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Young Archimedes
Journal
Journal and Proceedings of Young ArchimedesOnline Journal
Journal and Proceedings of Young Archimedes Volume 4, Number 1, 2019.
Febrary 25, 2019
Rei Yokoyama
(Kyoto prefectural Nan-you senior high school )
"A general solution of single counterfeit coin problem at the secondary education level"
(Abstract)
A solution of single counterfeit coin problem performed at the secondary education level is shown. In the problem, only one counterfeit coin exists in a set of coins and it is not known whether the counterfeit coin is lighter or heavier than genuine one. A general expression giving upper bound, f, of the number of initial coins among which a counterfeit coin can be identified by a given number of weighings, n, is derived to be f = (3n– 1)/2 by decomposing the initial set of coins into two subsets, each of which has different properties. Then, the upper bound for each subset is formulated and summed up to obtain the final expression, which is confirmed to be the same as appeared in literature.
Journal and Proceedings of Young Archimedes Volume 3, Number 1, 2018.
August 14, 2018
Ryuta Ito
(Takarazuka Higashi Senior High School, Japan)
"A generalization of a symmetry related to a certain class of prime numbers"
(Abstract)
Inoue classified the numbers of 0,...,9 in the recurring periods defined by special prime numbers. The classification show us a symmetry related to a certain class of prime numbers. We calculated the numbers of 0,...,9 in the recurring periods defined by general prime numbers, and revealed a more general symmetry which relates to the length of the recurring periods.
Masaharu Teramoto, Keito Tanemura, Shota Suzuki, Shintaro Kaimoto and Mariko Kashiwagi
(Kwansei Yakuin Senior High School, Japan)
"Some Formulas for Binomial Coefficients and Their Application for Probability Theory"
(Abstract)
In this article, the authors present equations of binomial coefficients and application of these equations for a problem of probability theory.
Journal and Proceedings of Young Archimedes Volume 2, Number 2, 2016.
February 05, 2016
Hinata Inoue
(Takarazuka Higashi Senior High School, Japan)
"On a regularity of a certain class of prime numbers"
(Abstract)
In general, all rational numbers are represented by recurring decimals and there exist remarkable numbers among them. It is well known that for a special prime number p, the natural number obtained by the recurring period of 1/p has a multiplicative cyclicity. Applying it, we revealed how to determine the numbers of 0, . . . , 9 in such a natural number.
February 05, 2016
Yuushi Nakaya, Shunsuke Nakamura
(Kwansei Gakuin Senior High School, Japan)
"A Necessary and Sufficient Condition for Chocolate Bars to Have the Grundy Number That Is Equal to Nim-Sum"
(Abstract)
Chocolate bar games are variants of the CHOMP game in which the goal is to leave your opponent with the single bitter part of the chocolate. The original chocolate bar game [2] consists of a rectangular bar of chocolate with one bitter corner. Since the horizontal and vertical grooves are independent, an m × n rectangular chocolate bar is equivalent to the game of NIM with a heap of m − 1 stones and a heap of n−1 stones. Since the Grundy number of the game of NIM with a heap of m−1 stones and a heap of n−1 stones is (m−1)⊕(n−1), the Grundy number of this m×n rectangular bar is (m−1)⊕(n−1). In this paper, we investigate step chocolate bars whose widths are determined by a fixed function of the horizontal distance from the bitter square. When the width of chocolate bar is proportional to the distance from the bitter square and the constant of proportionality is even, the authors have already proved that the Grundy number of this chocolate bar is (m − 1) ⊕ (n − 1), where m is is the largest width of the chocolate and n is the longest horizontal distance from the bitter part. This result was published in a mathematics journal (Integers, Volume 15, 2015). On the other hand, if the constant of proportionality is odd, the Grundy number of this chocolate bar is not (m−1)⊕(n−1). Therefore, it is natural to look for a necessary and sufficient condition for chocolate bars to have the Grundy number that is equal to (m − 1) ⊕ (n − 1), where m is is the largest width of the chocolate and n is the longest horizontal distance from the bitter part. In this paper, the authors present this necessary and sufficient condition.
February 05, 2016
Journal and Proceedings of Young Archimedes
Volume 2, Number 2, 2016. Full Text PDF
Journal and Proceedings of Young Archimedes Volume 2, Number 1, 2016.
February 05, 2016
Shane Scott
(Trinity Grammar School, Australia)
"Pythagorean Triples"
(Abstract)
From early Secondary School we are introduced to Pythagoras’ Theorem – The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides, and the associated formula c2 a2 b2. In school textbooks it has been used to find the length of a side of a triangle. A little part of the textbook is put aside to mention the converse of the theorem; that is if you have a triangle and the sum of the squares of two adjacent sides is equal to the square of the longest side then you must have a right angled triangle. The application of the converse has been used throughout history, from the Babylonians in which the tablet Plimpton 322 shows a collection of Pythagorean Triples. The Egyptians used them to ensure the blocks for the pyramids were perfectly square.
February 05, 2016
Jeb Reece H. Grabato, Azleya Trish C. Canuto, Pearly Jane A. Cortez, Keneth Noe H. Escueta, Agham Bayan B. Martinete, Gercel Therese R. Serafino
(UP High School in Iloilo, Philippines)
"Primes of the Form n^2 - 2"
(Abstract)
This paper presented the concept of primes with the form P = n^2 -2 and determined its properties. It found the following characteristics of P: (a) P is congruent to 3, 7 or 9 modulo 10; (b) any P ≠ 7 will never have a twin prime; (c) P > 2 are congruent to 3 modulo 4; and (d) P > 2 are congruent to 7 modulo 8. It also found the following characteristics of n: (a) are odd; (b) n ≠3 is not congruent to ±3 modulo 7; and (c) n is not congruent to ±6 modulo 17. Lastly, it found that the prime factor p of a non-prime n^2 -2
is congruent to ±1 modulo 8.
February 05, 2016
Marco Lee Solano
(Trinity Grammar School, Australia)
"Friends, Birthdays and Probability: Problems Concerning the Probability of Birthday Coincidences?"
(Abstract)
I have always been fascinated with coincidences. Viewing events in the world through the scope of chance and probability has become a habit, which may go back to my interest in Pokémon as a child. Pokémon is a game in which a major objective is “catching” an animal-type thing, which is greatly reliant on chance: a certain probability is associated with catching each animal on the first try, and this is often very unlikely. This means that a player must anticipate how many tries they will need to catch the animal (the Pokémon) because attempts must be purchased in advanced. A particular coincidence that was a source of constant bewilderment was that in my class of 31 boys (from year 8 to 10), there were two boys, Andrew and Adrian, who shared their birthday. I thought, “What are the chances of that”? We thought that this was a remarkable coincidence, since it seemed very unlikely and impressive. Since then, I have been introduced to the concept of the Birthday Problem, also called the birthday paradox. This math problem is attributed to Richard von Mises, who first mentioned it in 1939 [1]. This problem is not a paradox in the traditional meaning of a contradiction in logic, however the solution is counterintuitive to some, and I was tricked the first time. The original question asks “how many people are needed in a room such that the chance that some share a birthday is at least 50%?” The well-known answer to this question is 23. People often talk about the “significance”, or “size” of a coincidence. One might say that two people tossing coins with the same result was a relatively small coincidence, whereas meeting someone with the same birthday as you would be a relatively large coincidence. It seems that the “significance”, or the “size” of a coincidence is an expression for how unlikely a certain coincidence is to occur. I have always wondered what the chances of our classroom birthday coincidence occurring were, and how many friends you would usually need to have before you saw a coincidence like that.
February 05, 2016
Andrew Serb
(Trinity Grammar School, Australia)
"What is the Relationship between the Speed of a falling object and the size and shape of the Crater caused by its impact with the ground?"
(Abstract)
When I read about the meteor that supposedly killed the dinosaurs, I was astonished to find that an object only 10km wide could have caused a crater 193 km wide and 43 km deep, hurling out an estimated 90,000 km3 of debris into the air [1]. After looking at photographs of meteor craters on the moon, I wanted to explore the relationship of an objects speed at impact and the crater it creates. I also wanted to know whether or not a crater was more like a rounded bowl (a cap shape) or a paraboloid. Therefore, in this project, I decided to investigate the relationship between the speed of a falling object and the size and shape of the crater it makes when it impacts the ground. More specifically my aim was to see if there is a relationship between a crater’s volume, circumference and depth when compared to the causative object’s speed. In addition, I wanted to investigate what mathematical formula may best describe the shape of a crater when it hits the Earth. To do this I will set up an experimental rig that will drop a known object of consistent size and weight into a material and measure the crater that it makes. I will also explore whether or not the shape of the crater would change depending on how fast it hits the ground.
February 05, 2016
Flynn Innes
(Trinity Grammar School, Australia)
"There cannot be an Odd Perfect Number using the Properties of Odds and Evens"
(Abstract)
Mathematical inquiry can often lead to a jungle of unique questions and problems. In the field of Number Theory, there are a wide assortment of such mathematical creatures. Although these problems are easy to state, they can remain dormant for years with little sign of progress. In this article, we show that there cannot be an Odd Perfect Number using the Properties of Odds and Evens.
February 05, 2016
Journal and Proceedings of Young Archimedes
Volume 2, Number 1, 2016. Full Text PDF
Journal and Proceedings of Young Archimedes Volume 1, Number 1, 2015.
February 05, 2016
Dr Frederick Osman
(Trinity Grammar School, Australia)
"Solutions of Higher Order Dispersion Terms in the Nonlinear Schrödinger Equation"
(Abstract)
This paper presents the nonlinearity and dispersion effects involved in the propagation of optical solitons which can be understood by using a numerical routine to solve the Nonlinear Schrödinger Equation (NLSE). A sequence of code has been developed in Mathematica to explore in depth several features of the optical soliton’s formation and propagation. These numerical routines were implemented through the use with Mathematica and the results give a very clear idea of this interesting and important practical phenomenon. The resonant radiation of solitons due to higher order dispersive effects will be seen here to cause increasing turbulence which will ultimately lead to severe damping of the soliton, resulting in its diminished usefulness in telecommunications and other fields, including self-focusing. It is believed that these results will be of considerable use in any work or research that uses the self-focusing properties of the soliton [9] and this paper sets out to explain why the higher order optical soliton should be considered in such research.
February 05, 2016
Wayne Lester Ballador, Andrea Louise Gallaza, and Ignacio Lazaro III
(UP High School in Iloilo, Iloilo City Philippines)
"Pattern for Centre of Twin Primes"
(Abstract)
This paper presented the concept of the centre of twin primes and determined its properties. It proved that centre of twin primes that is greater than 7 is: (a) divisible by 6; and (b) congruent to 0, 2 or 8 modulo 10. These properties would enable researchers of twin primes to test a smaller set of numbers for primality without sacrificing accuracy.
February 05, 2016
Chris Millican
(Trinity Grammar School, Australia)
"Blu-Ray vs. DVD vs. CD: A Mathematical Exploration: Why Archimedean Spirals lie at the Centre of the differences between them?"
(Abstract)
As an avid musician and a fan of digital entertainment, my house is full of CD, DVD and Blu-ray discs. The questions addressed in this exploration are extremely relevant for me as a teenager who benefits daily from this technology and yet has little understanding about how it all actually works. I feel I have taken this technology for granted without having an understanding for what I am using. By the end of the exploration, my aim is to have quantitatively demonstrated the differences between CD, DVD and Blu-ray discs and therefore to have shown why Blu-Ray discs are arguably the best choice of the three. My exploration ultimately delved into various areas of mathematics including measurement, differential and integral calculus and trigonometric functions. The implications of my work would help me to realise the possible innovations that may further improve the capabilities of digital data storage in the future.
February 05, 2016
Yuzhe Zheng
(Trinity Grammar School, Australia)
"Bayes’ Theorem and Its Applications in Law"
(Abstract)
The aim of this paper is to clearly explain how Bayes’ Theorem can provide the correct interpretation of statistical evidence. This will be achieved through firstly, introducing Bayes’ Theorem in two different forms, and examining its use in two real-world cases, Regina v Sally Clark [1] and Regina v Denis John Adams [2]. In the first case, I will explain how Bayes’ Theorem accounts for one piece of evidence, and in the second case, I will explain how Bayes’ Theorem can also be used for multiple pieces of evidence. Through this process, I will also expose the logical fallacies of these cases and explore how this theorem avoids making such fallacies. Acknowledging that it is difficult for the layman to understand Bayes’ Theorem, I will also be using diagrams to represent Bayes’ Theorem for clarification.
February 05, 2016
Joshua Gereis and Victor Wu
(Trinity Grammar School, Australia)
"Effect of surface area and volume on rate of cooling"
(Abstract)
Newton’s Law of Cooling describes the rate of cooling of an object, given its temperature, the ambient temperature, and the object’s rate of cooling (represented by ‘k’ in Newton’s Law). This experiment aims to find the relationship between an object’s surface area to volume ratio and its rate of cooling. We find a linear relationship between these two values.
February 05, 2016
Takeru Kitagawa and Shunsuke Nakamura
(Kwansei Gakuin High School, Japan)
"New Chocolate Games"
(Abstract)
The authors studied a Chocolate Game that is a variant of the well known combinatorial game of Nim, and discovered new formulas for P-positions of the game. Chocolate Game studied in this paper can be cut in three directions, and each game has coordinates (x, y, z), where x, y, z are the maximum number of times we can cut the chocolate in each directions respectively. Rectangle chocolate games are mathematically equivalent to the game of Nim with three piles, but the coordinates (x,y,z) of Chocolate Game created by the authors satisfy inequalities, and this fact makes mathematical structures of these chocolates games different from that of the game of Nim. The result presented in this paper is carried out by a group of high school students using computer algebra system Mathematica.
February 05, 2016