1
DCM ATHEMATICA 2015
E DITORS Zihan Zhou Zhengyuan Zhu
Junxiao Shen S UPERVISOR Dr Purchase
C OVER D ESIGNED B Y Zhengyuan Zhu & Zihan Zhou
A RTICLE C ONTRIBUTORS
Alex Boiardi
Henry Bradley
Alexander Cartwright
Andrew Hong
Keesje ’t Hooft
David Jaffe
Leslie Leung
Zeb Micic
Mr. Ottewill
Jan Rybojad
Junxiao Shen
Xiaofeng Xu
Andy Zhang
Zhengyuan Zhu
3
E DITORS ’ N OTES
It is the time to look forward to Founder’s Day this year, whose theme is Persistence and Endurance. 2015 is the 100 th anniversary of Shackleton’s Great Imperial Trans Antarctic Expedition. Not surprisingly, this year’s DC Mathematica is tightly linked to the great spirits of a large number of mathematicians who devoted their whole lives to discover and explore mathematical principles behind simple questions. A large group of boys in the College have developed great enthusiasm to extend their vision beyond the syllabuses and they have made impressive progress. Not limited by their age, they eagerly show their discoveries by writing essays or short articles. During the editing process, we have received a considerable amount of articles and we have included some interesting mathematical puzzles for you to play with. Due to the limited length of this magazine, we are sorry that we cannot include all scripts we have received. Also thank you to Dr. Purchase for organising the editing again this year.
T HE E DITING T EAM . 9 TH J UNE 2015
5
C ONTENTS Fooled by Randomness.............................................................1
By Zhengyuan Zhu (Y12)
Game Theory .............................................................................5
By Henry Bradley (Y13)
Latin in Mathematics...............................................................10
By Zeb Micic (Y8)
Markov Chain in Shackleton’s Voyage..................................12
By Andy Zhang (Y12)
Dr. Purchase’s Multiplication Problem.................................16
By Alex Boiardi (Y9)
Persistence and Endurance ....................................................18
By Junxiao Shen (Y12)
Coding in Mathematics ...........................................................22
By Andrew Hong (Y8)
Reflections on Reflections......................................................23
By Mr. Ottewill
Shackleton, Sextant and the…quadratic regression ...........25
By Jan Rybojad (Y12)
The Golden Ratio....................................................................33
By Alexander Cartwright (Y8)
Are Fourier Series of great importance?..............................34
By Xiaofeng Xu (Y13)
The Pancake Problem .............................................................41
By Keesje 't Hooft (Y13)
Triangle and Square .................................................................46
By David Jaffe (Y9)
What is the God's Number of the Rubik's Cube?..............47
By Leslie Leung (Y13)
7
Fooled by Randomness
----Zhengyuan Zhu (Y12)
“Humans are often unaware of the existence of randomness. They tend to explain random outcomes as non-random. ” 1
People don’t understand how things are just random and try to make sense out of them. They would be making models out of stock data and look at how the pattern in the data is and forecast the future stock price, which exactly analysts and investors are doing in the investment banks. Is there really a sort of pattern and tendency behind the security price or just randomness? In this article, we would like to use some A-Level mathematics to make a sense of it. In order to understand how security prices work in the market, we need to understand the meaning of the term of the Efficient Market Hypothesis (EMH). It is a theory that states markets efficiently incorporate all public information. Security prices accurately reflect available information and respond rapidly to new information as soon as it becomes available 2 . The idea is that, if the market is really efficient, then any change from day to day has to be due only to news. However, news is essentially unpredictable. Thus, security prices have to do a random walk through time, which means any future movement in them is always unpredictable and changes are purely random. The random walk process can be described as follows: E(ε) = 0, for i ≠ s Using A-level statistics, we know that the mean of is zero and they are uncorrelated with any other previous values, meaning that is random. To test whether the stock market is really efficient, we are using the method called “auto-regression”, which is combined with PMCC (S1) and testing for zero correlation (S3). Autoregressive processes are used by investors to investigate the pattern in data. Auto-regressive models take into account past movements and future values are estimated based on the past values. E(ε i ε s ) = 0, Y t = Y t−1 + ε t ⟹ Y t − Y t−1 = ε t
1 Taleb, N. N. (2008) Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets. 2nd edn. New York: Random House Publishing Group 2 Efficient-market hypothesis (2015) in Wikipedia. Available at: http://en.wikipedia.org/wiki/Efficient- market_hypothesis (Accessed: 12 April 2015)
1
We investigate Shanghai Stock Exchange Index data (SSE) as an example (monthly closing price from 1991-2015). Let Shanghai Stock Exchange Index be 𝑖 , 𝑖 = 1, 2, 3 … . To find the differences (Y) of the original data (X), we take the following form.
𝑖 𝑖−1
) − ln( 𝑖−1 )
) = ln( 𝑖
𝑖
= ln (
Where X i
is approximately measured in percentage change. One observation is
lost due to differencing.
Figure 1. Shanghai Stock Exchange Index
for i = 2, 3 …, let 𝑖
=
In order to calculate the relationship between X i
and X i-1
𝑖+1 , and then the following formula (S1) can be used to calculate the sample correlation coefficient between X and Y:
𝑆
=
𝑆
√𝑆
− ̅ ) 2 𝑆
− ̅ ) 2