 Latest update: 18th Sept, 2010 Mathematics For Fun

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By topics Further Mathematics  Problem solving Binomial Theorem Interesting properties of Pascal triangle Derivatives Definition of Derivative Applet Differentiation Applets on Math Online Gallery (Austria) interactive exercises on the graphs of f(x), f'(x) and f"(x) by the Maths Online in Austria. Don't forget to try the derivative puzzle. Graphs of f, f' and f" in Derivatives and Tangent Line by MIT - move slidebar Derivatives checker - given y=f'(x), sketch y=f'(x) and y=f"(x) Math tools Browse - a library of excellent applets Calculus applet page by St. Louis University Application of differentiation in real life MaxMin Problems Optimization problems with applets for more than 10 problems quicktime Calculus I e.g. Exam III examples and animations the fastest route #11 Demos with positive effects 9 optimization problems move slider 5 optimization problems applet shows value but not the graph galleries of animations demos with positive effects (not yet)homepage Interesting max min problem on Math Tools browse Applications of derivatives The fastest route Visual Calculus Rowing or Walking variation of speeds Area of a rectangle inscribed in a semicircle Box Volume sketchpad on Math Tools Browse The longest ladder Shortest ladder over a fence by Visual Calculus Maximization of an area Cross section of gutter on www.ies.co.jp applet Solid of revolution 6 gif animations to illustrate how the volume of a solid of revolution is found. The web site is launched in January, 2002. An egg an excellent example of a surface of revolution plus may interesting observations all about chickens at this EggMath website. 3 dimensional graphics of the standard tori , apple ,lemon all found in the Eric Weisstein's World of Mathematics at the Wolfram Research mathworld website. Visualising Solid of revolution by MathServ Calculus Toolkit Locus Animation of cycloids and ellipse - Prolate cycloid (Why some points on a train wheel are moving backward as trains move forward?) Examples of Roulettes - animation for roulette, epitrochid Animation of regular pentagons rolling along truncated catenaries -guarantee a smooth ride The tautochrone problem - a curve along which an object under gravity takes equal time to slide from any point to the lowest point The brachistochrone problem - path of quickest descent under gravity Useful websites MathServ Calculus Toolkit by Vanderbilt University, USA Visual Calculus Calculus applets at www.jes.co.jp Math Tools Browse Thinkquest library algebra geometry trigonometry calculus Complex numbers A treasure hunt A mathematics puzzle a day keeps you smart all day!

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By topics  Mathematics  Numbers Transformations Mathsnet Interactive Interactve Tranformations - an online course eg Rotations, Reflections, etc. Funs activity - Drawing with Reflections Geometry Euclid's Elements online by Clark University Book I Defintions, Common Notions, Postulates, Propositions Book III Circles Definitions, Propositions More about Euclid of Alexandria Links Java Applets - Geometry on www.ies.co.jp HK EdCity Resources Dynamic Geometry - Exploring Geometry with Sketchpad Sketchpad Demonstrations and Conjecture List Angles in same segment Sketchpad files GeoGebraWiki International There are many beautiful theorems in geometry. Geometry step by step From the Land of the Incas offers beautiful illustrations of the triangle centres(such as the centroid, orthocenter, circumcenter, incenter, excenter) as well as the famous theorems Ceva's Theorem , Gergonne pt , Menelaus' Theorem . A clever use of the tangent properties can also be found in Semiperimeter and Incircle. Sketchpad files Angle in alternate segment and its proof case 1 , case 2 The nine point circle An exercise on test for concyclic points A triangle and its tangent circles A star and its circles 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 MAA Online From Lewis Carrol to Archimedes Breaking away from the Math Book projects and lesson plans Jigsaw Paradox 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 Interactive Histogram; Normal distribution; Interactive standard score By subject

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By topics  Advanced Mathematics  Logic An interactive exercise Problem solving Calculus Functions with special properties A plane filling curve - e.g. Peano curve A function which is not continuous at any point - Dirichlet function A continuous but nowhere differentiable function - Every differentiable function is continuous on IR. The converse is not true. There exist continuous functions on IR which is not necessarily differentable on IR. Actually, there exists a continuous function which is nowhere differentiable(e.g. Weierstrass function). Nowhere differentiable functions applet . Applets and Demos galleries of animations demos with positive effects - functions homepage Applets from www.ies.co.jp - Mean Value Theorem etc Maths Online Gallery - GeoGebra - intersections of a line and a sine curve, upper and lower sum of integral, wiki Mean Value Theorem - cut the knot , www.ies.co.jp . notes e Application of e Compound interest; Franklin Benjamin's will System of Linear Equations Matrices Partial Fractions Thinkquest library - created by pre-collegiate students Complex numbers A treasure hunt , applets Useful websites Calculus Toolkit MathServe Project by Vanderbilt University, USA Curve Sketching by Vanderbilt University, USA Function Plotter by Maths Online, University of Vienna Famous Curves by University of St. Andrews, Scotland Thinkquest library extensive material on differentiation and integration Java Applets by International Education Software MathsTools, Maths Online Gallery , Visual Calculus Calculus sites recommended by Math Forum By subject

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By topics  Brainteasers  Big Ben - angle between the clock hands A Simple Math Calculation - try this without using a calculator Regions of Equal Areas - surprise, surprise Angle in a semicircle - the easiest one Two probability problems 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 Hex 7 - invented independently in the 1940's by Piet Hein, born in Denmark and inventor of many mathematical games; and by John Nash, the American mathematician who won the Nobel Prize in Economic Science. Eric Solomon Online java - original java games and puzzles by Eric Solomon include the eye-catching and challenging Torus Twister (Rubik cube on a torus), Sheepdog Puzzle, Leapfrog Puzzle and many more. Penrose Tiling Clever game for clever people Warp image on Maths Online Gallery - a piece of Art Model Building (eg Game of Life) form Maths Online Gallery Game of life, Fractal, Julia set - Shoder Game Theory By subject

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By topics Websites Resourceful Get.to/Mathematics - a most resourceful website covering mathematics at HKCEE and HKAL level, exercises and notes as well as beautiful sketchpad diagrams included. The website is updated frequently and is a must for students. Good reading Cut the-knot -excellent website with puzzles, interactive applets, eye opening series, proofs etc. MAA online columns Feature Column MAA Mathematical Science Digital Library Covergence - online magazine Math Plus Magazine HKUST Math Excalibur Math Articles online - Excerpts from calculus textbooks and links to math journals More goodies... Mac Tutor History of Math Archives from University of St.Andrews, Scotland - An excellent website including a very useful famous curves index, mathematicians of the day, quotation, history topics, timelines and don't miss the Anniversaries Shodor interactive activities applet National Library of Virtual Manipulatives - Utah State University NCTM illuminations WIMS activities Mathsnet - interactive activities on transformations, A-level puzzles etc. Math Forum and Math Forum Internet Mathematics Library and java applets Maths Online (Austria) - excellent multimedia learning material, applets, interactive exercises and online tools - link by topics - collections Thinkquest Library created by pre-collegiate students MathTools Visual Calculus Geometric Sketchpad - Pythagoras Theorem, Rethinking Proof, Pre-calculus, Calculus, Algebra, Conic Sections, Geometry Activities, Shape makers, .s.. Math Applets by St. Louis University Manipula Math with Java www.ies.co.jp Applied Calculus by StefanWaner and Steven R. Costenoble True False quizzes 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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By topics 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 Maths Online gallery the 3rd applet - graphs of sine, cos and tan Mathsnet - the 4th applet - move the point Trig Review Applet of sine and cosine graph at St. Louis Univ Animations of sine graph, cosine graph Midshipman Online Calculus Lab y=Asinx, y=sinx+d, y=sin(x-c), y=bsinx Applets on Trigonometry on www.ies.co.jp - esp. y = sin x and y=sinax Practice Interactive exercises on trigonometric equations (questions only) Application Trig and Music Sum of Trig Functions - visualise how the shape of the sum of 3 or more sine curves Trig and Weather

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By topics 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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By topics 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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By topics 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!

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. 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

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By topics STATISTICS DATA

• collection
• organisation
• presentation
• analysis

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,… 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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By topics 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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By topics 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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By topics 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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By topics A mathematics puzzle a day keeps you smart all day!

2 