. Grades 4-8 ## How to Find Asymptotes of Graphs This post is all about finding Vertical and Horizontal asymptotes of graphs. ## Vertical Asymptotes Usually, vertical asymptotes come about when there is a rational function with a numerator and a denominator, for instance, $\displaystyle y=\frac{2}{x-3}$. When the denominator is 0, the function is undefined, and hence there is a vertical asymptote there. Hence, to find the asymptote, let the denominator be 0. E.g. $x-3=0$, so $x=3$. Another way vertical asymptotes can come about is via logarithmic graphs, e.g. $y=\ln (x+2)$. $\ln 0$ is undefined, so when $x+2=0$ or $x=-2$, there will be a vertical asymptote at $x=-2$. ## Horizontal Asymptote Horizontal asymptotes usually come about when one of the terms approaches zero as $x$ approaches infinity. To find the Horizontal Asymptote, find the value of y when x approaches infinity (i.e. when x becomes a very big number). For example, $\displaystyle y=\frac{1}{x}+1$. When x is a very big number, say x=10000, y will be close to 1 since 1/10000 is almost zero. Hence, the horizontal asymptote is $y=1$. Another time where Horizontal Asymptotes appear is for Exponential Graphs. For instance, $y=e^{-x}+1$. When x is very large, $e^{-x}$ will be very small, and hence $y$ approaches 1. This means that the Horizontal Asymptote will be $y=1$. Note: The graphs above were drawn using the software Geogebra. 🙂 Model-Centered Learning: Pathways to Mathematical Understanding Using GeoGebra ## Amazon Kindle for Reading Math Books Looking for an alternative to the Ipad? Why not try out the Amazon Kindle? Are your books taking up too much space and collecting dust? Want to switch to Ebooks in this Electronic Age? There are many free or affordable Kindle Math Books online for download/purchase. Other than Math books, the Kindle can also be used for reading other books, and also for playing games and using apps. It is a decent alternative to the Ipad, if you are not a fan of Apple.  Unlike tablets, Kindle Paperwhite is designed to deliver the best reading experience No screen glare in bright sunlight Read with one hand—over 30% lighter than iPad mini Battery lasts weeks, not hours Next-gen built-in light—read without eyestrain Here are some very afforadable Kindle Math Books/Apps:  Price:$0.99

Math BINGO Kindle Fire Edition!

The object of Math BINGO is to practice math facts while playing BINGO!

Features:
-Choose from 5 games: Addition, Subtraction, Multiplication, Division and Mixed
-Choose from 3 different levels of difficulty: Easy, Medium and Hard
-Create up to 5 player profiles
-Choose from 8 different fun cartoon avatars
-Keep track of number of games played by player profile
-The Scoreboard keeps track of scores for each game and level
-Collect and play with BINGO Bugs when you earn a high score!
-Fun bonus game: BINGO Bug Bungee

FREE Learning Games for Young Children!

Check out this post on Why the Kindle Fire HDX is a far better tablet than the iPad, if you are not convinced.

“I used to think that the iPad was the king of the tablets, but not any more. It’s time to whip the crown away from Apple and give it to its rightful holder – Amazon’s Kindle Fire HDX.”

## Free Math Movies

Want to watch movies about Mathematics? There is a nice website with Free Movies involving Math. It is not the full movie, but the portion of the movie that involves math.

The site is: http://www.math.harvard.edu/~knill/mathmovies/

This is a collection of movie clips in which Mathematics appears. The site is now in HTML5 video and should be accessible by all devices. If not, chose the direct video links. To include a clip into a presentation, chose the quicktime version.

Some interesting examples include: A Blackboard in linear algebra lecture at MIT on Parseval’s identity in Fourier theory. (http://www.math.harvard.edu/~knill/mathmovies/swf/goodwillhunting.html)

Another example: Young Spock learns Math. He memorizes the formula (4pi/3) r3 for the volume of the sphere, the square root of 2396324 and the definition of dimensionality log(n)/log(d). (http://www.math.harvard.edu/~knill/mathmovies/swf/startrek_spock.html)

If you are really interested in Math in the Movies, check out this book:

Math Goes to the Movies

Mel Gibson teaching Euclidean geometry, Meg Ryan and Tim Robbins acting out Zeno’s paradox, Michael Jackson proving in three different ways that 7 x 13 = 28. These are just a few of the intriguing mathematical snippets that occur in hundreds of movies. Burkard Polster and Marty Ross pored through the cinematic calculus to create this thorough and entertaining survey of the quirky, fun, and beautiful mathematics to be found on the big screen.

Math Goes to the Movies is based on the authors’ own collection of more than 700 mathematical movies and their many years using movie clips to inject moments of fun into their courses. With more than 200 illustrations, many of them screenshots from the movies themselves, this book provides an inviting way to explore math, featuring such movies as:

Good Will HuntingA Beautiful MindStand and DeliverPiDie HardThe Mirror Has Two Faces

The authors use these iconic movies to introduce and explain important and famous mathematical ideas: higher dimensions, the golden ratio, infinity, and much more. Not all math in movies makes sense, however, and Polster and Ross talk about Hollywood’s most absurd blunders and outrageous mathematical scenes. Interviews with mathematical consultants to movies round out this engaging journey into the realm of cinematic mathematics.

This fascinating behind-the-scenes look at movie math shows how fun and illuminating equations can be.

## Looking for a Math Board Game / Fun Math Game?

Look no further! In this post I will recommend the Top 5 Math Games for kids, on Amazon.com. Amazon is one of the biggest companies in the world, and is a highly trusted and respected online retailer. Sometimes, it is hard to find Math Games in the local area, the internet provides a convenient and hassle-free way to buy Fun Math Games.

Without further ado, these are the Top 5 Math Games for children:

• Takes kids on a journey through the Sum Swamp
• Endless loop of adding and subtracting
• Make your way over the crocodile shortcut
• For 2-4 players

222 people rated it 4.8 out of 5 stars. #301 selling item in Amazon (Office Products section)

Bars hold 10 penguins each and connect 2 ways

• Reinforces math concepts – numbers, counting, patterning, sorting, operations, hundreds and base ten
• Includes 10 plastic ice bars (12 x 1 Inches), 100 plastic penguins in 10 colors and Activity Guide

Rated 4.6 out of 5 stars.

Teaches fractions, problem solving, multiplication, division, algebraic thinking and more

• Offers problems designed to resemble questions on standardized math tests
• Aligned with NCTM standards and expectations
• For children between 10 to 11 years

Have fun creating equations crossword-style.

• Includes a laminated playing board, 190 tiles, and 4 racks behind which players hide their tiles.
• Ages 8 through adult. 2 to 4 players.
• Great for the family or classroom
• Easy to learn

Perfect for use in centers or for individual practice.

• Challenging games include What’s Scoop, Royal Patterns, Under the Sea, Facts in Flight and more!
• Explores concepts in place value, time, shapes, skip counting, addition and subtraction and more.
• Includes 16 full color game boards with 21 sheets of cards, and 24 page resource guide.

If you are interested in buying Singapore Math books, please check out our post on Singapore Math books.

# 2048 Strategy

2048 is a highly popular Math/Puzzle game that can be played on the computer or on mobile devices. The game is about adding two tiles together until you get the number 2048.

The link for the desktop version is: http://gabrielecirulli.github.io/2048/

Like most fun games, the concept of 2048 is deceptively simple, even a 5 year old kid could play it. However, it is hard to master it, and getting the coveted “2048” could prove quite tricky. Do not despair, for after reading this strategy guide, you have a much higher chance of winning the game!

## Strategy Guide / Walkthrough / FAQ

The 3 Top Priorities for 2048 game:

1) Keep your highest tile in the top left corner of the grid. This is your top priority.

2) Do not let low tiles, especially 2’s or 4’s, clog up the upper two rows. This is your second priority.

3) Keep your top row in the following order, from left to right, $2^{n+3}, 2^{n+2}, 2^{n+1}, 2^{n}$. An example would be, 512, 256, 128, 64.

The reason for Priority 1 is that this immensely increases your chances of successful merges of two higher numbers into 1. It synergises with Priority 3 to create a chain-effect. For example, imagine you have 512, 256, 128, 64 on the top row. After merging another 64 with the 64 on the top row, you will have 512, 256, 128, 128. The two 128’s can merge together, making 512, 256, 256. The two 256’s can merge together, making 512, 512. And then, we have a 1024!

The reason for Priority 2 is that letting 2 or 4’s clog up the top rows is very bad. It greatly reduces your mobility (the top 2 rows clogged up with even a single ‘2’ is hard to move). The 2 or 4’s up there have little to no chance to get merged since most of the numbers at the top are high numbers.

Top 3 Guidelines for 2048 game:

1) Press up and left arrows only. Only press right when the upper row is full. Press down only when you have utterly no other choice.

2) Keep the top row filled up, as far as possible.

3) Your general aim is to target the lowest tile on the upper row, to set up the chain effect described above.

Reason for Guideline 1: Pressing right when the upper row is not full has the chance of introducing a new tile on the upper left corner, so now your highest tile is no longer on the upper left corner. This is not good. (Violates Priority 1)

Reason for Guideline 2: Keeping the top row filled up enables you to press “right” without fear of introducing a new tile on the upper left corner.

Reason for Guideline 3: After reaching the late game, we need to think a few steps in advance, and think of which is the best move in accordance to the Top 3 Priorities, and also can target the lowest tile on the upper row to set up a chain effect.

Top 3 Time Saving Quick and Fast Tips for 2048 game

1) The first few steps do not require thinking. Just spam up and left until you get a moderately high number like 128 or 256. There is no harm done about this as the board is uncluttered and there is little chance of losing. You only need to start thinking deeper during the later part of the game, when your highest tile is 512 or more.

2) If Priority 1 is violated, i.e. your highest tile is no longer in the top left corner of the grid, try a few steps to see if you can salvage the situation and get it back to the top left corner. If no, it is better to quit and start a new game to save time. Same for Priority 2, if there is a 2 or 4 clogging the upper row, try a few more steps to see if you can salvage the situation, by merging to make a higher number. If no, we can restart to save time. Priority 3 is less crucial, if the numbers in the top row do not form $2^{n+3}, 2^{n+2}, 2^{n+1}, 2^{n}$, no need to restart. But keep it in mind and keep trying your best to achieve the ideal order.

3) When there is only one possible move, make that move without thinking to save time. (No other choice anyway)

This is the best video on youtube about 2048 Strategy. (Note: They put the highest tile on the bottom right instead. Should be no difference due to the symmetry of the board)

Note: Even the expert maker of this video only has a 30% winning rate! 2048 has some element of luck (the tiles arrive randomly). Personally, I took quite some time to beat the game too.

Good luck! 🙂

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## Math Weekly News

This is the list of Top Weekly Math News around the world. To view more Math News, you can scroll down to the bottom of this website, and there will be more Free Math News for your viewing.

 Math, reading scores tick up; low proficiency levels persist Milwaukee Journal Sentinel Results from the final administration of an annual standardized reading and math exam show student achievement has ticked upward very slightly in … Math homework gives some parents trouble Sturgis Journal Some educators say more and more parents are having trouble helping their young children with basic math. They say Common Core State Standards … Math Minions are market masters: Oak Grove sixth-graders’ stock picks beat teams at top US … WDAY FARGO –- Oak Grove Lutheran School has some sixth-grade minions worth investing in. Math teacher Dave Carlson’s “Math Minions” posted 21.6 … Math Field Day draws almost 700 junior, senior high school students Southeast Missourian Students from 28 area schools compete during the Southeast Missouri State University Department of Mathematics’ 37th annual Math Field Day … New math app shortcuts jet-lag recovery Financial Express SummaryA new smartphone app uses math to help travellers alter their internal body clocks to new time zones more quickly in order to overcome jet … Math team from UNO finishes in top 10 percent Omaha World-Herald Some math students at the University of Nebraska at Omaha are celebrating their finish in the top 10 percent in the annual William Lowell Putnam … Math Academy expands program in county Shelby Star The summer program that serves Cleveland County Schools students to enhance their math skills is planning to have a permanent location in Kings … Math-O-Vision Calls for Video Contest Entries DigitalJournal.com Here’s some math for you: High school students could win $4,000 simply by entering Math-O-Vision’s video contest to show how they see and … Doing the math on Red Sox’ poor start Boston Globe NEW YORK — The Red Sox took two of three games from Baltimore to open the season and it appeared the defending World Series champions were … STEM Heroines: Math role models for girls Christian Science Monitor The number of women in science, technology, engineering, and math fields is growing, but women are still a minority overall in STEM career paths. ## Singapore Education News April 2014  Singapore education minister shares 4 principles on ICT use in education FutureGov Magazine The Ministry of Education in Singapore is now preparing for the fourth Education Master Plan. According to Heng Swee Keat, Minister for Education, … Despite criticisms, education system delivers, says PM Lee TODAYonline SINGAPORE — Although the education system here has often been criticised, it is essentially a good system that has delivered good results, said … Oman, Singapore sign MoU for cooperation in education Times of Oman The Ministry of Education hopes through this MoU to cooperate in the future with the National Institute of Education in Singapore for teacher training, … Teachers urged to innovate to bring out the best in students Channel News Asia SINGAPORE: Education Minister Heng Swee Keat has said that whether innovation in education is successful depends on whether a child is able to … S’pore will raise quality of all schools: PM Lee Channel News Asia Be it special needs or neighbourhood schools, tertiary institutions or those for special talents — Singapore will raise the quality of every educational … Huge response for UniSIM degrees in arts, sports and music education TODAYonline SINGAPORE — More than 250 applications have been received for SIM … music and sports education, surpassing the number of places available. Masagos Zulkifli meets Omani ministers in Muscat Channel News Asia A statement from Singapore’s Ministry of Foreign Affairs (MFA) said they … collaboration in education on teacher training and assessment systems. Bridge students’ digital interests to school learning: Heng Swee Keat Channel News Asia Minister for Education Heng Swee Keat made this point on Wednesday as he … by the Ministry of Education and Academy of Principals, Singapore. Universities need to offer ideas with ‘great impact’ TODAYonline SINGAPORE — Senior Minister of State (Education and Law) Indranee Rajah yesterday threw down the gauntlet to Singapore universities to come up … Jurong library sets up children’s corner to encourage early reading TODAYonline Minister of State, Ministry of Communications and Information and Ministry of Education, Ms Sim Ann with her youngest child, at the Early Literacy … ## HCI Confession Page Math Joke “Today I asked the girl I like on FB to help me do math prob 9x-7i>3(3x-7u) 9x-7i>9x-21u -7i>-3(7u) -i>-3u = i<3u but she go put the ans as 3u>i ruining my whole plan T.T” -HCJC Student (M) Math isn’t hard. Love is. Currently in its eighteenth printing in Japan, this best-selling novel is available in English at last. Combining mathematical rigor with light romance, Math Girls is a unique introduction to advanced mathematics, delivered through the eyes of three students as they learn to deal with problems seldom found in textbooks. Math Girls has something for everyone, from advanced high school students to math majors and educators. Praise for Math Girls! “…the type of book that might inspire teens to realize how much interesting mathematics there is in the world—not just the material that is forced upon them for some standardized test.” “Recommended” —CHOICE: Current Reviews for Academic Libraries “Imagine the improbable: high-school students getting together on their own — not in a Math Club or Math Circle, not in preparation for any Math Olympiad or “regular” test, not on the advice of any of their teachers, not as part of any organized program — to talk about pure math, math more interesting than the math found in their textbooks. The three students in this book do that for the sheer love of it. That to me is the beauty and fascination of this novel for young people, mostly young people interested in math.” —Marion Cohen, Arcadia University, MAA Reviews “Sometimes the math goes over your head—or at least my head. But that hardly matters. The focus here is the joy of learning, which the book conveys with aplomb.” —Daniel Pink, NYT and WSJ best-selling author of Drive and A Whole New Mind “if you have a…teenager who’s really into math, this is a really interesting choice” —Carol Zall, Public Radio International, The World “Math Girls provides a fun and engaging way to learn and review mathematical concepts…the characters’ joy as they explore and discover new and old ideas is infectious.” —review, “Experiments in Manga” blog Reviews from amazon.co.jp “As a physics major, math has always been a painful tool to use and nothing more. But Math Girls changed the way I look at mathematics. Now I actually find it interesting!” — “Au” “Math Girls is a fun read, but I was surprised to find that it’s also a serious math book chock full of careful explanations. I hope that people who think they don’t like math will read it. Even when the formulas go over your head, just following the story gives you a great feel for how fun math can be.” — “Nyanta” “I got hooked on this book during summer vacation, and had a great time reading it by the pool. It was so good that I read it twice, the second time while working out the problems on the hotel stationary.” — “Kei0210” “Advanced math, explained in a playful way. But it’s not just a textbook, with dry solutions to problems. It’s a bittersweet story, with mathematics telling part of the tale. A brilliant comparison between the uncertainties of youth and the absolute proofs of symbols and numbers.” — Shiori Oguchi ## Paul Erdos’ Proof that there are Infinite Primes # Paul Erdos’ Proof that there are Infinite Primes (with Examples) Every integer can be uniquely written as $rs^2$, where $r$ is square-free (not divisible by any square numbers). For instance, 6 is square-free but 18 is not, since 18 can be divided by $3^2$. We can do this by letting $s^2$ to be the largest square number that divides $n$, and then let $r=n/s^2$. For instance, if $n=108$, $6^2$ is the largest square number that divides 108, so we let $r=108/6^2 = 3$. Now, suppose to the contrary that a finite number k of prime numbers exists. We fix a positive integer $N$, and try to over-estimate the number of integers between 1 and $N$. Using our previous argument, each of these numbers can be written as $rs^2$, where $r$ is square-free and $r$ and $s^2$ are both less than $N$. By the fundamental theorem of arithmetic, there are only $2^k$ square free numbers. (The number of subsets of a set with k elements is 2k) Since $s^2, we have $s<\sqrt{N}$. Hence, the number of integers less than N is at most $2^k \sqrt{N}$. ($2^k$ choices for $r$ and $\sqrt{N}$ choices for $s$) i.e. $\boxed{ 2^k \sqrt{N} \geq N}$, for all N. This inequality does not hold for $N$ sufficiently large. For instance, we can let $N=2^{4k}$, then $2^k \sqrt{2^{4k}}=2^{3k} < 2^{4k}$. Hence, this is a contradiction, and there are infinitely many primes! An example of how the above argument works: Suppose the only prime numbers are 2, 3, 5. (k=3) Then, there are only $2^3=8$ square-free numbers, namely, 1, 2, 3, 5, 2×3=6, 2×5=10, 3×5=15, 2x3x5=30. For example, if we fix $N=2^{12}$, $s< \sqrt{2^{12}}=2^6$. $2^k \sqrt{2^{12}} = 2^3 \cdot 2^6 = 2^9 < N$, which is a contradiction. References: http://en.wikipedia.org/wiki/Euclid’s_theorem#Erd.C5.91s.27s_proof An elegant proof from Erdős PAUL ERDOS – N IS A NUMBER ## E Maths Tuition Centre E Maths Challenging Question Sometimes, E Maths can seem more difficult than A Maths! Nevertheless, exposure and practising E Maths questions will make it easier as the student gets used to the problem solving techniques required to solve “tricky” E Maths questions. Can you solve the following E Maths Problem? 🙂 It takes 8 workers to complete a certain job. The job can be completed 2 days earlier if 8 more workers are added. How many workers are needed if the house has to be completed 3 days earlier? E Maths Group Tuition at Bishan: https://mathtuition88.com/group-tuition/ https://mathtuition88.com/ ## The story of Euclid and the infinitude of primes Once upon a time, there lived a Mathematician named Euclid. Euclid came up with an ingenious method of proving that there are infinitely many prime numbers. Prime numbers are whole numbers that only have two factors, one and itself. For instance, 7 is a prime number while 6=2×3 is not. The proof is by contradiction. Suppose that there are only a finite number of prime numbers, $p_1, p_2, \cdots, p_n$. Now, we consider the number $P=p_1 \cdot p_2 \cdots p_n+1$. P is either a prime number, or it is a composite number. If P is a prime number, then we have just contradicted the assumption that there are only a finite number of prime numbers $p_1, p_2, \cdots, p_n$! If P is a composite number, then, it has a factor (smaller than P). However, none of $p_1, p_2, \cdots, p_n$ can be a factor since P divided by those primes will leave a remainder of 1! Hence, P has another factor that is not in the original list, contradicting the initial assumption once again. (Strictly speaking, Euclid’s proof is not by contradiction, instead he used a similar argument to show that given any finite list of primes, there is at least one other prime.) As of now, the largest known prime to mankind is 257,885,161 − 1, a number with 17,425,170 digits! (http://en.wikipedia.org/wiki/Largest_known_prime_number) ## Tuition for All Subjects If you are looking for tuition for other subjects, be it English Tuition, Social Studies Tuition, Geography Tuition, Physics Tuition, Chemistry Tuition, Biology Tuition, Chinese Tuition, Economics Tuition, GP Tuition or Piano/Violin Lessons … ## Check out our page on Other Subjects Tuition! We have a recommendation of a top tuition agency listed on that page, as well as top tutors in Singapore. If you are looking for O Level E Maths and A Maths Tuition, … ## Check out our page on O Level Group Tuition at Bishan! ## Math Joke from China Translation： After class, Xiaogang asks Xiaodong what is the Math homework for today. Xiaodong replied, “The integer solutions of $\displaystyle \frac{x-2}{2}+3\geq x+1$ $1-3(x-1)<8-x$ are the question numbers of the Math Homework that are due.” A Netizen posted: “If I had such a classmate, I would say, “Sorry I don’t know you, bye”” （10 分）放学时，小刚问小东今天数学作业是哪儿题，小东回答说：“不等式组 $\displaystyle \frac{x-2}{2}+3\geq x+1$ $1-3(x-1)<8-x$ 的整数解就是今天数学作业的题号”，聪明的你知道今天的数学作业是哪几题吗？ ## E Maths Group Tuition and Why E Maths is actually more difficult than A Maths # https://mathtuition88.com/group-tuition/ Come for E Maths Tuition at Bishan! Conveniently located at Block 230 Bishan Street 23 #B1-35 S(570230), near Catholic High School. Schedule: • Monday 7pm-9pm (E Maths) (Perfect for students who have CCA in the afternoon, or students who want to keep their weekends free.) ## Do not underestimate E Maths! Those who truly understand the Maths Syllabus at Secondary Level will know that E Maths is in some ways, more difficult than A Maths. Not to mention the infamous E Maths Bell Curve, which is definitely well above 75 marks to get a chance of A1. (Some sources say the Bell Curve for E Maths is 90 marks. See http://forums.sgclub.com/singapore/questions_about_o_378705.html ) “Amaths is good, like ditzy said, I tell you, if you practise amaths everyday and do past years papers, you can see amaths quite okay one. During my secondary school years i felt that emaths more difficult than amaths.” “You can ask many people out there, which one is more tedious and require thinking more, they’ll say emaths. Reason simple very simple: Emaths: They throw an entire paragraph at you which hidden clues here and there, slight errors is almost unavoidable. Amaths: Short and simple, they throw question at you. Just solve. End of story.” My year some speculated that E Maths A1 was as high as 90?” ## How to calculate Chinese Zodiac # How to use Math to calculate Chinese Zodiac and Impress your Friends The Shengxiao (Chinese: 生肖, literally “birth likeness”), also known in English as the Chinese zodiac (“zodiac” derives from the similar concept in Western Astrology and means “circle of animals”), is a scheme and systematic plan of future action, that relates each year to an animal and its reputed attributes, according to a 12-year cycle. It remains popular in several East Asian countries, such as China, Vietnam, Korea and Japan. (Wikipedia: Chinese Zodiac) ## Simple Mental Calculation: Years after 1900 1. Subtract 1900 from the year you are finding. E.g. 1988-1900=88 2. Divide your answer by 12 and find the remainder. E.g. 88/12=7R4 3. Add 1 to your answer. E.g. 4+1=5 4. The answer represents the Chinese Zodiac of that year! (1-Rat 鼠, 2-Ox 牛, 3-Tiger 虎, 4-Rabbit 兔, 5-Dragon 龙, 6-Snake 蛇, 7-Horse 马, 8-Goat 羊, 9-Monkey 猴, 10-Rooster 鸡, 11-Dog 狗, 12-Pig 猪) Hence, 1988 is the Year of the Dragon, since 5 represents Dragon. ## Calculation for Years after 2000 Very similar to the above, with some slight changes. 1. Subtract 2000 from the year you are finding. E.g. 2015-2000=15 2. Divide your answer by 12 and find the remainder. E.g. 15/12=1R3 3. Add 5 to your answer. E.g. 3+5=8 (Note: If your answer is greater than 12, subtract 12 from it) 4. The answer represents the Chinese Zodiac of that year! (1-Rat, 2-Ox, 3-Tiger, 4-Rabbit, 5-Dragon, 6-Snake, 7-Horse, 8-Goat, 9-Monkey, 10-Rooster, 11-Dog, 12-Pig) Hence, 2015 is the Year of the Goat, since 8 represents Goat. ## Chinese Zodiac List Test your new skills and check the Chinese Zodiac for the following years from 1900 to 2100! You may also refer to the table at: http://www.prokerala.com/general/calendar/chinese-years.php Year 1900: Rat Year 1901: Ox Year 1902: Tiger Year 1903: Rabbit Year 1904: Dragon Year 1905: Snake Year 1906: Horse Year 1907: Goat Year 1908: Monkey Year 1909: Rooster Year 1910: Dog Year 1911: Pig Year 1912: Rat Year 1913: Ox Year 1914: Tiger Year 1915: Rabbit Year 1916: Dragon Year 1917: Snake Year 1918: Horse Year 1919: Goat Year 1920: Monkey Year 1921: Rooster Year 1922: Dog Year 1923: Pig Year 1924: Rat Year 1925: Ox Year 1926: Tiger Year 1927: Rabbit Year 1928: Dragon Year 1929: Snake Year 1930: Horse Year 1931: Goat Year 1932: Monkey Year 1933: Rooster Year 1934: Dog Year 1935: Pig Year 1936: Rat Year 1937: Ox Year 1938: Tiger Year 1939: Rabbit Year 1940: Dragon Year 1941: Snake Year 1942: Horse Year 1943: Goat Year 1944: Monkey Year 1945: Rooster Year 1946: Dog Year 1947: Pig Year 1948: Rat Year 1949: Ox Year 1950: Tiger Year 1951: Rabbit Year 1952: Dragon Year 1953: Snake Year 1954: Horse Year 1955: Goat Year 1956: Monkey Year 1957: Rooster Year 1958: Dog Year 1959: Pig Year 1960: Rat Year 1961: Ox Year 1962: Tiger Year 1963: Rabbit Year 1964: Dragon Year 1965: Snake Year 1966: Horse Year 1967: Goat Year 1968: Monkey Year 1969: Rooster Year 1970: Dog Year 1971: Pig Year 1972: Rat Year 1973: Ox Year 1974: Tiger Year 1975: Rabbit Year 1976: Dragon Year 1977: Snake Year 1978: Horse Year 1979: Goat Year 1980: Monkey Year 1981: Rooster Year 1982: Dog Year 1983: Pig Year 1984: Rat Year 1985: Ox Year 1986: Tiger Year 1987: Rabbit Year 1988: Dragon Year 1989: Snake Year 1990: Horse Year 1991: Goat Year 1992: Monkey Year 1993: Rooster Year 1994: Dog Year 1995: Pig Year 1996: Rat Year 1997: Ox Year 1998: Tiger Year 1999: Rabbit Year 2000: Dragon Year 2001: Snake Year 2002: Horse Year 2003: Goat Year 2004: Monkey Year 2005: Rooster Year 2006: Dog Year 2007: Pig Year 2008: Rat Year 2009: Ox Year 2010: Tiger Year 2011: Rabbit Year 2012: Dragon Year 2013: Snake Year 2014: Horse Year 2015: Goat Year 2016: Monkey Year 2017: Rooster Year 2018: Dog Year 2019: Pig Year 2020: Rat Year 2021: Ox Year 2022: Tiger Year 2023: Rabbit Year 2024: Dragon Year 2025: Snake Year 2026: Horse Year 2027: Goat Year 2028: Monkey Year 2029: Rooster Year 2030: Dog Year 2031: Pig Year 2032: Rat Year 2033: Ox Year 2034: Tiger Year 2035: Rabbit Year 2036: Dragon Year 2037: Snake Year 2038: Horse Year 2039: Goat Year 2040: Monkey Year 2041: Rooster Year 2042: Dog Year 2043: Pig Year 2044: Rat Year 2045: Ox Year 2046: Tiger Year 2047: Rabbit Year 2048: Dragon Year 2049: Snake Year 2050: Horse Year 2051: Goat Year 2052: Monkey Year 2053: Rooster Year 2054: Dog Year 2055: Pig Year 2056: Rat Year 2057: Ox Year 2058: Tiger Year 2059: Rabbit Year 2060: Dragon Year 2061: Snake Year 2062: Horse Year 2063: Goat Year 2064: Monkey Year 2065: Rooster Year 2066: Dog Year 2067: Pig Year 2068: Rat Year 2069: Ox Year 2070: Tiger Year 2071: Rabbit Year 2072: Dragon Year 2073: Snake Year 2074: Horse Year 2075: Goat Year 2076: Monkey Year 2077: Rooster Year 2078: Dog Year 2079: Pig Year 2080: Rat Year 2081: Ox Year 2082: Tiger Year 2083: Rabbit Year 2084: Dragon Year 2085: Snake Year 2086: Horse Year 2087: Goat Year 2088: Monkey Year 2089: Rooster Year 2090: Dog Year 2091: Pig Year 2092: Rat Year 2093: Ox Year 2094: Tiger Year 2095: Rabbit Year 2096: Dragon Year 2097: Snake Year 2098: Horse Year 2099: Goat Year 2100: Monkey ## April Fools Math! Check out this video on April Fools Math! The content is about Complex Numbers. Really funny. 🙂 Over a million views! Also, check out this post on Math Jokes: https://mathtuition88.com/2014/03/08/math-jokes/ Also, we have a series of Best Fake Proofs on Math Stackexchange: $\displaystyle x^2=\underbrace{x+x+\cdots+x}_{(x\text{ times})}$ $\displaystyle\frac{d}{dx}x^2=\frac{d}{dx}[\underbrace{x+x+\cdots+x}_{(x\text{ times})}]$ $2x=1+1+\cdots+1=x$ $2=1$ What went wrong? Amazing Endless Growing Chocolate! Can you spot the magic? Math Jokes 4 Mathy Folks ## What is 1 divided by 0? What is 1 divided by 0? Is it infinity? Contrary to popular opinion, 1 divided by 0 is not infinity! Wikipedia states that “the expression has no meaning, as there is no number which, multiplied by 0, gives a (assuming a≠0), and so division by zero is undefined“. ## How to show that division by zero is undefined $\displaystyle \lim_{x\to 0^+} \frac{1}{x}=+\infty$ The limit of 1/x as x approaches zero from the right is positive infinity. However, $\displaystyle \lim_{x\to 0^-} \frac{1}{x}=-\infty$ The limit of 1/x as x approaches zero from the left is negative infinity. Since the left limit and right limit are different, the limit of 1/x as x approaches infinity does not exist! Note: There are mathematical structures in which a/0 is defined for some a (see Riemann sphere, real projective line, and section 4 for examples); however, such structures cannot satisfy every ordinary rule of arithmetic (the field axioms). ## Which is bigger, 0.9999999…. or 1? One way to argue that 0.99999… is equal to 1 is the following: 1/3=0.33333… Multiply the above equation by 3, 1=0.99999… Is the above convincing? If you are still not convinced, we can let x=0.99999… Then, 10x=9.99999… 10x-x=9.99999…-0.99999…=9 9x=9 There are also more advanced methods of proving 0.999…=1, listed here on Wikipedia. (http://en.wikipedia.org/wiki/0.999…) Some of the techniques include Infinite series and sequences, Dedekind cuts, and Cauchy sequences. ## Math Rap Video and How not knowing Math can cost you$15,000

Math Rap covering many Math topics:

• Pythagoras’ Theorem
• Equation of a straight line
• Probability
• Logic
• Truth Tables

“Come to class everyday, and do your homework every night, and everything is going to be alright.” 🙂

If this guy had studied basic Pythagoras’ Theorem, he would have been $15,000 dollars richer! In Math, the “majority” may not be right! 🙂 ## US “Common Core” Math gets a bit too convoluted How would you solve 427-316? Most people educated in the past would simply use the usual subtraction working to arrive at the answer of 111. But lately, in the USA, the Common Core Math approach gets very complicated and teaches an approach that even the father who has a “Bachelor of Science Degree in Electronics Engineering, which included extensive study in differential equations and other higher math applications” cannot explain it, nor get the answer correct. See also the Common Core method of adding 26+17: Is it really necessary to use “number bonds” to “skip-count by seven” to add 7+7? I agree with the father above that in Maths, “simplification is valued over complication”. This is why I always teach the easiest to understand method and the shortest method to solve questions to my students. No point using a complicated method when a shorter and simpler method can give the same answer! ## How to save money as a student # How to save money as a student Education post: What tips/frugal tips would you give to an international student moving to Singapore? As an international student, I assume your lodgings are taken care of, hence the bulk of your money will go to food and entertainment. 1. Eat at the university as much as possible 2. Eat at kopitiams rather than restaurants- and gives you a sense of the ‘authentic Singapore’, as it were. 3. Get this card. It gives you a 10% discount at every Kopitiam (the chain, not the general type of coffee shop). You can get back the deposit later. 4. Get second-hand textbooks. Sometimes, they have sales in the canteen. I got my marketing textbook for$5, and my biochemistry one for $20. Campbell and Reece went for$21. Genetics was $20, too, down from the original prices of$45-80.

5. Or don’t get textbooks at all. Go to the library.

## Functions and Graphs

In this post, we will discuss how to sketch the graphs of $y=ax^n$, for y=-2 to 3.

First, we will look at Quadratic Graphs ($y=ax^2+bx+c$)

If a>0, the graph is a “U” shape or “happy face”. 🙂

If a<0, the graph is a “n” shape or “sad face”. 😦

## Intercepts

• To find x-intercept: Let y=0
• To find y-intercept: Let x=0

## Completing the square

• E.g. $2x^2+8x+4=2(x^2+4x+2)$ (take out common factor of $x^2$)
• =$2(x^2+4x \mathbf{+2^2-2^2}+2)$ (Key step: Divide coefficient of x by 2, add & subtract the square of it)
• Check your answer using substitution method! (Sub. in $x=9$ into your initial and final answer)

## Turning Points

$y=\pm (x-p)^2+q$

* (p,q) is the turning point of the graph

For example, $y=(x-2)^2+3$

Minimum pt: (2,3)

## Graphs of $y=ax^n$

### $\displaystyle y=ax^{-2}=\frac{a}{x^2}$, where a>0 (Volcano shape)

Not touching the x-axis (Asymptote)

If a<0, the graph becomes upside down.

($\displaystyle y=\frac{-1}{x^2}$)

### $\displaystyle y=ax^{-1}=\frac{a}{x}$, where a>0. (Hyperbola, Slanted Hourglass)

If a<0, it will be upside down (reflected about x-axis)

($\displaystyle y=\frac{-1}{x}$)

# Partial Fractions

Step 1:

Firstly, we have to check that the fraction $\displaystyle \frac{P(x)}{Q(x)}$ is a proper fraction. (Degree of P(x) strictly smaller than Degree of Q(x))

For example,

$\displaystyle \frac{3}{2x+9}$: Proper fraction

$\displaystyle \frac{8x^2}{x^2+3}$: Improper fraction

$\displaystyle \frac{2x^5+1}{3x^3+x}$: Improper fraction

If the partial fraction is an improper fraction, we need to use long division.

Step 2:

Secondly, we have to see which type of partial fractions it is. There are three main types of partial fractions.

## Distinct Linear Factors:

$\displaystyle\boxed{\frac{px+q}{(ax+b)(cx+d)}=\frac{A}{ax+b}+\frac{B}{cx+d}}$

## Repeated Linear Factors:

$\displaystyle\boxed{\frac{px+q}{(ax+b)(cx+d)^2}=\frac{A}{ax+b}+\frac{B}{cx+d}+\frac{C}{(cx+d)^2}}$

(Quadratic Factor must be irreducible, i.e. cannot be factorized further)

E.g.

$\displaystyle\boxed{\frac{2x-1}{(x^2+3)(x+1)}=\frac{Ax+B}{x^2+3}+\frac{C}{x+1}}$

## Worked Example

Q: Express $\displaystyle\frac{8}{2x^2-5x-3}$ in partial fractions.

Step 1: Factorize denominator

$\displaystyle\frac{8}{2x^2-5x-3}=\frac{8}{(2x+1)(x-3)}$

Step 2: Use Partial Fractions formula for Distinct Linear Factors

$\displaystyle\frac{8}{(2x+1)(x-3)}=\frac{A}{2x+1}+\frac{B}{x-3}$

Step 3: Multiply throughout by $(2x+1)(x-3)$

$8=A(x-3)+B(2x+1)$

Step 4: Choose two suitable values of x.

Let x=3, then 8=B(7), thus B=8/7

Let x=-1/2, then 8=A(-3.5), thus A=-16/7

Step 5: Write down the partial fraction and check using substition (very important to check your answer)

$\displaystyle\frac{8}{2x^2-5x-3}=\frac{-16}{7(2x+1)}+\frac{8}{7(x-3)}$ (Ans)

Checking:

Let x=9,

$\displaystyle\frac{8}{2x^2-5x-3}=\frac{8}{114}=\frac{4}{57}$

$\displaystyle\frac{-16}{7(2x+1)}+\frac{8}{7(x-3)}=\frac{4}{57}$

Since both give the same value, the check is a success!

## Can Dogs do Mathematics?

Today, our post will be about Man’s best friend and how dogs can do Mathematics. Or can they?

Some people doubt that dogs are capable of even the most rudimentary form of quantitative thinking. The most basic form of analyzing the world in a quantitative way involves the judgment of size, namely, answering the question of whether one thing larger than another. Early researchers would put out two balls of hamburger, one large and the other small, and when they found that dogs were as likely to choose the small one as the large, they concluded that dogs could not estimate size. However there is a flaw in this test. Dogs think in an opportunistic manner, a sort of “A bird in the hand is worth two in the bush” mentality. If the two plates were a different distances, the dog would always grab the closest. However if they were at equal distances, the dog would show that he understood the notion of size by going after the larger one. Norton Milgram, at the University of Toronto confirmed that dogs can judge size well using a tray which contains two objects of different sizes. If the dog pushes the correct object, then underneath it he will find a food treat. Dogs can be taught always to pick the larger (or smaller) of two objects, regardless of the shape or identity of the objects, and they learn this fairly easily.

Watch the above video, about Maggie the Jack Russell that can do Arithmetic!

You can also watch the video at: http://www.oprah.com/oprahshow/Maggie-the-Dog-Does-Math

Maggie’s a Jack Russell terrier who can add, subtract, multiply and divide! Watch her do math for Oprah.

Maggie the Jack Russell is a maths genius (for a dog!), watch her compete with a class of 7 year olds.

However, there has been the case of Clever Hans, a horse that seemingly can do Mathematics.

However, it was “not actually performing these mental tasks, but was watching the reaction of his human observers”. (Wikipedia) Could Maggie be observing the subconscious signals of her owner? The owner might not even be aware of that, like in the case of Clever Hans.

Is Maths Dog Maggie a Fraud? Robert De Franco devises a series of tests to see if owner Jessie is somehow prompting Maggie the right answers.

Wikipedia:

Clever Hans (in German, der Kluge Hans) was an Orlov Trotter horse that was claimed to have been able to perform arithmetic and other intellectual tasks.

After a formal investigation in 1907, psychologist Oskar Pfungst demonstrated that the horse was not actually performing these mental tasks, but was watching the reaction of his human observers. Pfungst discovered this artifact in the research methodology, wherein the horse was responding directly to involuntary cues in the body language of the human trainer, who had the faculties to solve each problem. The trainer was entirely unaware that he was providing such cues.[1] In honour of Pfungst’s study, the anomalous artifact has since been referred to as the Clever Hans effect and has continued to be important knowledge in the observer-expectancy effect and later studies in animal cognition. Hans was studied by the famous German philosopher and psychologist Carl Stumpf in the early 20th century. Stumpf was observing the sensational phenomena of the horse, which also added to his impact on phenomenology.

## 1000 Multiplication Questions

These are some Multiplication Questions and Answers generated by the Multiplication Worksheet Generator for Grade 1 – Grade 3 listed on the Maths Resources for Sale page.

Q1) 13 x 7 =

Q2) 10 x 9 =

Q3) 10 x 8 =

Q4) 5 x 10 =

Q5) 9 x 13 =

Q6) 7 x 7 =

Q7) 11 x 12 =

Q8) 8 x 3 =

Q9) 9 x 12 =

Q10) 6 x 2 =

Q11) 8 x 2 =

Q12) 10 x 2 =

Q13) 11 x 12 =

Q14) 10 x 5 =

Q15) 14 x 5 =

Q16) 12 x 11 =

Q17) 6 x 13 =

Q18) 12 x 9 =

Q19) 13 x 3 =

Q20) 6 x 5 =

Q21) 3 x 5 =

Q22) 7 x 14 =

Q23) 12 x 14 =

Q24) 9 x 11 =

Q25) 11 x 12 =

Q26) 7 x 11 =

Q27) 14 x 10 =

Q28) 13 x 4 =

Q29) 13 x 6 =

Q30) 5 x 10 =

Q31) 14 x 2 =

Q32) 8 x 12 =

Q33) 4 x 5 =

Q34) 3 x 3 =

Q35) 13 x 12 =

Q36) 7 x 9 =

Q37) 3 x 11 =

Q38) 9 x 9 =

Q39) 14 x 9 =

Q40) 7 x 4 =

Q41) 4 x 8 =

Q42) 5 x 10 =

Q43) 4 x 3 =

Q44) 6 x 9 =

Q45) 5 x 8 =

Q46) 7 x 6 =

Q47) 6 x 13 =

Q48) 6 x 3 =

Q49) 4 x 4 =

Q50) 2 x 8 =

Q51) 6 x 5 =

Q52) 14 x 5 =

Q53) 2 x 3 =

Q54) 4 x 10 =

Q55) 10 x 6 =

Q56) 11 x 8 =

Q57) 13 x 4 =

Q58) 3 x 5 =

Q59) 8 x 9 =

Q60) 2 x 2 =

Q61) 7 x 6 =

Q62) 4 x 7 =

Q63) 3 x 13 =

Q64) 12 x 10 =

Q65) 14 x 2 =

Q66) 6 x 5 =

Q67) 3 x 2 =

Q68) 3 x 11 =

Q69) 5 x 8 =

Q70) 10 x 13 =

Q71) 13 x 2 =

Q72) 10 x 3 =

Q73) 6 x 14 =

Q74) 7 x 2 =

Q75) 7 x 8 =

Q76) 4 x 12 =

Q77) 8 x 11 =

Q78) 3 x 13 =

Q79) 4 x 9 =

Q80) 2 x 3 =

Q81) 2 x 12 =

Q82) 12 x 10 =

Q83) 9 x 13 =

Q84) 11 x 10 =

Q85) 9 x 8 =

Q86) 6 x 10 =

Q87) 2 x 3 =

Q88) 14 x 7 =

Q89) 9 x 2 =

Q90) 13 x 12 =

Q91) 12 x 9 =

Q92) 8 x 8 =

Q93) 12 x 8 =

Q94) 12 x 12 =

Q95) 9 x 10 =

Q96) 3 x 6 =

Q97) 7 x 9 =

Q98) 5 x 8 =

Q99) 14 x 11 =

Q100) 14 x 5 =

Q101) 7 x 5 =

Q102) 10 x 10 =

Q103) 12 x 8 =

Q104) 2 x 5 =

Q105) 12 x 7 =

Q106) 9 x 14 =

Q107) 9 x 6 =

Q108) 11 x 3 =

Q109) 14 x 5 =

Q110) 9 x 11 =

Q111) 12 x 5 =

Q112) 14 x 3 =

Q113) 3 x 8 =

Q114) 13 x 10 =

Q115) 13 x 4 =

Q116) 4 x 5 =

Q117) 13 x 2 =

Q118) 10 x 4 =

Q119) 12 x 8 =

Q120) 7 x 13 =

Q121) 6 x 11 =

Q122) 9 x 4 =

Q123) 4 x 10 =

Q124) 4 x 10 =

Q125) 4 x 7 =

Q126) 8 x 2 =

Q127) 7 x 8 =

Q128) 5 x 13 =

Q129) 14 x 5 =

Q130) 11 x 13 =

Q131) 13 x 8 =

Q132) 3 x 8 =

Q133) 6 x 5 =

Q134) 4 x 4 =

Q135) 9 x 4 =

Q136) 4 x 9 =

Q137) 4 x 3 =

Q138) 4 x 13 =

Q139) 4 x 6 =

Q140) 5 x 7 =

Q141) 5 x 11 =

Q142) 10 x 7 =

Q143) 5 x 7 =

Q144) 2 x 14 =

Q145) 13 x 12 =

Q146) 8 x 11 =

Q147) 11 x 4 =

Q148) 13 x 9 =

Q149) 13 x 9 =

Q150) 6 x 11 =

Q151) 9 x 13 =

Q152) 2 x 10 =

Q153) 12 x 9 =

Q154) 8 x 3 =

Q155) 2 x 11 =

Q156) 5 x 3 =

Q157) 10 x 8 =

Q158) 8 x 5 =

Q159) 11 x 7 =

Q160) 11 x 5 =

Q161) 7 x 6 =

Q162) 10 x 10 =

Q163) 9 x 7 =

Q164) 13 x 4 =

Q165) 10 x 11 =

Q166) 13 x 2 =

Q167) 3 x 14 =

Q168) 8 x 4 =

Q169) 6 x 12 =

Q170) 12 x 9 =

Q171) 12 x 9 =

Q172) 3 x 10 =

Q173) 7 x 5 =

Q174) 8 x 11 =

Q175) 6 x 3 =

Q176) 3 x 8 =

Q177) 3 x 6 =

Q178) 11 x 8 =

Q179) 3 x 9 =

Q180) 8 x 9 =

Q181) 11 x 14 =

Q182) 6 x 6 =

Q183) 10 x 12 =

Q184) 14 x 8 =

Q185) 9 x 10 =

Q186) 11 x 12 =

Q187) 4 x 13 =

Q188) 2 x 13 =

Q189) 13 x 10 =

Q190) 10 x 13 =

Q191) 13 x 9 =

Q192) 7 x 8 =

Q193) 9 x 2 =

Q194) 9 x 7 =

Q195) 3 x 10 =

Q196) 2 x 2 =

Q197) 13 x 7 =

Q198) 11 x 4 =

Q199) 6 x 6 =

Q200) 11 x 11 =

Q201) 10 x 6 =

Q202) 5 x 7 =

Q203) 6 x 11 =

Q204) 3 x 13 =

Q205) 9 x 11 =

Q206) 13 x 2 =

Q207) 4 x 11 =

Q208) 8 x 14 =

Q209) 8 x 5 =

Q210) 12 x 8 =

Q211) 12 x 12 =

Q212) 4 x 11 =

Q213) 14 x 10 =

Q214) 6 x 11 =

Q215) 9 x 5 =

Q216) 10 x 8 =

Q217) 11 x 14 =

Q218) 5 x 9 =

Q219) 3 x 4 =

Q220) 8 x 5 =

Q221) 12 x 12 =

Q222) 5 x 6 =

Q223) 11 x 3 =

Q224) 6 x 9 =

Q225) 6 x 5 =

Q226) 12 x 4 =

Q227) 4 x 12 =

Q228) 11 x 8 =

Q229) 3 x 2 =

Q230) 3 x 13 =

Q231) 4 x 14 =

Q232) 4 x 2 =

Q233) 6 x 8 =

Q234) 14 x 3 =

Q235) 3 x 13 =

Q236) 14 x 8 =

Q237) 5 x 13 =

Q238) 6 x 14 =

Q239) 5 x 7 =

Q240) 13 x 12 =

Q241) 9 x 12 =

Q242) 5 x 10 =

Q243) 12 x 4 =

Q244) 14 x 4 =

Q245) 12 x 14 =

Q246) 7 x 3 =

Q247) 7 x 8 =

Q248) 4 x 8 =

Q249) 2 x 5 =

Q250) 3 x 3 =

Q251) 10 x 9 =

Q252) 5 x 4 =

Q253) 11 x 9 =

Q254) 3 x 10 =

Q255) 6 x 4 =

Q256) 12 x 9 =

Q257) 12 x 14 =

Q258) 9 x 7 =

Q259) 8 x 9 =

Q260) 4 x 2 =

Q261) 9 x 14 =

Q262) 6 x 11 =

Q263) 13 x 14 =

Q264) 14 x 8 =

Q265) 9 x 11 =

Q266) 2 x 3 =

Q267) 11 x 13 =

Q268) 2 x 2 =

Q269) 2 x 3 =

Q270) 9 x 12 =

Q271) 2 x 11 =

Q272) 3 x 13 =

Q273) 3 x 5 =

Q274) 8 x 11 =

Q275) 3 x 13 =

Q276) 11 x 11 =

Q277) 7 x 12 =

Q278) 13 x 6 =

Q279) 11 x 14 =

Q280) 11 x 14 =

Q281) 4 x 9 =

Q282) 2 x 7 =

Q283) 8 x 7 =

Q284) 5 x 2 =

Q285) 10 x 4 =

Q286) 12 x 10 =

Q287) 8 x 4 =

Q288) 8 x 10 =

Q289) 2 x 3 =

Q290) 8 x 2 =

Q291) 14 x 6 =

Q292) 11 x 12 =

Q293) 2 x 8 =

Q294) 12 x 12 =

Q295) 3 x 7 =

Q296) 4 x 13 =

Q297) 3 x 10 =

Q298) 2 x 5 =

Q299) 9 x 4 =

Q300) 4 x 5 =

Q301) 6 x 13 =

Q302) 13 x 5 =

Q303) 10 x 14 =

Q304) 14 x 3 =

Q305) 9 x 9 =

Q306) 10 x 3 =

Q307) 11 x 14 =

Q308) 12 x 10 =

Q309) 10 x 7 =

Q310) 14 x 3 =

Q311) 4 x 10 =

Q312) 9 x 4 =

Q313) 3 x 9 =

Q314) 8 x 12 =

Q315) 5 x 14 =

Q316) 12 x 12 =

Q317) 9 x 3 =

Q318) 13 x 12 =

Q319) 12 x 8 =

Q320) 2 x 2 =

Q321) 12 x 7 =

Q322) 12 x 7 =

Q323) 4 x 8 =

Q324) 5 x 4 =

Q325) 13 x 10 =

Q326) 4 x 8 =

Q327) 10 x 8 =

Q328) 9 x 11 =

Q329) 2 x 12 =

Q330) 10 x 6 =

Q331) 14 x 10 =

Q332) 10 x 9 =

Q333) 2 x 6 =

Q334) 3 x 12 =

Q335) 14 x 2 =

Q336) 14 x 5 =

Q337) 4 x 14 =

Q338) 7 x 3 =

Q339) 6 x 5 =

Q340) 4 x 5 =

Q341) 5 x 4 =

Q342) 13 x 7 =

Q343) 10 x 2 =

Q344) 10 x 5 =

Q345) 8 x 2 =

Q346) 14 x 2 =

Q347) 4 x 11 =

Q348) 11 x 14 =

Q349) 12 x 7 =

Q350) 7 x 11 =

Q351) 4 x 11 =

Q352) 3 x 3 =

Q353) 4 x 12 =

Q354) 9 x 6 =

Q355) 5 x 4 =

Q356) 6 x 3 =

Q357) 6 x 9 =

Q358) 2 x 13 =

Q359) 6 x 3 =

Q360) 4 x 10 =

Q361) 12 x 11 =

Q362) 8 x 7 =

Q363) 8 x 3 =

Q364) 9 x 5 =

Q365) 12 x 10 =

Q366) 3 x 4 =

Q367) 3 x 7 =

Q368) 13 x 2 =

Q369) 6 x 7 =

Q370) 10 x 10 =

Q371) 3 x 9 =

Q372) 10 x 4 =

Q373) 8 x 4 =

Q374) 3 x 8 =

Q375) 10 x 11 =

Q376) 10 x 14 =

Q377) 8 x 13 =

Q378) 5 x 4 =

Q379) 13 x 6 =

Q380) 13 x 10 =

Q381) 14 x 3 =

Q382) 13 x 10 =

Q383) 14 x 11 =

Q384) 10 x 6 =

Q385) 11 x 4 =

Q386) 3 x 8 =

Q387) 11 x 11 =

Q388) 8 x 13 =

Q389) 14 x 14 =

Q390) 12 x 3 =

Q391) 7 x 2 =

Q392) 5 x 4 =

Q393) 9 x 2 =

Q394) 2 x 9 =

Q395) 4 x 13 =

Q396) 3 x 5 =

Q397) 14 x 10 =

Q398) 2 x 3 =

Q399) 7 x 8 =

Q400) 2 x 12 =

Q401) 12 x 6 =

Q402) 4 x 9 =

Q403) 5 x 13 =

Q404) 3 x 3 =

Q405) 4 x 12 =

Q406) 11 x 12 =

Q407) 6 x 2 =

Q408) 4 x 12 =

Q409) 10 x 13 =

Q410) 14 x 13 =

Q411) 3 x 12 =

Q412) 9 x 9 =

Q413) 12 x 5 =

Q414) 13 x 11 =

Q415) 7 x 8 =

Q416) 7 x 10 =

Q417) 13 x 9 =

Q418) 4 x 10 =

Q419) 3 x 4 =

Q420) 12 x 9 =

Q421) 12 x 13 =

Q422) 14 x 5 =

Q423) 13 x 4 =

Q424) 4 x 4 =

Q425) 7 x 8 =

Q426) 3 x 6 =

Q427) 3 x 4 =

Q428) 2 x 13 =

Q429) 5 x 8 =

Q430) 10 x 10 =

Q431) 3 x 6 =

Q432) 9 x 7 =

Q433) 13 x 12 =

Q434) 3 x 7 =

Q435) 4 x 12 =

Q436) 3 x 7 =

Q437) 14 x 12 =

Q438) 12 x 3 =

Q439) 6 x 11 =

Q440) 7 x 8 =

Q441) 10 x 4 =

Q442) 8 x 4 =

Q443) 10 x 3 =

Q444) 5 x 5 =

Q445) 14 x 12 =

Q446) 3 x 3 =

Q447) 8 x 7 =

Q448) 14 x 13 =

Q449) 3 x 14 =

Q450) 4 x 11 =

Q451) 3 x 14 =

Q452) 10 x 7 =

Q453) 11 x 14 =

Q454) 12 x 11 =

Q455) 12 x 8 =

Q456) 12 x 7 =

Q457) 13 x 11 =

Q458) 2 x 4 =

Q459) 3 x 14 =

Q460) 8 x 12 =

Q461) 13 x 3 =

Q462) 7 x 12 =

Q463) 14 x 10 =

Q464) 10 x 5 =

Q465) 5 x 5 =

Q466) 10 x 13 =

Q467) 3 x 5 =

Q468) 10 x 5 =

Q469) 13 x 9 =

Q470) 7 x 8 =

Q471) 5 x 2 =

Q472) 4 x 3 =

Q473) 2 x 11 =

Q474) 6 x 10 =

Q475) 2 x 12 =

Q476) 7 x 12 =

Q477) 8 x 11 =

Q478) 12 x 12 =

Q479) 2 x 13 =

Q480) 4 x 7 =

Q481) 4 x 10 =

Q482) 9 x 11 =

Q483) 12 x 14 =

Q484) 7 x 11 =

Q485) 10 x 6 =

Q486) 5 x 11 =

Q487) 5 x 13 =

Q488) 13 x 2 =

Q489) 11 x 13 =

Q490) 2 x 3 =

Q491) 12 x 12 =

Q492) 11 x 5 =

Q493) 12 x 7 =

Q494) 2 x 3 =

Q495) 8 x 2 =

Q496) 13 x 14 =

Q497) 6 x 6 =

Q498) 10 x 4 =

Q499) 9 x 3 =

Q500) 3 x 10 =

Q501) 4 x 14 =

Q502) 4 x 6 =

Q503) 10 x 13 =

Q504) 14 x 5 =

Q505) 3 x 13 =

Q506) 11 x 8 =

Q507) 5 x 10 =

Q508) 11 x 2 =

Q509) 4 x 14 =

Q510) 13 x 13 =

Q511) 14 x 14 =

Q512) 5 x 11 =

Q513) 9 x 14 =

Q514) 4 x 13 =

Q515) 13 x 11 =

Q516) 11 x 13 =

Q517) 13 x 9 =

Q518) 6 x 11 =

Q519) 6 x 6 =

Q520) 2 x 2 =

Q521) 14 x 13 =

Q522) 5 x 5 =

Q523) 2 x 6 =

Q524) 3 x 13 =

Q525) 12 x 9 =

Q526) 10 x 14 =

Q527) 5 x 14 =

Q528) 5 x 4 =

Q529) 7 x 10 =

Q530) 4 x 14 =

Q531) 13 x 7 =

Q532) 12 x 9 =

Q533) 3 x 8 =

Q534) 7 x 9 =

Q535) 13 x 10 =

Q536) 11 x 12 =

Q537) 10 x 3 =

Q538) 5 x 5 =

Q539) 10 x 10 =

Q540) 14 x 9 =

Q541) 11 x 11 =

Q542) 6 x 7 =

Q543) 8 x 11 =

Q544) 8 x 12 =

Q545) 10 x 7 =

Q546) 5 x 10 =

Q547) 6 x 9 =

Q548) 8 x 7 =

Q549) 5 x 2 =

Q550) 6 x 5 =

Q551) 9 x 2 =

Q552) 7 x 2 =

Q553) 5 x 3 =

Q554) 8 x 5 =

Q555) 2 x 2 =

Q556) 10 x 12 =

Q557) 8 x 9 =

Q558) 13 x 6 =

Q559) 14 x 12 =

Q560) 3 x 11 =

Q561) 7 x 3 =

Q562) 5 x 6 =

Q563) 2 x 4 =

Q564) 13 x 11 =

Q565) 11 x 2 =

Q566) 6 x 6 =

Q567) 10 x 8 =

Q568) 13 x 5 =

Q569) 10 x 3 =

Q570) 2 x 7 =

Q571) 9 x 7 =

Q572) 11 x 13 =

Q573) 2 x 3 =

Q574) 8 x 6 =

Q575) 11 x 14 =

Q576) 13 x 11 =

Q577) 13 x 14 =

Q578) 3 x 9 =

Q579) 2 x 4 =

Q580) 8 x 3 =

Q581) 12 x 7 =

Q582) 8 x 3 =

Q583) 13 x 4 =

Q584) 11 x 7 =

Q585) 3 x 4 =

Q586) 3 x 14 =

Q587) 13 x 3 =

Q588) 3 x 14 =

Q589) 10 x 11 =

Q590) 8 x 7 =

Q591) 2 x 11 =

Q592) 8 x 7 =

Q593) 9 x 2 =

Q594) 8 x 12 =

Q595) 5 x 10 =

Q596) 3 x 2 =

Q597) 10 x 7 =

Q598) 9 x 5 =

Q599) 2 x 14 =

Q600) 8 x 11 =

Q601) 4 x 11 =

Q602) 2 x 10 =

Q603) 6 x 11 =

Q604) 8 x 4 =

Q605) 9 x 7 =

Q606) 5 x 3 =

Q607) 8 x 10 =

Q608) 13 x 6 =

Q609) 7 x 9 =

Q610) 14 x 11 =

Q611) 11 x 2 =

Q612) 5 x 13 =

Q613) 6 x 4 =

Q614) 9 x 5 =

Q615) 6 x 4 =

Q616) 9 x 5 =

Q617) 6 x 12 =

Q618) 6 x 12 =

Q619) 7 x 12 =

Q620) 4 x 8 =

Q621) 4 x 7 =

Q622) 7 x 13 =

Q623) 11 x 12 =

Q624) 5 x 5 =

Q625) 3 x 11 =

Q626) 4 x 10 =

Q627) 7 x 6 =

Q628) 11 x 3 =

Q629) 12 x 12 =

Q630) 5 x 9 =

Q631) 4 x 10 =

Q632) 11 x 7 =

Q633) 14 x 4 =

Q634) 3 x 13 =

Q635) 12 x 14 =

Q636) 13 x 9 =

Q637) 6 x 3 =

Q638) 7 x 11 =

Q639) 3 x 6 =

Q640) 11 x 9 =

Q641) 2 x 9 =

Q642) 10 x 4 =

Q643) 13 x 6 =

Q644) 13 x 14 =

Q645) 8 x 10 =

Q646) 12 x 13 =

Q647) 14 x 12 =

Q648) 11 x 12 =

Q649) 2 x 4 =

Q650) 12 x 7 =

Q651) 3 x 5 =

Q652) 2 x 10 =

Q653) 3 x 5 =

Q654) 2 x 6 =

Q655) 6 x 4 =

Q656) 14 x 9 =

Q657) 6 x 3 =

Q658) 13 x 6 =

Q659) 11 x 13 =

Q660) 2 x 12 =

Q661) 5 x 5 =

Q662) 7 x 13 =

Q663) 13 x 8 =

Q664) 4 x 9 =

Q665) 9 x 3 =

Q666) 14 x 3 =

Q667) 7 x 13 =

Q668) 3 x 14 =

Q669) 14 x 14 =

Q670) 13 x 12 =

Q671) 4 x 13 =

Q672) 6 x 14 =

Q673) 9 x 11 =

Q674) 8 x 10 =

Q675) 2 x 13 =

Q676) 13 x 9 =

Q677) 10 x 8 =

Q678) 12 x 7 =

Q679) 9 x 11 =

Q680) 8 x 8 =

Q681) 4 x 12 =

Q682) 9 x 3 =

Q683) 9 x 11 =

Q684) 8 x 11 =

Q685) 13 x 13 =

Q686) 12 x 6 =

Q687) 10 x 2 =

Q688) 3 x 11 =

Q689) 3 x 3 =

Q690) 6 x 7 =

Q691) 13 x 3 =

Q692) 5 x 11 =

Q693) 14 x 8 =

Q694) 4 x 5 =

Q695) 11 x 8 =

Q696) 12 x 13 =

Q697) 12 x 6 =

Q698) 5 x 9 =

Q699) 8 x 13 =

Q700) 4 x 7 =

Q701) 12 x 11 =

Q702) 4 x 12 =

Q703) 7 x 12 =

Q704) 14 x 8 =

Q705) 4 x 2 =

Q706) 2 x 13 =

Q707) 5 x 7 =

Q708) 9 x 12 =

Q709) 2 x 6 =

Q710) 14 x 12 =

Q711) 7 x 5 =

Q712) 14 x 6 =

Q713) 10 x 6 =

Q714) 7 x 8 =

Q715) 11 x 7 =

Q716) 5 x 6 =

Q717) 11 x 8 =

Q718) 14 x 9 =

Q719) 5 x 6 =

Q720) 10 x 6 =

Q721) 13 x 6 =

Q722) 13 x 10 =

Q723) 7 x 5 =

Q724) 9 x 14 =

Q725) 8 x 13 =

Q726) 14 x 11 =

Q727) 11 x 6 =

Q728) 12 x 9 =

Q729) 9 x 2 =

Q730) 7 x 13 =

Q731) 12 x 11 =

Q732) 2 x 10 =

Q733) 4 x 10 =

Q734) 6 x 10 =

Q735) 4 x 4 =

Q736) 4 x 11 =

Q737) 10 x 12 =

Q738) 13 x 6 =

Q739) 4 x 6 =

Q740) 8 x 7 =

Q741) 5 x 9 =

Q742) 12 x 13 =

Q743) 14 x 11 =

Q744) 2 x 10 =

Q745) 11 x 3 =

Q746) 7 x 14 =

Q747) 3 x 9 =

Q748) 3 x 5 =

Q749) 13 x 4 =

Q750) 4 x 5 =

Q751) 7 x 6 =

Q752) 14 x 9 =

Q753) 3 x 9 =

Q754) 13 x 11 =

Q755) 12 x 12 =

Q756) 6 x 4 =

Q757) 4 x 11 =

Q758) 2 x 6 =

Q759) 13 x 6 =

Q760) 4 x 10 =

Q761) 13 x 2 =

Q762) 6 x 11 =

Q763) 9 x 12 =

Q764) 2 x 9 =

Q765) 6 x 3 =

Q766) 8 x 4 =

Q767) 10 x 8 =

Q768) 8 x 11 =

Q769) 10 x 14 =

Q770) 5 x 10 =

Q771) 8 x 2 =

Q772) 8 x 3 =

Q773) 12 x 4 =

Q774) 2 x 7 =

Q775) 13 x 14 =

Q776) 5 x 2 =

Q777) 3 x 10 =

Q778) 13 x 3 =

Q779) 5 x 7 =

Q780) 9 x 8 =

Q781) 14 x 10 =

Q782) 14 x 2 =

Q783) 14 x 5 =

Q784) 9 x 9 =

Q785) 9 x 3 =

Q786) 12 x 11 =

Q787) 3 x 8 =

Q788) 9 x 12 =

Q789) 4 x 2 =

Q790) 9 x 6 =

Q791) 4 x 14 =

Q792) 10 x 13 =

Q793) 7 x 5 =

Q794) 14 x 8 =

Q795) 10 x 4 =

Q796) 3 x 13 =

Q797) 10 x 12 =

Q798) 3 x 14 =

Q799) 6 x 14 =

Q800) 4 x 3 =

Q801) 14 x 12 =

Q802) 6 x 2 =

Q803) 3 x 7 =

Q804) 5 x 8 =

Q805) 10 x 6 =

Q806) 3 x 2 =

Q807) 9 x 2 =

Q808) 8 x 14 =

Q809) 10 x 3 =

Q810) 10 x 8 =

Q811) 14 x 8 =

Q812) 9 x 6 =

Q813) 3 x 12 =

Q814) 9 x 6 =

Q815) 2 x 8 =

Q816) 13 x 2 =

Q817) 11 x 11 =

Q818) 9 x 13 =

Q819) 11 x 10 =

Q820) 2 x 10 =

Q821) 2 x 11 =

Q822) 14 x 4 =

Q823) 9 x 5 =

Q824) 10 x 7 =

Q825) 4 x 6 =

Q826) 10 x 7 =

Q827) 4 x 8 =

Q828) 3 x 11 =

Q829) 12 x 14 =

Q830) 5 x 5 =

Q831) 13 x 7 =

Q832) 2 x 3 =

Q833) 13 x 2 =

Q834) 4 x 3 =

Q835) 7 x 10 =

Q836) 5 x 2 =

Q837) 6 x 10 =

Q838) 4 x 5 =

Q839) 4 x 7 =

Q840) 10 x 14 =

Q841) 14 x 14 =

Q842) 11 x 2 =

Q843) 3 x 5 =

Q844) 6 x 7 =

Q845) 4 x 14 =

Q846) 8 x 12 =

Q847) 12 x 8 =

Q848) 14 x 13 =

Q849) 8 x 11 =

Q850) 14 x 11 =

Q851) 6 x 11 =

Q852) 9 x 9 =

Q853) 14 x 8 =

Q854) 4 x 7 =

Q855) 5 x 7 =

Q856) 8 x 11 =

Q857) 11 x 6 =

Q858) 3 x 10 =

Q859) 8 x 14 =

Q860) 2 x 12 =

Q861) 10 x 14 =

Q862) 10 x 5 =

Q863) 14 x 5 =

Q864) 7 x 10 =

Q865) 7 x 14 =

Q866) 7 x 5 =

Q867) 10 x 10 =

Q868) 6 x 5 =

Q869) 6 x 14 =

Q870) 13 x 11 =

Q871) 2 x 3 =

Q872) 11 x 6 =

Q873) 9 x 7 =

Q874) 3 x 7 =

Q875) 13 x 10 =

Q876) 7 x 11 =

Q877) 8 x 11 =

Q878) 11 x 13 =

Q879) 7 x 2 =

Q880) 8 x 10 =

Q881) 7 x 12 =

Q882) 9 x 14 =

Q883) 7 x 12 =

Q884) 8 x 13 =

Q885) 10 x 2 =

Q886) 2 x 13 =

Q887) 13 x 7 =

Q888) 13 x 7 =

Q889) 9 x 13 =

Q890) 2 x 5 =

Q891) 13 x 9 =

Q892) 7 x 7 =

Q893) 9 x 11 =

Q894) 7 x 3 =

Q895) 14 x 12 =

Q896) 7 x 5 =

Q897) 9 x 6 =

Q898) 3 x 14 =

Q899) 3 x 6 =

Q900) 11 x 9 =

Q901) 8 x 13 =

Q902) 14 x 12 =

Q903) 14 x 8 =

Q904) 3 x 12 =

Q905) 3 x 2 =

Q906) 8 x 2 =

Q907) 14 x 8 =

Q908) 11 x 8 =

Q909) 8 x 4 =

Q910) 3 x 9 =

Q911) 7 x 14 =

Q912) 4 x 2 =

Q913) 2 x 7 =

Q914) 8 x 12 =

Q915) 6 x 7 =

Q916) 11 x 10 =

Q917) 7 x 14 =

Q918) 6 x 9 =

Q919) 3 x 5 =

Q920) 2 x 9 =

Q921) 14 x 3 =

Q922) 6 x 3 =

Q923) 2 x 3 =

Q924) 7 x 9 =

Q925) 11 x 9 =

Q926) 12 x 7 =

Q927) 12 x 13 =

Q928) 8 x 6 =

Q929) 10 x 8 =

Q930) 9 x 11 =

Q931) 10 x 7 =

Q932) 11 x 9 =

Q933) 4 x 13 =

Q934) 4 x 2 =

Q935) 8 x 7 =

Q936) 2 x 13 =

Q937) 11 x 2 =

Q938) 3 x 12 =

Q939) 12 x 3 =

Q940) 6 x 9 =

Q941) 9 x 14 =

Q942) 7 x 9 =

Q943) 2 x 8 =

Q944) 2 x 9 =

Q945) 11 x 6 =

Q946) 6 x 8 =

Q947) 13 x 7 =

Q948) 5 x 14 =

Q949) 8 x 12 =

Q950) 5 x 14 =

Q951) 7 x 2 =

Q952) 5 x 8 =

Q953) 14 x 13 =

Q954) 4 x 8 =

Q955) 8 x 14 =

Q956) 6 x 12 =

Q957) 9 x 6 =

Q958) 14 x 6 =

Q959) 10 x 2 =

Q960) 6 x 11 =

Q961) 9 x 8 =

Q962) 3 x 7 =

Q963) 7 x 13 =

Q964) 7 x 9 =

Q965) 5 x 10 =

Q966) 3 x 2 =

Q967) 3 x 4 =

Q968) 10 x 6 =

Q969) 9 x 4 =

Q970) 8 x 10 =

Q971) 3 x 12 =

Q972) 6 x 14 =

Q973) 13 x 3 =

Q974) 12 x 5 =

Q975) 7 x 2 =

Q976) 11 x 13 =

Q977) 12 x 4 =

Q978) 10 x 2 =

Q979) 2 x 8 =

Q980) 3 x 8 =

Q981) 5 x 3 =

Q982) 13 x 9 =

Q983) 2 x 9 =

Q984) 10 x 14 =

Q985) 13 x 7 =

Q986) 8 x 2 =

Q987) 11 x 7 =

Q988) 11 x 9 =

Q989) 10 x 5 =

Q990) 2 x 8 =

Q991) 14 x 13 =

Q992) 7 x 5 =

Q993) 2 x 14 =

Q994) 10 x 10 =

Q995) 12 x 10 =

Q996) 8 x 8 =

Q997) 3 x 10 =

Q998) 10 x 8 =

Q999) 10 x 13 =

Q1000) 7 x 14 =

Q1) 91

Q2) 90

Q3) 80

Q4) 50

Q5) 117

Q6) 49

Q7) 132

Q8) 24

Q9) 108

Q10) 12

Q11) 16

Q12) 20

Q13) 132

Q14) 50

Q15) 70

Q16) 132

Q17) 78

Q18) 108

Q19) 39

Q20) 30

Q21) 15

Q22) 98

Q23) 168

Q24) 99

Q25) 132

Q26) 77

Q27) 140

Q28) 52

Q29) 78

Q30) 50

Q31) 28

Q32) 96

Q33) 20

Q34) 9

Q35) 156

Q36) 63

Q37) 33

Q38) 81

Q39) 126

Q40) 28

Q41) 32

Q42) 50

Q43) 12

Q44) 54

Q45) 40

Q46) 42

Q47) 78

Q48) 18

Q49) 16

Q50) 16

Q51) 30

Q52) 70

Q53) 6

Q54) 40

Q55) 60

Q56) 88

Q57) 52

Q58) 15

Q59) 72

Q60) 4

Q61) 42

Q62) 28

Q63) 39

Q64) 120

Q65) 28

Q66) 30

Q67) 6

Q68) 33

Q69) 40

Q70) 130

Q71) 26

Q72) 30

Q73) 84

Q74) 14

Q75) 56

Q76) 48

Q77) 88

Q78) 39

Q79) 36

Q80) 6

Q81) 24

Q82) 120

Q83) 117

Q84) 110

Q85) 72

Q86) 60

Q87) 6

Q88) 98

Q89) 18

Q90) 156

Q91) 108

Q92) 64

Q93) 96

Q94) 144

Q95) 90

Q96) 18

Q97) 63

Q98) 40

Q99) 154

Q100) 70

Q101) 35

Q102) 100

Q103) 96

Q104) 10

Q105) 84

Q106) 126

Q107) 54

Q108) 33

Q109) 70

Q110) 99

Q111) 60

Q112) 42

Q113) 24

Q114) 130

Q115) 52

Q116) 20

Q117) 26

Q118) 40

Q119) 96

Q120) 91

Q121) 66

Q122) 36

Q123) 40

Q124) 40

Q125) 28

Q126) 16

Q127) 56

Q128) 65

Q129) 70

Q130) 143

Q131) 104

Q132) 24

Q133) 30

Q134) 16

Q135) 36

Q136) 36

Q137) 12

Q138) 52

Q139) 24

Q140) 35

Q141) 55

Q142) 70

Q143) 35

Q144) 28

Q145) 156

Q146) 88

Q147) 44

Q148) 117

Q149) 117

Q150) 66

Q151) 117

Q152) 20

Q153) 108

Q154) 24

Q155) 22

Q156) 15

Q157) 80

Q158) 40

Q159) 77

Q160) 55

Q161) 42

Q162) 100

Q163) 63

Q164) 52

Q165) 110

Q166) 26

Q167) 42

Q168) 32

Q169) 72

Q170) 108

Q171) 108

Q172) 30

Q173) 35

Q174) 88

Q175) 18

Q176) 24

Q177) 18

Q178) 88

Q179) 27

Q180) 72

Q181) 154

Q182) 36

Q183) 120

Q184) 112

Q185) 90

Q186) 132

Q187) 52

Q188) 26

Q189) 130

Q190) 130

Q191) 117

Q192) 56

Q193) 18

Q194) 63

Q195) 30

Q196) 4

Q197) 91

Q198) 44

Q199) 36

Q200) 121

Q201) 60

Q202) 35

Q203) 66

Q204) 39

Q205) 99

Q206) 26

Q207) 44

Q208) 112

Q209) 40

Q210) 96

Q211) 144

Q212) 44

Q213) 140

Q214) 66

Q215) 45

Q216) 80

Q217) 154

Q218) 45

Q219) 12

Q220) 40

Q221) 144

Q222) 30

Q223) 33

Q224) 54

Q225) 30

Q226) 48

Q227) 48

Q228) 88

Q229) 6

Q230) 39

Q231) 56

Q232) 8

Q233) 48

Q234) 42

Q235) 39

Q236) 112

Q237) 65

Q238) 84

Q239) 35

Q240) 156

Q241) 108

Q242) 50

Q243) 48

Q244) 56

Q245) 168

Q246) 21

Q247) 56

Q248) 32

Q249) 10

Q250) 9

Q251) 90

Q252) 20

Q253) 99

Q254) 30

Q255) 24

Q256) 108

Q257) 168

Q258) 63

Q259) 72

Q260) 8

Q261) 126

Q262) 66

Q263) 182

Q264) 112

Q265) 99

Q266) 6

Q267) 143

Q268) 4

Q269) 6

Q270) 108

Q271) 22

Q272) 39

Q273) 15

Q274) 88

Q275) 39

Q276) 121

Q277) 84

Q278) 78

Q279) 154

Q280) 154

Q281) 36

Q282) 14

Q283) 56

Q284) 10

Q285) 40

Q286) 120

Q287) 32

Q288) 80

Q289) 6

Q290) 16

Q291) 84

Q292) 132

Q293) 16

Q294) 144

Q295) 21

Q296) 52

Q297) 30

Q298) 10

Q299) 36

Q300) 20

Q301) 78

Q302) 65

Q303) 140

Q304) 42

Q305) 81

Q306) 30

Q307) 154

Q308) 120

Q309) 70

Q310) 42

Q311) 40

Q312) 36

Q313) 27

Q314) 96

Q315) 70

Q316) 144

Q317) 27

Q318) 156

Q319) 96

Q320) 4

Q321) 84

Q322) 84

Q323) 32

Q324) 20

Q325) 130

Q326) 32

Q327) 80

Q328) 99

Q329) 24

Q330) 60

Q331) 140

Q332) 90

Q333) 12

Q334) 36

Q335) 28

Q336) 70

Q337) 56

Q338) 21

Q339) 30

Q340) 20

Q341) 20

Q342) 91

Q343) 20

Q344) 50

Q345) 16

Q346) 28

Q347) 44

Q348) 154

Q349) 84

Q350) 77

Q351) 44

Q352) 9

Q353) 48

Q354) 54

Q355) 20

Q356) 18

Q357) 54

Q358) 26

Q359) 18

Q360) 40

Q361) 132

Q362) 56

Q363) 24

Q364) 45

Q365) 120

Q366) 12

Q367) 21

Q368) 26

Q369) 42

Q370) 100

Q371) 27

Q372) 40

Q373) 32

Q374) 24

Q375) 110

Q376) 140

Q377) 104

Q378) 20

Q379) 78

Q380) 130

Q381) 42

Q382) 130

Q383) 154

Q384) 60

Q385) 44

Q386) 24

Q387) 121

Q388) 104

Q389) 196

Q390) 36

Q391) 14

Q392) 20

Q393) 18

Q394) 18

Q395) 52

Q396) 15

Q397) 140

Q398) 6

Q399) 56

Q400) 24

Q401) 72

Q402) 36

Q403) 65

Q404) 9

Q405) 48

Q406) 132

Q407) 12

Q408) 48

Q409) 130

Q410) 182

Q411) 36

Q412) 81

Q413) 60

Q414) 143

Q415) 56

Q416) 70

Q417) 117

Q418) 40

Q419) 12

Q420) 108

Q421) 156

Q422) 70

Q423) 52

Q424) 16

Q425) 56

Q426) 18

Q427) 12

Q428) 26

Q429) 40

Q430) 100

Q431) 18

Q432) 63

Q433) 156

Q434) 21

Q435) 48

Q436) 21

Q437) 168

Q438) 36

Q439) 66

Q440) 56

Q441) 40

Q442) 32

Q443) 30

Q444) 25

Q445) 168

Q446) 9

Q447) 56

Q448) 182

Q449) 42

Q450) 44

Q451) 42

Q452) 70

Q453) 154

Q454) 132

Q455) 96

Q456) 84

Q457) 143

Q458) 8

Q459) 42

Q460) 96

Q461) 39

Q462) 84

Q463) 140

Q464) 50

Q465) 25

Q466) 130

Q467) 15

Q468) 50

Q469) 117

Q470) 56

Q471) 10

Q472) 12

Q473) 22

Q474) 60

Q475) 24

Q476) 84

Q477) 88

Q478) 144

Q479) 26

Q480) 28

Q481) 40

Q482) 99

Q483) 168

Q484) 77

Q485) 60

Q486) 55

Q487) 65

Q488) 26

Q489) 143

Q490) 6

Q491) 144

Q492) 55

Q493) 84

Q494) 6

Q495) 16

Q496) 182

Q497) 36

Q498) 40

Q499) 27

Q500) 30

Q501) 56

Q502) 24

Q503) 130

Q504) 70

Q505) 39

Q506) 88

Q507) 50

Q508) 22

Q509) 56

Q510) 169

Q511) 196

Q512) 55

Q513) 126

Q514) 52

Q515) 143

Q516) 143

Q517) 117

Q518) 66

Q519) 36

Q520) 4

Q521) 182

Q522) 25

Q523) 12

Q524) 39

Q525) 108

Q526) 140

Q527) 70

Q528) 20

Q529) 70

Q530) 56

Q531) 91

Q532) 108

Q533) 24

Q534) 63

Q535) 130

Q536) 132

Q537) 30

Q538) 25

Q539) 100

Q540) 126

Q541) 121

Q542) 42

Q543) 88

Q544) 96

Q545) 70

Q546) 50

Q547) 54

Q548) 56

Q549) 10

Q550) 30

Q551) 18

Q552) 14

Q553) 15

Q554) 40

Q555) 4

Q556) 120

Q557) 72

Q558) 78

Q559) 168

Q560) 33

Q561) 21

Q562) 30

Q563) 8

Q564) 143

Q565) 22

Q566) 36

Q567) 80

Q568) 65

Q569) 30

Q570) 14

Q571) 63

Q572) 143

Q573) 6

Q574) 48

Q575) 154

Q576) 143

Q577) 182

Q578) 27

Q579) 8

Q580) 24

Q581) 84

Q582) 24

Q583) 52

Q584) 77

Q585) 12

Q586) 42

Q587) 39

Q588) 42

Q589) 110

Q590) 56

Q591) 22

Q592) 56

Q593) 18

Q594) 96

Q595) 50

Q596) 6

Q597) 70

Q598) 45

Q599) 28

Q600) 88

Q601) 44

Q602) 20

Q603) 66

Q604) 32

Q605) 63

Q606) 15

Q607) 80

Q608) 78

Q609) 63

Q610) 154

Q611) 22

Q612) 65

Q613) 24

Q614) 45

Q615) 24

Q616) 45

Q617) 72

Q618) 72

Q619) 84

Q620) 32

Q621) 28

Q622) 91

Q623) 132

Q624) 25

Q625) 33

Q626) 40

Q627) 42

Q628) 33

Q629) 144

Q630) 45

Q631) 40

Q632) 77

Q633) 56

Q634) 39

Q635) 168

Q636) 117

Q637) 18

Q638) 77

Q639) 18

Q640) 99

Q641) 18

Q642) 40

Q643) 78

Q644) 182

Q645) 80

Q646) 156

Q647) 168

Q648) 132

Q649) 8

Q650) 84

Q651) 15

Q652) 20

Q653) 15

Q654) 12

Q655) 24

Q656) 126

Q657) 18

Q658) 78

Q659) 143

Q660) 24

Q661) 25

Q662) 91

Q663) 104

Q664) 36

Q665) 27

Q666) 42

Q667) 91

Q668) 42

Q669) 196

Q670) 156

Q671) 52

Q672) 84

Q673) 99

Q674) 80

Q675) 26

Q676) 117

Q677) 80

Q678) 84

Q679) 99

Q680) 64

Q681) 48

Q682) 27

Q683) 99

Q684) 88

Q685) 169

Q686) 72

Q687) 20

Q688) 33

Q689) 9

Q690) 42

Q691) 39

Q692) 55

Q693) 112

Q694) 20

Q695) 88

Q696) 156

Q697) 72

Q698) 45

Q699) 104

Q700) 28

Q701) 132

Q702) 48

Q703) 84

Q704) 112

Q705) 8

Q706) 26

Q707) 35

Q708) 108

Q709) 12

Q710) 168

Q711) 35

Q712) 84

Q713) 60

Q714) 56

Q715) 77

Q716) 30

Q717) 88

Q718) 126

Q719) 30

Q720) 60

Q721) 78

Q722) 130

Q723) 35

Q724) 126

Q725) 104

Q726) 154

Q727) 66

Q728) 108

Q729) 18

Q730) 91

Q731) 132

Q732) 20

Q733) 40

Q734) 60

Q735) 16

Q736) 44

Q737) 120

Q738) 78

Q739) 24

Q740) 56

Q741) 45

Q742) 156

Q743) 154

Q744) 20

Q745) 33

Q746) 98

Q747) 27

Q748) 15

Q749) 52

Q750) 20

Q751) 42

Q752) 126

Q753) 27

Q754) 143

Q755) 144

Q756) 24

Q757) 44

Q758) 12

Q759) 78

Q760) 40

Q761) 26

Q762) 66

Q763) 108

Q764) 18

Q765) 18

Q766) 32

Q767) 80

Q768) 88

Q769) 140

Q770) 50

Q771) 16

Q772) 24

Q773) 48

Q774) 14

Q775) 182

Q776) 10

Q777) 30

Q778) 39

Q779) 35

Q780) 72

Q781) 140

Q782) 28

Q783) 70

Q784) 81

Q785) 27

Q786) 132

Q787) 24

Q788) 108

Q789) 8

Q790) 54

Q791) 56

Q792) 130

Q793) 35

Q794) 112

Q795) 40

Q796) 39

Q797) 120

Q798) 42

Q799) 84

Q800) 12

Q801) 168

Q802) 12

Q803) 21

Q804) 40

Q805) 60

Q806) 6

Q807) 18

Q808) 112

Q809) 30

Q810) 80

Q811) 112

Q812) 54

Q813) 36

Q814) 54

Q815) 16

Q816) 26

Q817) 121

Q818) 117

Q819) 110

Q820) 20

Q821) 22

Q822) 56

Q823) 45

Q824) 70

Q825) 24

Q826) 70

Q827) 32

Q828) 33

Q829) 168

Q830) 25

Q831) 91

Q832) 6

Q833) 26

Q834) 12

Q835) 70

Q836) 10

Q837) 60

Q838) 20

Q839) 28

Q840) 140

Q841) 196

Q842) 22

Q843) 15

Q844) 42

Q845) 56

Q846) 96

Q847) 96

Q848) 182

Q849) 88

Q850) 154

Q851) 66

Q852) 81

Q853) 112

Q854) 28

Q855) 35

Q856) 88

Q857) 66

Q858) 30

Q859) 112

Q860) 24

Q861) 140

Q862) 50

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Q871) 6

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Q873) 63

Q874) 21

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Q876) 77

Q877) 88

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Q879) 14

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Q883) 84

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Q885) 20

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Q888) 91

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Q890) 10

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Q892) 49

Q893) 99

Q894) 21

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Q899) 18

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Q901) 104

Q902) 168

Q903) 112

Q904) 36

Q905) 6

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Q911) 98

Q912) 8

Q913) 14

Q914) 96

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Q916) 110

Q917) 98

Q918) 54

Q919) 15

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Q921) 42

Q922) 18

Q923) 6

Q924) 63

Q925) 99

Q926) 84

Q927) 156

Q928) 48

Q929) 80

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Q931) 70

Q932) 99

Q933) 52

Q934) 8

Q935) 56

Q936) 26

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Q938) 36

Q939) 36

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Q941) 126

Q942) 63

Q943) 16

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Q946) 48

Q947) 91

Q948) 70

Q949) 96

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Q951) 14

Q952) 40

Q953) 182

Q954) 32

Q955) 112

Q956) 72

Q957) 54

Q958) 84

Q959) 20

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Q961) 72

Q962) 21

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Q964) 63

Q965) 50

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Q968) 60

Q969) 36

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Q971) 36

Q972) 84

Q973) 39

Q974) 60

Q975) 14

Q976) 143

Q977) 48

Q978) 20

Q979) 16

Q980) 24

Q981) 15

Q982) 117

Q983) 18

Q984) 140

Q985) 91

Q986) 16

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Q988) 99

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Q991) 182

Q992) 35

Q993) 28

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Q995) 120

Q996) 64

Q997) 30

Q998) 80

Q999) 130

Q1000) 98