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Mathematics For Fun

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Further Mathematics
 
                                                                
Logic
Problem Solving


Binomial Theorem



  Interesting properties of Pascal triangle
Limitsnew
Derivatives






 
 
  Application of differentiation in real life


MaxMin Problems
   

Interesting max min problem on Math Tools browse Applications of derivatives

Solid of revolution









Locus










 



More on Calculus








Functions with special properties


Applets and Demos

Useful websites
Calculus Toolkit MathServe Project by Vanderbilt University, USA


Curve Sketching by Vanderbilt University, USA


Function Plotter by Maths Online, University of Vienna


MathServ Calculus Toolkit by Vanderbilt University, USA
    Visual Calculus Calculus applets at www.jes.co.jp Math Tools Browse


Famous Curves by University of St. Andrews, Scotland


Java Applets by International Education Software


MathsTools, Maths Online Gallery , Visual Calculus


Calculus sites recommended by Math Forum


Thinkquest library algebra geometry trigonometry calculus
e
Application of e Compound interest; Franklin Benjamin's will
Matrices









System of Linear Equations

Partial Fractions
Thinkquest library - created by pre-collegiate students
Complex numbers

A treasure hunt

   Applets
 

A mathematics puzzle a day keeps you smart all day!

 

 

 

 

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Mathematics
 
Numbers  
Quadratic Equationsnew
Functionsnew
Transformations

 
Geometry


 

 
Links

 

There are many beautiful theorems in geometry.

Interesting links   X+Y Files various problems Other Links
    Puzzles All triangles are isosceles
    Math Pages animated illustrations All triangles are isosceles
    Cut the Knot Interactive Math Activities

newShoder.Interactivate: Clock Arithmetic     Suitcase of dreams: interesting facts and numbers
    MAA Online From Lewis Carrol to Archimedes
    Breaking away from the Math Book projects and lesson plans
    Jigsaw Paradox

newNrich problems for discussion
Sketchpad   Triangle sketchpad page by St. Louis University,
Trigonometry   How to generate sine, cosine and tangent curves; interactive exercises
3-dim trigonometry   Online Exercises sketchup tutorials
Polynomials  

Interactive exercises on remainder theorem and factor theorem (questions only)


  Synthetic Division - finding the quotient and remainder without using long division (powerpoint)

 

A surprising application of polynomials in daily life (powerpoint)


  Proof by dissection- a visual proof of (a+b)^2 (powerpoint)
AP and GP   A brainteaser about the sum of AP and GP Part I, II, III
Linear Programming   Important points to note
Probability  

Birthday problem

    Monty Hall problem
    One problem involving infinite series
Statistics   Shoder. Interactive Histogram; Normal distribution; Interactive standard score

 

 

 

 

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Brainteasers    
   
Games  
 
  • 24-point - Playing Tips
    • K stands for 13, Q stands for 12, J stand for 11, A stands for 1
    • Some hands have no solutions and marks will be awarded if you guess correctly
   
  • Tic Tac Toe - more challenging than the conventional version
Famous Math Games  
Videos
Youtube

 

 

 

 

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Websites    
Good reading  
More goodies...




















 

Eye opener

 

 

 

 

 

 

 

 

 

 

 

 
  • Numbers and Patterns Fun with Curves and Topology recommend excellent website to investigate the properties of prime numbers, Pascal triangle, to name a few.
  • Math puzzles.com - A website which reports interesting news, competitions, recommended websites and examples of math puzzles. Materials are added every 15-20 days. Archived Pages (in chronological order) and More mathpuzzle pages (in alphabetical order) are available in the right column. Puzzler's picks include Math Games in MAA online. The website offers a wide coverage of mathematical delights at different levels and pursues a topic of interest in great depth. For instance, sudoko variations, orthogonal sudoku and chess sudoku are all mentioned in recent updates.
  • Mathsnet Daily Puzzle - including one from "536 puzzles and Curious Problems" by Henry Earnest Dudeney , 1968
  • Eye opening series
  • g4g4 - This is the website originally set up for the G4G conferences held in honour of Martin Gardner. The website includes interesting topics in Mathematics, puzzles and games and links. In the recreational mathematics section, a pdf version of the book The mathmagician and pied puzzler is also available.
  • Numericana
  • Art of Problem Solving - A large online math community with the following features:
    • Math Jam - a 60-min problem solving session held, at least once a week, in a virtual classroom. Each transcript of Math Jams consists of a series of problems to be solved by the collaboration of the participants.
    • The community forum - Math problems on various topics are posted
    • Lists of books on problem solving
    • Flash animation on the left column to illustrate various mathematical formula
  • Projects (origami, coin sliding) 2001-2004 at McGill University -excellent library of projects, each explaining the mathematical problem, possibly with historical background, algorithm, applet

Enrichment




 
Math Links



 

 

 

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Exercises on trigonometry

Definition




Draw a circle with cenre (0,0) and radius 1. P is a point on the circle which makes an angle x with the positive x-axis. Idenitify the lengths sinx, cosx, tanx, cotx.

Trig graphs
   

Practice

 

 
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AP and GP

A brainteaser about the sum of an A.P.

Part I

In a kingdom far, far away, there once lived a King and 10 wealthy jewelers, very well-known in their trade. Every year the 10 jewelers paid tax to the King, each gave the King a stack of 10 gold coins. The real coins weighed exactly 1 oz each. This year the King received report that one and only one stack contained 10"light" coins, each having exactly 0.1 oz of gold shaved off the edge. The King now ordered his personal adviser, Mr. Fischer, to IDENTIFY the crooked jeweller and the stack of light coins with JUST ONE SINGLE weighing on a scale.

It took Mr. Fischer, intelligent as he was, the whole afternoon to think of a perfect solution. Even then, he wouldn't reveal much to his aides. All he asked them to do was label the stacks of the coins 1, 2, 3, ..., 10. Take one coin from stack 1, two coins from stack 2, three coins from stack 3, four coins from stack 4, and so on up to stack 10.

"Weigh the coins you just collected from the stacks," he instructed his aides.

"54.3 oz was the reading, sir!" The aides said after the ONE and ONLY ONE weighing allowed by the King.

"Very well, the crooked jeweler was the who gave stack number ...!" Mr. Fischer whispered to himself.

"What did you say, sir?" The aides were eager to know.

"See how many coins you just weighed and you will know which stack was faulty!" Mr. Fischer wanted to test his aides.

 

Now which stack was the one from the crooked jeweller?

 

Part II

 

The King was so impressed with Mr. Fischer's many achievements (finding the crooked jeweller was just one) that he told Mr. Fischer,"Ask me for anything you want. Whatever you ask I will give you, up to half my kingdom!"

 

Mr. Fischer immediately took the King to the royal court whose marble floor was exactly an 8x8 chessboard. "Your majesty, if you so wish, please ask one of your servants to put one grain on the first square, two grains on the second square, four grains on the third square, eight grains on the fourth square and so on, doubling the amount of the grains when it comes to the next square until the last square of the chessboard is filled, then let your humble servant have all the grains placed on the chessboard!"

 

The King was again very pleased with Mr. Fischer. "How modest this Mr. Fischer is! How considerate! He could have asked a lot more!"

The King was about to grant Mr. Fischer what he asked for. Nonetheless, reason got the better of him. As a mere formality, the King summoned his book-keeper, Mr. Anderson, to calculate how much all this would cost him. When Mr. Anderson came back and showed the King his calculations, the King simply couldn't believe his eyes. Why?

 

Can you calculate how many grains there are on the chessboard?

(At that time, annual grain production of the whole world is roughly 10 16grains.)

 

What should the King do with this Mr. Fischer? Should he honour his word and let Mr. Fischer be his biggest creditor or should he take back his word and ...?

 

Part III

 

Using similar calculations, which of the following is the best deal for Tom's pocket money for the next 2 weeks?
(a) $ 7 every day
(b) $ 1 for the first day, $ 2 for the second day, $ 3 for the third day, ...
(c) 1 c for the first day, 2 c for the second day, 4 c for the third day, 8 c for the fourth day, ...
Afterthought
(1) How different are (a)(b)(c) if the period of 2 weeks is extended to a month?
(2) How will (a)(b)(c) look in a graph of total amount of pocket money vs time? Each of them represent a standard type of growth, can you name all of them?

 

 

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LINEAR PROGRAMMING

  IMPORTANT POINTS TO NOTE IN LINEAR PROGRAMMING
1 Define clearly what x and y represent.
Clue can usually be found from the last sentence of the question.
2 Set up appropriate constraints . Some favourite ones are
x >= 0 , y >= 0 ( or > 0)
x , y are integers (check if applicable)
3 State clearly what is to be maximised or minimised.
To minimise P = $ (3k x + k y)
Remember to include k whenever we don't know the exact production cost of X and Y.
4 Select the best scale by examining the constraints.
If x >= 0 , y >= 0
3x + y > 240
x + 2y < 300
then the x-axis must include at least up to ____
and the y-axis must include at least up to ____ .
5 Label the x,y-axes and write down the scale of the axes.
Draw the corresponding line for each constraint, decide whether it should be a dotted line (if > ) or a solid line (if >= ). Label the line by an equation (e.g. 3x + y = 240). Depending on the inequality sign, add the appropriate arrows and update the region .
6 Shade the feasible region.
If x, y are restricted to be integers, make sure you add
"solution are points with integral coordinates in the shaded region."
7 To max $ (k x + 3k y) or max $ (2x + 6y - 200) .

In both cases, we only need to draw x+ 3y = 0
and then parallel lines x +3 y = C.
To draw x+3y=0, rewrite as y = -x/3 and plot (0,0) and (3,-1).

Note that A,B,C,D are the only
vertices for this feasible region.
P ,Q, R are not vertices. Why ?

Furthermore, if x, y must be integers and
the vertex D does not have integral coordinates.
Select another point closest to D but
with integral coordinates
.

8 Lastly, answer the question wisely.

Find the number of cars and minibuses that minimise the running cost.
The answer should be "5 cars and 8 minibuses"
instead of "the minimum cost is $ 10,000".

 

 

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PROBABILITY

3 interesting problems - Birthday problem, Monty Hall problem and one involving infinite series

 

The Birthday problem Surprise - Surprise !

A challenge to your intutive sense but quite simple in a mathematical sense!

(A) Let's start with an easy one first.

At least how many people must be gathered so that we can be 100% certain that some of them share the same birthday?

(B) Now comes the serious part ...

How many people should there be in a class so that there is at least a 50% chance that some of them share the same birthday?

Should it be half of what we have in (A) above? How about one third? say 180 people? 120 people? Think about your own class ( around 40 in size ), do you know any 2 people having the same birthday? Ask another class, is it common to have 2 or more people having the same birthday?

Discussion

(1) A computer simulation

(2) How to approach the problem mathematically? (Assume there are only 365 different birthdays.)

(a) Suppose there are 3 people in a class. Find the probability that

(i) all have different probability (ii) at least some have same birthdays

(b) Repeat (a) with 5 people in a class. How about n people? Guess how large should n be so that the probability that some of them have the same birthday is greater than 0.5.

(3) A full explanation Does the answer agree with your intuition?

   

A probability problem involving an infinite series NOT YET

2 monkeys throwing coconuts at each other !

   
  The famous Monty Hall Problem *

Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors?

If you were the contestant, which of the following would have a better chance to win the big prize?

Strategy 1 (stick): Stick with the original door

Strategy 2 (switch): Switch to the other door

or it doesn't matter since the two strategies have equal chance of winning the big prize

*This problem was named Monty Hall in honor of the long time host of the American television game show "Let's Make a Deal." During the show, contestants are shown three closed doors. One of the doors has a big prize behind it, and the other two have junk behind them. The contestants are asked to pick a door, which remains closed to them. Then the game show host, Monty, opens one of the other two doors and reveals the contents to the contestant. Monty always chooses a door with a gag gift behind it. The contestants are then given the option to stick with their original choice or to switch to the other unopened door.

Discussion

(1) A computer simulation for the Monty Hall problem - "Let's make a deal" applet
  just scroll down the page until you see 3 large doors #1, #2 and #3. Try the simulation repeatedly with your friends : one using the always STICK door strategy and the other using the always SWITCH door strategy. Do the statistics shown in small print below the 3 doors deviate significantly between the 2 strategies? Is there a clear winner and does that agree with your intuition?
   
(2) The winning strategy is ... with FULL explanation below
  There is a 1/3 chance that you'll hit the prize door, and a 2/3 chance that you'll miss the prize. If you do not switch, 1/3 is your probability to get the prize. However, if you missed (and this with the probability of 2/3) then the prize is behind one of the remaining two doors. Furthermore, of these two, the host will open the empty one, leaving the prize door closed. Therefore, if you miss and then switch, you are certain to get the prize! Summing up, if you do not switch your chance of winning is 1/3 whereas if you do switch your chance of winning is 2/3! More discussion
   
(3) History of the Monty Hall problem
 

This problem has aroused a heated debate for quite some time when it first appeared in 1991 and has never failed to vex and amaze people with its counter-intuitive solution ever since.

In 1991, Marylin Vos Savant** received the Monty Hall problem from Craig. F. Whitaker (Columbia, MD). She carefully explained the logic of the correct solution in a number of subsequent columns, but never completely convinced the doubters. Marylin's response caused an avalanche of correspondence, mostly from people who would not accept her solution (particularly advocates of "50-50" school). Eventually, she issued a call to Math teachers among her readers to organize experiments and send her the charts. Some readers with access to computers ran computer simulations. At long last, the truth was established and accepted. The matter continued at such length that it eventually became a notable news story in the New York Times and elsewhere.

**Marylin Vos Savant ran the popular "Ask Marylin" question-and-answer column of the U.S. Parade magazine. According to Parade, Marilyn vos Savant was listed in the "Guinness Book of World Records Hall of Fame" for "Highest IQ" with IQ score of 228.

 

   
 

 

 

 

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STATISTICS

 

DATA

  • collection
  • organisation
  • presentation
  • analysis
 

Histogram - real data Interactive bar chart

 

Normal Distribution (Gaussian distribution)

 

Normal distributions are important because a lot of data in real life (e.g. social data) are distributed approximately normally, especially when it involves
(1) a large population and
(2) the data produced by a member of the population is independent of that produced by another member.

 

Examples :

  • Marks of a Mathematics test of 100,000 candidates
  • Weights of the 5 kg packets of rice in a supermarket
  • IQ score for a large group of people and many more

Animation of falling balls by the University of Toronto

  The normal curve is bell-shaped and is given by the equation
 
  Unlike the functions sinx, cosx,íK where the primitive functions can be easily found, the primitive function of f(x) cannot be expressed in a closed form (i.e. as a finite number of sum, difference, product and quotient of elementary functions ).The area under the bell curve cannot be found by integration as we know it. Instead, it will be looked up from the z (standard score) table:
  (1) traditional z table
  (2) interactive z table
  With the z table, many examples involving normal distributed data can be solved.
  More about normal distribution

 

   

 

 

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e and the natural logarithm


Application

(1)

 

 

It is known that for the same principal deposited in a bank, the more frequent the compound interest is credited the more one gets in the end. In other words, daily compounding yields more interest than monthly compounding which in turn yields more than yearly compunding. If a bank offers continuous compounding, is it as good as it sounds? Does it mean that the total amount will increase without bound?

(a)

 

 

 

Compare the amount in a bank account at the end of 10 years for a deposit of $P at an interest rate of 6 % compounded (a) yearly (b) half-yearly (c) monthly (d) daily (e) continuously. Answer

In general, find the total amount in the bank account for a deposit of $P at an interest rate r% compounded continuously over n years.

(Hint: assume the interest is compounded n times a year and then take n to infinity. Note that no matter how frequent interest is compounded, the total amount is bounded by the limit value Per%t)

(b)

 

 

 

 

 

The rule of 70:

Suppose it takes n years for an amount of money to double when invested at the rate of r% compounded continuously. The rule of thumb states that nr is approximately 70.

The proof is quite short. Try it.

In fact, even if the amount is compounded annually, the rule of thumb still works provided that r% is small compared to 1. Try proving it! (You may use the approximation that ln (1+x) is close to x when x is small.)

 

(2) Franklin Benjamin's will

Franklin Benjamin (1706-1790) American printer, author, diplomat, philosopher, and scientist, inventor of the lighting rod and bifocal glasses

Summary

 

 

 

 

Franklin Benjamin would give 1000 pounds to Boston (and another 1000 to Philadelphia). The plan was to lend money to young apprentices in these cities at 5% interest with the provision that each borrower should return the interest and part of the principal each year. Franklin predicted that if the plan was exceuted without interruption, the sum would reach 131,000 pounds at the end of 100 years, of which 100,000 pounds were to be allocated to public works in Boston and the remaining 31,000 pounds would continue to be lent to young people in the same manner for another 100 years. He predicted that if there was no unfortunate accident to prevent the operation, the sum would be 4,610,000 pounds.

Fact

 

Though it was not always possible to find as many borrowers in Boston as Franklin had planned, the managers of the trust did the best they could. At the end of 100 years from the reception of the Franklin gift, in January 1894, the fund had grown from 1000 pounds to 90,000 pounds.

Question


 

In the first 100 years since the will, the original capital had mutiplied about 90 times instead of 131 times Franklin had imagined. What rate of interest, compounded continuously, would have multipled the capital by 90?


(Answer: 4.5%)

 

 

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System of Linear Equations

   
  Amy, Ben and Calvin play a game as follows. The player who loses each round must give each of the other players as much money as the player has at that time. In round 1, Amy loses and gives Ben and Calvin as much money as they each have. In round 2, Ben loses, and gives Amy and Calvin as much money as they each then have. Calvin loses in round 3 and gives Amy and Ben as much money as they each have. They decide to quit at this point and discover that they each have $24. How much money did they each start with?
   
Sol This problem can be solved using the (1) top down strategy or (2) bottom up strategy

Method 1
 
Amy
Ben
Calvin
At start
$x
$y
$z
after round 1



after round 2      
after round 3      

By completing the above table, you should obtain the following equations

x - y - z = 6, 3y - x - z = 12 and 7z - x - y = 24; from which you can solve for x, y and z. Answer

Linear Equation Solver


Method 2

Instead of considering how much money Amy, Ben and Calvin had originally, work backwards from the moment when each of them has $24 each. Complete the following table and see how easily you can reach exactly the same conclusion as in method 1.

 
Amy
Ben
Calvin
At the end
$24
$24
$24
before round 3



before round 2      
before round 1      

 

 

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Complex numbers

 

Harry found in his attic the old treasure map his great grandma inherited a long long time ago. His great grandma cherished the map over all these years and kept it a secret until she thought Harry was wise enough to recover the treasure and let the whole family enjoy the great wealth.

Sail to 114.5o North, 23.1oEast, there lies a small island. At the south of the island is a pasture where an oak tree and a pine tree and a gallows stand. Start walking from the gallows to the oak tree, counting carefully the number of steps taken. At the oak tree, turn right through 90 degrees and walk exactly the same number of steps. Make a mark there (P). Go back to th gallows and walk towards the pine tree, again counting the number of steps needed. From the pine tree, turn left through 90 degrees and again walk exactly the same number of steps. Make a mark there (Q). Start digging midway between P and Q. There lies the treasure.

Unfortunately, the wood gallows had vanished completely after all these years. On the other hand, the oak and the pine tree have survived numerous storms and become landmarks of the island. "Why didn't great grandma tell me earlier about the map? I could have easily found the treasure back then!" With great disappointment, Harry threw the map into the fire.

  This is a sad story. It is made even sadder by the fact that has Harry known more about mathematics (notably about complex numbers), he would have easily calculated the exact location of the treasure. If only he has paid more attention in class!
Sol

Imagine that the island lies on the Argand plane. The line joining the oak and the pine is the real axis. Without loss of generality, let O be the midpoint of the two trees and the trees' location are represented by the numbers 1 and -1 respectively. The gallows, P and Q are represented by z, p and q in the Argand plane.

Can you use the geometry of complex numbers to find p, q and hence z? Answer

BACK

Ans

(a)P(1+0.06)^10 (b)P(1+0.03)^20 (c)P(1+0.005)^120 (d)P(1+0.06/365)^3650

(e)limit value of P(1+0.06/n)^10n as n tends to infinity, on simplifying, this is Pe^0.6. Compare the results of (a) and (e).

Ans

Solving the equations, Amy has $39, Ben $21, Calvin $12 before the game starts.

Ans

p = i(z+1) + 1, q = i(1-z)-1, z = i !

In this case, Harry could have measured the distance between the oak and the pine trees (say x m) and then from midway of the two trees, he should walk in the direction making exactly a right angle with the line joining the two trees for x/2 m. There lies the treasure.

   

 

 

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