Jsms newsletter jul 2015

Hong Kong Joint School Mathematics Society HKJSMS Newsletter

Issue 2

14-15

July 2015

From the editors: Dear readers, while you are enjoying the summer vacation, please don’t forget to read our newsletters. This time, besides the symbols and formulas, there are also movies about mathematics. Please enjoy reading.

Content: Movies About Mathematics ●

x+y

P.2-3

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21

P.3-4

More About Pythagoras’ Theorem ●

Proof of the theorem

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Pythagorean Triples

Game Section

P.5 P.6-8 P.8

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Movies About Mathematics

By Allie Poon

x+y An upcoming movie about mathematics called X+Y tells the story of an autistic teenage math prodigy who tries out in the International Mathematical Olympiad (IMO) with the help of h i s s t r a ng e l y in s p i r a t io n a l teacher, finding friendship and love along the ride. Not only does the heartwarming story touch every mathematician with the bewitching yet comforting presence of mathematics, but also lets many more confront their autistic realities and fully emb ra ce them co mfo rtab ly.

In the movie, Nathan struggles to connect with people, often pushing away those who want to be closest to him, including his mother, Julie. Without the ability to understand love or affection, Nathan finds the comfort and security he needs in numbers and mathematics. Mentored by his strange, anarchic teacher, Mr. Humphreys, it becomes clear that Nathan’s talents are enough to win him a place on the British team competing at the highly revered International Mathematics Olympiad. Being part of a team and one which has a real chance of winning seems like it could change Nathan’s life forever. But when the team attends training in Taiwan, Nathan is faced with a multitude of unexpected challenges, including the new and unfamiliar feelings he begins to experience for one of the Chinese competitors, the beautiful Zhang Mei. From England to Taipei and back again, this inspiring and life-affirming story follows the unconventional and hilarious relationship between student and teacher, whose roles are often reversed, and the unfathomable experience of first love.

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I am hopeful that this film can and will encourage more young people to appreciate and become actively involved in math, not competitions, as well as attract a larger audience to and organizations to step forward as volunteers to help develop and expand events related to mathematics in Hong Kong. This increased awareness and support of the study of mathematics is especially important as Hong Kong is hosting the 2016 International Mathematical Olympiad.

To this day, mathematics is misrepresented as a boring, purely academic school subject. The idea of books, movies, even art, related to math may seem absurd, but indeed there are quite a lot of movies talking about mathematics, and here are just one of many more:

21

21 is an American heist drama film is inspired by the true story of the MIT Blackjack Team as told in Bringing Down the House, the best-selling book by Ben Mezrich. Released in 2008.

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The movie 21 is the story of MIT students who count cards to improve their probability of winning the card game Blackjack at casinos. Not surprisingly, this movie has a lot of mathematics in it. Most obvious is the "counting of the cards", which is based on the techniques published in Edward O. Thorpe's 1962 book Beat the Dealer. Discussions of the method and mathematics of "card counting" are described on various other websites. You can learn about other mathematical ideas which appear in the movie.

In "21", when Ben Campbell (played by Jim Sturgess) is celebrating his birthday, the cake says 0, 1, 1, 2, 3, 5, 8, 13, ...

These are the first terms in the Fibonacci Series, which was used as an example in the book Liber Abaci published in 1202 by Leonardo Fibonacci. This is obtained by first writing the numbers "0, 1", then defining each subsequent number as the sum of the previous two numbers in the series. Thus, the third number in the series is 1 = 1+0, the fourth number is 2 = 1+1, the fifth number is 3 = 2 + 1, etc. The next number on the cake would be 21=13+8, for Ben's 21st birthday.

Clever, huh? (Hmmm, does "21" refer to Blackjack or Ben's age?) Ben will have to wait until he is 34 = 21+13 for his next "Fibonacci birthday".

One can define other Fibonacci Series by specifying different numbers in the first two slots. For example, the Fibonacci Series starting with "2, 5" is 2, 5, 7, 12, 19, 31, 50, ...

The Fibonacci Series

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More About Pythagoras’ Theorem

By Andy Leung In secondary school curriculum, Pythagoras' Theorem is one of the most frequently used theorem related to geometry:

For △ABC,

∠ABC=90°→ AB2 + BC2 = AC2

Mostly we only focus at how to make use of the theorem, but WHY this theorem holds? According to the book The Pythagorean Proposition, there are at least 370 proofs. In this article, proofs from different nations and in different perspectives would be mentioned.

(1) Solving sides algebraically In the figure, ∠DAB=∠BAC (common) ∠ADB=∠ABC (given) ∴ △ADB~△ABC (AA) AD/AB=AB/AC (ratio of 2 sides, ~△s) ∠BCA=∠DCB (common) ∠DBC=∠ABC (given) ∴ △DBC~△ABC (AA) DC/BC=BC/AC (ratio of 2 sides, ~△s) AB2=AC×AD BC2=AC×DC AB2 + BC2 =AC(AD+DC)=AC2 □

(2) Using Heron’s formula

There are two congruent right-angled triangles with height side coincide with each other. Area of the large triangle S = a(2b)/2 = ab Heron’s formula: S = p(p-x)(p-y)(p-z) where p=(x+y+z)/2; x,y,z are the lengths of the three sides of the triangle.

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→S2 =b2(b+c)(c-b) substitute S2=(ab)2: a2b2=b2(c2-b2) divide both sides by b2, a2=c2-b2 → a2+b2=c2 □

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Feeling tired of complicated proofs? Let us see something more interesting related to this well-known triples: Pythagorean Triples!

If a,b and c are all positive integers for a certain triangle, they are called the Pythagorean triples.

For example: (5,12,13) is a Pythagorean triple. i.e. 25 + 144 = 169

There are some formulas to generate Pythagorean Triples:

(1)Dickson's method This method is generated by Eugene Dickson in 1920.

To find integer solutions to such that is a square.

, find positive integers r, s, and t

r is an even r is an even number; s and t are factors of e.g. Let r=12, st=72; one set of factors are (8,9) when s=8 and t=9, a triple of [20, 21, 29] will be generated. x=12+8=20; y=12+9=21: z=12+8+9=29.

202+212=400+441=841=292 Try This: given r=26, Find the Pythagorean triple which x is the smallest.

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(2) The water containers: an analogy This is similar to the typical deductive approach of the Pythagoras’ theorem.

Originally there are three cuboids with same height. At first water is filled into two smaller square containers, then the whole object is turned with the largest square container at the bottom. It is claimed that the container has contained all water from the two smaller containers. ha2 + hb2 = hc2 → a2 + b2 = c2

Though, this formula cannot generate all Pythagorean Triples. Therefore a constant k is added:

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These formulas looks quie simple and handy, but the easiest one is like this: (2n+1)2 + [2n(n+1)]2 = [2n(n+1) + 1]2

Yet, this formula cannot be used to find all primitive Pythagorean’ Triples.

Game Section Need a real relaxing time right? Let’s play a game called KenKen, which means “smarty boxes”.

Same as Sudoku, numbers in every column and row cannot be repeated, but for KenKen calculations within boxes bounded with bould lines are required.

Solution:

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