The Unreasonably Beautiful World of Numbers
|
|
|
- Emil Randall
- 5 years ago
- Views:
Transcription
1 The Unreasonably Beautiful World of Numbers Sunil K. Chebolu Illinois State University Presentation for Math Club, March 3rd, /28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
2 Why are numbers beautiful? It s like asking why is Beethoven s Ninth Symphony beautiful. If you don t see why, someone can t tell you. I know numbers are beautiful. If they aren t beautiful, nothing is. Paul Erdös 2/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
3 Outline Hindu-Arabic numerals Tricks and puzzles with numbers Some important landmarks in number theory Open problems 3/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
4 The Hindu-Arabic numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 The Hindu-Arabic numerals is a positional decimal number system. These were introduced by the Indian mathematicians in the 8th century. The Arabs, however, played an essential part in the dissemination of this numeral system. 4/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
5 Shunya The concept of shunya originated in the Vedas in ancient India has a long history and varied manifestations in different dimensions, in mathematics, in philosophy and in mysticism. In mathematical literature it is used in the sense of zero having no numerical value of its own but playing the key role in the system of decimal notation (dasa). This discovery of shunya (a symbol for nothing) and the place value system were unique to the Indian civilization. These ideas have escaped some of the greatest minds of antiquity, including Archimedes. 5/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
6 Numerals, a time travel from India to Europe 6/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
7 The base 10 notation Why the number one is 1 and the number two is 2 etc.? Where did these symbols come from? Here is the answer... 7/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
8 A mind reading trick 1. Choose a single digit number (not zero). 2. Multiply it by If the answer has two digits, add them. 4. Subtract 5 from what you have. 5. Turn this number into a letter by the rule A = 1, B = 2, C = 3, and so on. 6. Think of a country beginning with this letter. 7. Finally, take the last letter of this country and think of an animal begining with that letter. 8/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
9 9/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
10 How does this work? Look at the multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81 The digits of these numbers always add up to 9. Therefore when you subtract a 5 in step 4, you will always get a 4. There are not many countries that start with the letter D. The first country that comes to mind is Denmark. There are not many animals with starting letter K. Kangaroo is probably the only one. However, one other possibility is Dominican Republic, which gives Cat in the final step of the trick. But that is very rare! 10/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
11 A trick with match sticks A magician claims that he can tell how many matches there are in a box just by listening to the rattle of the contents. 1. You (the magician) hand a match box to a member of the audience. The match box contains a known number of matches 29 is a good number, as you will see. 2. The audience participant is asked to take the matches out and replace as many as they wish, counting as they go. 3. You ask them to add the digits of this number and remove those many matches from the box. Then they return the box to you. 4. You then shake the box and tell exactly how many matches remain. 11/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
12 How does this work? One line answer: The test of divisibility for the number 9. Every number is congruent modulo 9 to the sum of its digits. This means, a number N minus the sum of the digits of N is always divisible by 9. For a match box with at most 29 matches in it, the number of matches left in the match box can only be 9 or 18. With a little practice, it is not hard to tell whether the number is 9 or 18 by listening to the rattle carefully. In fact, you can try a similar trick based on the test of divisibility for the number /28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
13 A matrix puzzle Fill the matrix below with integers such that the sum of any five integers which do not belong to the same row or column is 57 13/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
14 How do we construct such a matrix? Answer: Really simple. Start with any 10 numbers which add up to 57. Say, 12, 1, 4, 18, 0, 7, 0, 4, 9, Addition Table 14/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
15 This reveals the Magic Circle any 5 numbers in the above addition matrix such that no two belong to the same row or column. What is the sum of these 5 numbers? = 57 15/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
16 Circular sudoko Fill in the grid so that every ring and every pair of neighboured circle segments contains the digits 1 through 8. Invented by Professor Peter Higgins, at Essex Univeristy, UK. 16/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
17 The Taxicab number Hardy: The number of my taxicab was It seemed to me rather a dull number. Ramanujan: No, Hardy! No! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways = = /28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
18 Ramanujan Numbers Since then, integer solutions to I 3 + J 3 = K 3 + L 3 have been called Ramanujan Numbers. The first five of these are: I J K L /28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
19 The Collatz problem 1. Pick a positive integer n 2. If n is even divide by 2, and if n is odd replace it with 3n repeat this process Lothar Collatz in 1937 conjectured that this process will eventually reach the number 1. For instance, starting with n = 7, we get /28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
20 This conjecture is wide open although a lot of work has been done on it. It has been verified by computers for n up to a million million! For instance, it is known that of the first 1000 integers more than 350 have a hailstone maximum height of 9232 before collapsing to 1. Paul Erdös ( ) regarded this as a problem for which mathematics was not ready yet. 20/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
21 The irrational numbers The most unexpected theorem in mathematics: 2 is irrational. The Greeks were quite disturbed to discover this fact. For, it has ruined the famous slogan of the Pythagorean school: All is number. 21/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
22 Fermat s last theorem Fermat s Last Theorem states that the equation x n + y n = z n has no positive integer solutions for x, y and z when n 3. Fermat, in his copy of Diophantus s Arithmetica, wrote in 1637 what became the most enigmatic note in the history of mathematics: I have discovered a truly remarkable proof which this margin is too small to contain. 22/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
23 FLT was proved in 1995 by Sir Andrew Wiles from Princeton University. Professor Wiles proved the Taniyama-Shimura conjecture which implies FLT. 23/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
24 The most hilarious proof Theorem: 2 1/n is irrational if n 3 Proof: Suppose to the contrary 2 1/n is rational. Let 2 1/n = a b. Then we have This gives 2 = an b n. 2b n = a n. From there we get b n + b n = a n A contradition to Fermat s last theorem. QED. Mathematics is consistent! 24/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
25 2 in paper industry Why is the standard A 4 size paper 210 mm x 297 mm? The dimensions of the A 4 paper are chosen from a good rational approximation of 2. But why chose 2? 25/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
26 Here is the answer A rectangular paper when folded down the middle on its longer side should give two smaller sheets which are similar to the original sheet. Suppose the sides of a rectangular sheet have lengths a and b (b > a). a: b = b 2 : a This means a b = b 2a = 2a2 = b 2 = 2 = b a. 26/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
27 Problems The Goldbach conjecture: Every even integer greater than 2 is a sum of two primes. The Twin prime conjecture: There exist infinitely many twin primes? The Catalan conjecture: The only non-trivial solution in positive integers for the equation x y y x = 1 is x = 3 and y = 2. Primes in arithmetic progression: Are there arbitrarily long arithmetic progressions consisting of prime numbers? 27/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
28 Wir müssen wissen Wir werden wissen 28/28 Sunil Chebolu The Unreasonably Beautiful World of Numbers
Table of Contents. Table of Contents 1
Table of Contents 1) The Factor Game a) Investigation b) Rules c) Game Boards d) Game Table- Possible First Moves 2) Toying with Tiles a) Introduction b) Tiles 1-10 c) Tiles 11-16 d) Tiles 17-20 e) Tiles
Modular Arithmetic and Doomsday
Modular Arithmetic and Doomsday Blake Thornton Much of this is due directly to Joshua Zucker and Paul Zeitz. 1. Subtraction Magic Trick. While blindfolded, a magician asks a member from the audience to
UK JUNIOR MATHEMATICAL CHALLENGE. April 26th 2012
UK JUNIOR MATHEMATICAL CHALLENGE April 6th 0 SOLUTIONS These solutions augment the printed solutions that we send to schools. For convenience, the solutions sent to schools are confined to two sides of
The Real Number System and Pythagorean Theorem Unit 9 Part B
The Real Number System and Pythagorean Theorem Unit 9 Part B Standards: 8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion;
Chapter 4 Number Theory
Chapter 4 Number Theory Throughout the study of numbers, students Á should identify classes of numbers and examine their properties. For example, integers that are divisible by 2 are called even numbers
Mathematics of Magic Squares and Sudoku
Mathematics of Magic Squares and Sudoku Introduction This article explains How to create large magic squares (large number of rows and columns and large dimensions) How to convert a four dimensional magic
Grade 6 Math Circles. Divisibility
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Introduction Grade 6 Math Circles November 12/13, 2013 Divisibility A factor is a whole number that divides exactly into another number without a remainder.
Launchpad Maths. Arithmetic II
Launchpad Maths. Arithmetic II LAW OF DISTRIBUTION The Law of Distribution exploits the symmetries 1 of addition and multiplication to tell of how those operations behave when working together. Consider
An elementary study of Goldbach Conjecture
An elementary study of Goldbach Conjecture Denise Chemla 26/5/2012 Goldbach Conjecture (7 th, june 1742) states that every even natural integer greater than 4 is the sum of two odd prime numbers. If we
The Pythagorean Theorem
! The Pythagorean Theorem Recall that a right triangle is a triangle with a right, or 90, angle. The longest side of a right triangle is the side opposite the right angle. We call this side the hypotenuse
MST125. Essential mathematics 2. Number theory
MST125 Essential mathematics 2 Number theory This publication forms part of the Open University module MST125 Essential mathematics 2. Details of this and other Open University modules can be obtained
Modular arithmetic Math 2320
Modular arithmetic Math 220 Fix an integer m 2, called the modulus. For any other integer a, we can use the division algorithm to write a = qm + r. The reduction of a modulo m is the remainder r resulting
A natural number is called a perfect cube if it is the cube of some. some natural number.
A natural number is called a perfect square if it is the square of some natural number. i.e., if m = n 2, then m is a perfect square where m and n are natural numbers. A natural number is called a perfect
Modular Arithmetic. Kieran Cooney - February 18, 2016
Modular Arithmetic Kieran Cooney - kieran.cooney@hotmail.com February 18, 2016 Sums and products in modular arithmetic Almost all of elementary number theory follows from one very basic theorem: Theorem.
Three of these grids share a property that the other three do not. Can you find such a property? + mod
PPMTC 22 Session 6: Mad Vet Puzzles Session 6: Mad Veterinarian Puzzles There is a collection of problems that have come to be known as "Mad Veterinarian Puzzles", for reasons which will soon become obvious.
THE ASSOCIATION OF MATHEMATICS TEACHERS OF NEW JERSEY 2018 ANNUAL WINTER CONFERENCE FOSTERING GROWTH MINDSETS IN EVERY MATH CLASSROOM
THE ASSOCIATION OF MATHEMATICS TEACHERS OF NEW JERSEY 2018 ANNUAL WINTER CONFERENCE FOSTERING GROWTH MINDSETS IN EVERY MATH CLASSROOM CREATING PRODUCTIVE LEARNING ENVIRONMENTS WEDNESDAY, FEBRUARY 7, 2018
Practice Midterm 2 Solutions
Practice Midterm 2 Solutions May 30, 2013 (1) We want to show that for any odd integer a coprime to 7, a 3 is congruent to 1 or 1 mod 7. In fact, we don t need the assumption that a is odd. By Fermat s
repeated multiplication of a number, for example, 3 5. square roots and cube roots of numbers
NUMBER 456789012 Numbers form many interesting patterns. You already know about odd and even numbers. Pascal s triangle is a number pattern that looks like a triangle and contains number patterns. Fibonacci
UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST
UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST First Round For all Colorado Students Grades 7-12 October 31, 2009 You have 90 minutes no calculators allowed The average of n numbers is their sum divided
SOLUTIONS TO PROBLEM SET 5. Section 9.1
SOLUTIONS TO PROBLEM SET 5 Section 9.1 Exercise 2. Recall that for (a, m) = 1 we have ord m a divides φ(m). a) We have φ(11) = 10 thus ord 11 3 {1, 2, 5, 10}. We check 3 1 3 (mod 11), 3 2 9 (mod 11), 3
One of the useful things about numbers is that they re conceptual, which
In This Chapter Chapter 1 Playing the Numbers Game Finding out how numbers were invented Looking at a few familiar number sequences Examining the number line Understanding four important sets of numbers
CMPSCI 250: Introduction to Computation. Lecture #14: The Chinese Remainder Theorem David Mix Barrington 4 October 2013
CMPSCI 250: Introduction to Computation Lecture #14: The Chinese Remainder Theorem David Mix Barrington 4 October 2013 The Chinese Remainder Theorem Infinitely Many Primes Reviewing Inverses and the Inverse
FOM 11 Ch. 1 Practice Test Name: Inductive and Deductive Reasoning
FOM 11 Ch. 1 Practice Test Name: Inductive and Deductive Reasoning Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Justin gathered the following evidence.
Applications of Fermat s Little Theorem and Congruences
Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4
N umber theory provides a rich source of intriguing
c05.qxd 9/2/10 11:58 PM Page 181 Number Theory CHAPTER 5 FOCUS ON Famous Unsolved Problems N umber theory provides a rich source of intriguing problems. Interestingly, many problems in number theory are
What I can do for this unit:
Unit 1: Real Numbers Student Tracking Sheet Math 10 Common Name: Block: What I can do for this unit: After Practice After Review How I Did 1-1 I can sort a set of numbers into irrationals and rationals,
An interesting class of problems of a computational nature ask for the standard residue of a power of a number, e.g.,
Binary exponentiation An interesting class of problems of a computational nature ask for the standard residue of a power of a number, e.g., What are the last two digits of the number 2 284? In the absence
MATH 135 Algebra, Solutions to Assignment 7
MATH 135 Algebra, Solutions to Assignment 7 1: (a Find the smallest non-negative integer x such that x 41 (mod 9. Solution: The smallest such x is the remainder when 41 is divided by 9. We have 41 = 9
Goldbach conjecture (1742, june, the 7 th )
Goldbach conjecture (1742, june, the 7 th ) We note P the prime numbers set. P = {p 1 = 2, p 2 = 3, p 3 = 5, p 4 = 7, p 5 = 11,...} remark : 1 P Statement : Each even number greater than 2 is the sum of
Pythagorean Theorem Unit
Pythagorean Theorem Unit TEKS covered: ~ Square roots and modeling square roots, 8.1(C); 7.1(C) ~ Real number system, 8.1(A), 8.1(C); 7.1(A) ~ Pythagorean Theorem and Pythagorean Theorem Applications,
1.6 Congruence Modulo m
1.6 Congruence Modulo m 47 5. Let a, b 2 N and p be a prime. Prove for all natural numbers n 1, if p n (ab) and p - a, then p n b. 6. In the proof of Theorem 1.5.6 it was stated that if n is a prime number
CALCULATING SQUARE ROOTS BY HAND By James D. Nickel
By James D. Nickel Before the invention of electronic calculators, students followed two algorithms to approximate the square root of any given number. First, we are going to investigate the ancient Babylonian
MULTIPLES, FACTORS AND POWERS
The Improving Mathematics Education in Schools (TIMES) Project MULTIPLES, FACTORS AND POWERS NUMBER AND ALGEBRA Module 19 A guide for teachers - Years 7 8 June 2011 7YEARS 8 Multiples, Factors and Powers
Number Relationships. Chapter GOAL
Chapter 1 Number Relationships GOAL You will be able to model perfect squares and square roots use a variety of strategies to recognize perfect squares use a variety of strategies to estimate and calculate
Intermediate Mathematics League of Eastern Massachusetts
Meet #5 March 2009 Intermediate Mathematics League of Eastern Massachusetts Meet #5 March 2009 Category 1 Mystery 1. Sam told Mike to pick any number, then double it, then add 5 to the new value, then
Grades 7 & 8, Math Circles 27/28 February, 1 March, Mathematical Magic
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Card Tricks Grades 7 & 8, Math Circles 27/28 February, 1 March, 2018 Mathematical Magic Have you ever
MAT Modular arithmetic and number theory. Modular arithmetic
Modular arithmetic 1 Modular arithmetic may seem like a new and strange concept at first The aim of these notes is to describe it in several different ways, in the hope that you will find at least one
Two congruences involving 4-cores
Two congruences involving 4-cores ABSTRACT. The goal of this paper is to prove two new congruences involving 4- cores using elementary techniques; namely, if a 4 (n) denotes the number of 4-cores of n,
Quantitative Aptitude Preparation Numbers. Prepared by: MS. RUPAL PATEL Assistant Professor CMPICA, CHARUSAT
Quantitative Aptitude Preparation Numbers Prepared by: MS. RUPAL PATEL Assistant Professor CMPICA, CHARUSAT Numbers Numbers In Hindu Arabic system, we have total 10 digits. Namely, 0, 1, 2, 3, 4, 5, 6,
Solutions to the European Kangaroo Pink Paper
Solutions to the European Kangaroo Pink Paper 1. The calculation can be approximated as follows: 17 0.3 20.16 999 17 3 2 1000 2. A y plotting the points, it is easy to check that E is a square. Since any
Mathematical Magic Tricks
Mathematical Magic Tricks T. Christine Stevens, American Mathematical Society Project NExT workshop, Chicago, Illinois, 7/25/17 Here are some magic tricks that I have used with students
Constructions of Coverings of the Integers: Exploring an Erdős Problem
Constructions of Coverings of the Integers: Exploring an Erdős Problem Kelly Bickel, Michael Firrisa, Juan Ortiz, and Kristen Pueschel August 20, 2008 Abstract In this paper, we study necessary conditions
#A3 INTEGERS 17 (2017) A NEW CONSTRAINT ON PERFECT CUBOIDS. Thomas A. Plick
#A3 INTEGERS 17 (2017) A NEW CONSTRAINT ON PERFECT CUBOIDS Thomas A. Plick tomplick@gmail.com Received: 10/5/14, Revised: 9/17/16, Accepted: 1/23/17, Published: 2/13/17 Abstract We show that out of the
NUMBER THEORY AMIN WITNO
NUMBER THEORY AMIN WITNO.. w w w. w i t n o. c o m Number Theory Outlines and Problem Sets Amin Witno Preface These notes are mere outlines for the course Math 313 given at Philadelphia
Workout 5 Solutions. Peter S. Simon. Quiz, December 8, 2004
Workout 5 Solutions Peter S. Simon Quiz, December 8, 2004 Problem 1 Marika shoots a basketball until she makes 20 shots or until she has made 60% of her shots, whichever happens first. After she has made
Twenty-fourth Annual UNC Math Contest Final Round Solutions Jan 2016 [(3!)!] 4
![Twenty-fourth Annual UNC Math Contest Final Round Solutions Jan 2016 [(3!)!] 4](/thumbs/83/88926850.jpg "Twenty-fourth Annual UNC Math Contest Final Round Solutions Jan 2016 [(3!)!] 4")
Twenty-fourth Annual UNC Math Contest Final Round Solutions Jan 206 Rules: Three hours; no electronic devices. The positive integers are, 2, 3, 4,.... Pythagorean Triplet The sum of the lengths of the
Grade 6/7/8 Math Circles April 1/2, Modular Arithmetic
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Modular Arithmetic Centre for Education in Mathematics and Computing Grade 6/7/8 Math Circles April 1/2, 2014 Modular Arithmetic Modular arithmetic deals
Solutions for the Practice Questions
Solutions for the Practice Questions Question 1. Find all solutions to the congruence 13x 12 (mod 35). Also, answer the following questions about the solutions to the above congruence. Are there solutions
Goldbach Conjecture (7 th june 1742)
Goldbach Conjecture (7 th june 1742) We note P the odd prime numbers set. P = {p 1 = 3, p 2 = 5, p 3 = 7, p 4 = 11,...} n 2N\{0, 2, 4}, p P, p n/2, q P, q n/2, n = p + q We call n s Goldbach decomposition
Year 5 Problems and Investigations Spring
Year 5 Problems and Investigations Spring Week 1 Title: Alternating chains Children create chains of alternating positive and negative numbers and look at the patterns in their totals. Skill practised:
NRP Math Challenge Club
Week 7 : Manic Math Medley 1. You have exactly $4.40 (440 ) in quarters (25 coins), dimes (10 coins), and nickels (5 coins). You have the same number of each type of coin. How many dimes do you have? 2.
Meet #5 March Intermediate Mathematics League of Eastern Massachusetts
Meet #5 March 2008 Intermediate Mathematics League of Eastern Massachusetts Meet #5 March 2008 Category 1 Mystery 1. In the diagram to the right, each nonoverlapping section of the large rectangle is
1 = 3 2 = 3 ( ) = = = 33( ) 98 = = =
Math 115 Discrete Math Final Exam December 13, 2000 Your name It is important that you show your work. 1. Use the Euclidean algorithm to solve the decanting problem for decanters of sizes 199 and 98. In
Latin Squares for Elementary and Middle Grades
Latin Squares for Elementary and Middle Grades Yul Inn Fun Math Club email: Yul.Inn@FunMathClub.com web: www.funmathclub.com Abstract: A Latin square is a simple combinatorial object that arises in many
Solutions for the Practice Final
Solutions for the Practice Final 1. Ian and Nai play the game of todo, where at each stage one of them flips a coin and then rolls a die. The person who played gets as many points as the number rolled
GPLMS Revision Programme GRADE 6 Booklet
GPLMS Revision Programme GRADE 6 Booklet Learner s name: School name: Day 1. 1. a) Study: 6 units 6 tens 6 hundreds 6 thousands 6 ten-thousands 6 hundredthousands HTh T Th Th H T U 6 6 0 6 0 0 6 0 0 0
Assignment 5 unit3-4-radicals. Due: Friday January 13 BEFORE HOMEROOM
Assignment 5 unit3-4-radicals Name: Due: Friday January 13 BEFORE HOMEROOM Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Write the prime factorization
Applications. 30 Prime Time
Applications For Exercises 1 6, give the dimensions of each rectangle that can be made from the given number of tiles. Then use the dimensions of the rectangles to list all the factor pairs for each number.
Outline Introduction Big Problems that Brun s Sieve Attacks Conclusions. Brun s Sieve. Joe Fields. November 8, 2007
Big Problems that Attacks November 8, 2007 Big Problems that Attacks The Sieve of Eratosthenes The Chinese Remainder Theorem picture Big Problems that Attacks Big Problems that Attacks Eratosthene s Sieve
Grade 6 Math Circles February 21/22, Patterns - Solutions
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles February 21/22, 2017 Patterns - Solutions Tower of Hanoi The Tower of Hanoi is a
PRIMES IN SHIFTED SUMS OF LUCAS SEQUENCES. Lenny Jones Department of Mathematics, Shippensburg University, Shippensburg, Pennsylvania
#A52 INTEGERS 17 (2017) PRIMES IN SHIFTED SUMS OF LUCAS SEQUENCES Lenny Jones Department of Mathematics, Shippensburg University, Shippensburg, Pennsylvania lkjone@ship.edu Lawrence Somer Department of
Magic Squares. Lia Malato Leite Victoria Jacquemin Noemie Boillot
Magic Squares Lia Malato Leite Victoria Jacquemin Noemie Boillot Experimental Mathematics University of Luxembourg Faculty of Sciences, Tecnology and Communication 2nd Semester 2015/2016 Table des matières
6.2 Modular Arithmetic
6.2 Modular Arithmetic Every reader is familiar with arithmetic from the time they are three or four years old. It is the study of numbers and various ways in which we can combine them, such as through
Grade 6 Math Circles February 21/22, Patterns
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles February 21/22, 2017 Patterns Tower of Hanoi The Tower of Hanoi is a puzzle with
Math 319 Problem Set #7 Solution 18 April 2002
Math 319 Problem Set #7 Solution 18 April 2002 1. ( 2.4, problem 9) Show that if x 2 1 (mod m) and x / ±1 (mod m) then 1 < (x 1, m) < m and 1 < (x + 1, m) < m. Proof: From x 2 1 (mod m) we get m (x 2 1).
Math236 Discrete Maths with Applications
Math236 Discrete Maths with Applications P. Ittmann UKZN, Pietermaritzburg Semester 1, 2012 Ittmann (UKZN PMB) Math236 2012 1 / 43 The Multiplication Principle Theorem Let S be a set of k-tuples (s 1,
Student Book SAMPLE CHAPTERS
Student Book SAMPLE CHAPTERS Nelson Student Book Nelson Math Focus... Eas Each lesson starts with a Lesson Goal. Chapter 6 You will need base ten blocks GOAL Multiply using a simpler, related question.
FOM 11 Ch. 1 Practice Test Name: Inductive and Deductive Reasoning
FOM 11 Ch. 1 Practice Test Name: Inductive and Deductive Reasoning Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Justin gathered the following evidence.
Square Roots and the Pythagorean Theorem
UNIT 1 Square Roots and the Pythagorean Theorem Just for Fun What Do You Notice? Follow the steps. An example is given. Example 1. Pick a 4-digit number with different digits. 3078 2. Find the greatest
CMPSCI 250: Introduction to Computation. Lecture #14: The Chinese Remainder Theorem David Mix Barrington 24 February 2012
CMPSCI 250: Introduction to Computation Lecture #14: The Chinese Remainder Theorem David Mix Barrington 24 February 2012 The Chinese Remainder Theorem Infinitely Many Primes Reviewing Inverses and the
Course Syllabus - Online Prealgebra
Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 1.1 Whole Numbers, Place Value Practice Problems for section 1.1 HW 1A 1.2 Adding Whole Numbers Practice Problems for section 1.2 HW 1B 1.3 Subtracting Whole Numbers
4th Pui Ching Invitational Mathematics Competition. Final Event (Secondary 1)
4th Pui Ching Invitational Mathematics Competition Final Event (Secondary 1) 2 Time allowed: 2 hours Instructions to Contestants: 1. 100 This paper is divided into Section A and Section B. The total score
Math 127: Equivalence Relations
Math 127: Equivalence Relations Mary Radcliffe 1 Equivalence Relations Relations can take many forms in mathematics. In these notes, we focus especially on equivalence relations, but there are many other
Göttlers Proof of the Collatz Conjecture
Göttlers Proof of the Collatz Conjecture Henry Göttler, Chantal Göttler, Heinrich Göttler, Thorsten Göttler, Pei-jung Wu goettlercollatzproof@gmail.com March 8, 2018 Abstract Over 80 years ago, the German
Grade 6 Math Circles March 7/8, Magic and Latin Squares
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles March 7/8, 2017 Magic and Latin Squares Today we will be solving math and logic puzzles!
2-1 Inductive Reasoning and Conjecture
Write a conjecture that describes the pattern in each sequence. Then use your conjecture to find the next item in the sequence. 18. 1, 4, 9, 16 1 = 1 2 4 = 2 2 9 = 3 2 16 = 4 2 Each element is the square
Class 8 Cubes and Cube Root
ID : in-8-cubes-and-cube-root [1] Class 8 Cubes and Cube Root For more such worksheets visit www.edugain.com Answer the questions (1) Find the value of A if (2) If you subtract a number x from 15 times
17. Symmetries. Thus, the example above corresponds to the matrix: We shall now look at how permutations relate to trees.
7 Symmetries 7 Permutations A permutation of a set is a reordering of its elements Another way to look at it is as a function Φ that takes as its argument a set of natural numbers of the form {, 2,, n}
Notes for Recitation 3
6.042/18.062J Mathematics for Computer Science September 17, 2010 Tom Leighton, Marten van Dijk Notes for Recitation 3 1 State Machines Recall from Lecture 3 (9/16) that an invariant is a property of a
Sheet 1: Introduction to prime numbers.
Option A Hand in at least one question from at least three sheets Sheet 1: Introduction to prime numbers. [provisional date for handing in: class 2.] 1. Use Sieve of Eratosthenes to find all prime numbers
Primitive Roots. Chapter Orders and Primitive Roots
Chapter 5 Primitive Roots The name primitive root applies to a number a whose powers can be used to represent a reduced residue system modulo n. Primitive roots are therefore generators in that sense,
Fall. Spring. Possible Summer Topics
Fall Paper folding: equilateral triangle (parallel postulate and proofs of theorems that result, similar triangles), Trisect a square paper Divisibility by 2-11 and by combinations of relatively prime
Meet # 1 October, Intermediate Mathematics League of Eastern Massachusetts
Meet # 1 October, 2000 Intermediate Mathematics League of Eastern Massachusetts Meet # 1 October, 2000 Category 1 Mystery 1. In the picture shown below, the top half of the clock is obstructed from view
Category A: Estimating Square Roots and Cube Roots - 3
Category A: Estimating Square Roots and Cube Roots When estimating irrational numbers, the easiest way to compare values is by squaring (or cubing) the given values. Ex: Between which two consecutive numbers
Puzzles ANIL KUMAR C P. The Institute of Mathematical Sciences, Chennai. Puzzles for kids. Date: May 4, 2014
Puzzles By ANIL KUMAR C P The Institute of Mathematical Sciences, Chennai Puzzles for kids Date: May 4, 2014 To my School Teachers Gurur Brahma Gurur V ishnu, Gurur Devoh M aheswaraha Gurur Sakshath P
Think Of A Number. Page 1 of 10
Think Of A Number Tell your audience to think of a number (and remember it) Then tell them to double it. Next tell them to add 6. Then tell them to double this answer. Next tell them to add 4. Then tell
March 5, What is the area (in square units) of the region in the first quadrant defined by 18 x + y 20?
March 5, 007 1. We randomly select 4 prime numbers without replacement from the first 10 prime numbers. What is the probability that the sum of the four selected numbers is odd? (A) 0.1 (B) 0.30 (C) 0.36
Math 1111 Math Exam Study Guide
Math 1111 Math Exam Study Guide The math exam will cover the mathematical concepts and techniques we ve explored this semester. The exam will not involve any codebreaking, although some questions on the
CPM EDUCATIONAL PROGRAM
CPM EDUCATIONAL PROGRAM SAMPLE LESSON: ALGEBRA TILES PART 1: INTRODUCTION TO ALGEBRA TILES The problems in Part 1 introduce algebra tiles to students. These first eleven problems will probably span two
PRIMES STEP Plays Games
PRIMES STEP Plays Games arxiv:1707.07201v1 [math.co] 22 Jul 2017 Pratik Alladi Neel Bhalla Tanya Khovanova Nathan Sheffield Eddie Song William Sun Andrew The Alan Wang Naor Wiesel Kevin Zhang Kevin Zhao
The Chinese Remainder Theorem
The Chinese Remainder Theorem 8-3-2014 The Chinese Remainder Theorem gives solutions to systems of congruences with relatively prime moduli The solution to a system of congruences with relatively prime
Paper 1. Calculator not allowed. Mathematics test. First name. Last name. School. Remember KEY STAGE 3 TIER 5 7. satspapers.org
Ma KEY STAGE 3 Mathematics test TIER 5 7 Paper 1 Calculator not allowed First name Last name School 2009 Remember The test is 1 hour long. You must not use a calculator for any question in this test. You
Final Math Paper. James Marden. December 3, Introduction. Mersenne primes are a set of primes which are of the form: (2 p ) 1 = M
Final Math Paper James Marden December 3, 2016 Introduction Mersenne primes are a set of primes which are of the form: (2 p ) 1 = M There are currently only 49 known Mersenne primes, the smallest of which
MODULAR ARITHMETIC II: CONGRUENCES AND DIVISION
MODULAR ARITHMETIC II: CONGRUENCES AND DIVISION MATH CIRCLE (BEGINNERS) 02/05/2012 Modular arithmetic. Two whole numbers a and b are said to be congruent modulo n, often written a b (mod n), if they give
Meet #3 January Intermediate Mathematics League of Eastern Massachusetts
Meet #3 January 2009 Intermediate Mathematics League of Eastern Massachusetts Meet #3 January 2009 Category 1 Mystery 1. How many two-digit multiples of four are there such that the number is still a
Public Key Encryption
Math 210 Jerry L. Kazdan Public Key Encryption The essence of this procedure is that as far as we currently know, it is difficult to factor a number that is the product of two primes each having many,
The congruence relation has many similarities to equality. The following theorem says that congruence, like equality, is an equivalence relation.
Congruences A congruence is a statement about divisibility. It is a notation that simplifies reasoning about divisibility. It suggests proofs by its analogy to equations. Congruences are familiar to us
0:00:07.150,0:00: :00:08.880,0:00: this is common core state standards support video in mathematics
0:00:07.150,0:00:08.880 0:00:08.880,0:00:12.679 this is common core state standards support video in mathematics 0:00:12.679,0:00:15.990 the standard is three O A point nine 0:00:15.990,0:00:20.289 this
1 Introduction. 2 An Easy Start. KenKen. Charlotte Teachers Institute, 2015
1 Introduction R is a puzzle whose solution requires a combination of logic and simple arithmetic and combinatorial skills 1 The puzzles range in difficulty from very simple to incredibly difficult Students
Your Task. Unit 3 (Chapter 1): Number Relationships. The 5 Goals of Chapter 1
Unit 3 (Chapter 1): Number Relationships The 5 Goals of Chapter 1 I will be able to: model perfect squares and square roots use a variety of strategies to recognize perfect squares use a variety of strategies