On Hardy's Apology Numbers
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4 VI June 2016
4 VI June 2016 www.ijraset.com Volume 4 Issue VI, June 2016 IC Value: 13.98 ISSN: 2321-9653 International Journal for Research in Applied Science & Engineering Technology (IJRASET) Special Rectangles and Narcissistic Numbers of Order 3 And 4 G.Janaki1, P.Saranya2 1,2Department of Mathematics, Cauvery College for women, Trichy-620018 Abstract— We search for infinitely many rectangles such that x2 y2 3A S 2 k2 SK Narcissistic numbers of order 3 and 4 respectively, in which x, y represents the length and breadth of the rectangle. Also the total number of rectangles satisfying the relation under consideration as well as primitive and non-primitive rectangles are also present. Keywords—Rectangle, Narcissistic numbers of order 3 and 4, primitive,non-primitive. I. INTRODUCTION The older term for number theory is arithmetic, which was superseded as number theory by early twentieth century. The first historical find of an arithmetical nature is a fragment of a table, the broken clay tablet containing a list of Pythagorean triples. Since then the finding continues. For more ideas and interesting facts one can refer [1].In [2] one can get ideas on pairs of rectangles dealing with non-zero integral pairs representing the length and breadth of rectangle. [3,4] has been studied for knowledge on rectangles in connection with perfect squares , Niven numbers and kepriker triples.[5-10] was referred for connections between Special rectangles and polygonal numbers, jarasandha numbers and dhuruva numbers Recently in [11,12] special pythagorean triangles in connections with Narcissistic numbers are obtained. In this communication, we search for infinitely many rectangles such that x2 y2 3A S 2 k2 SK Narcissistic numbers of order 3 and 4 respectively, in which x,y represents the length and breadth of the rectangle. -
Maths Secrets of Simpsons Revealed in New Book
MONDAY 7 OCTOBER 2013 WWW.THEDAY.CO.UK Maths secrets of Simpsons revealed in new book The most successful TV show of all time is written by a team of brilliant ‘mathletes’, says writer Simon Singh, and full of obscure mathematical jokes. Can numbers really be all that funny? MATHEMATICS Nerd hero: The smartest girl in Springfield was created by a team of maths wizards. he world’s most popular cartoon a perfect number, a narcissistic number insist that their love of maths contrib- family has a secret: their lines are and a Mersenne Prime. utes directly to the more obvious humour written by a team of expert mathema- Another of these maths jokes – a black- that has made the show such a hit. Turn- Tticians – former ‘mathletes’ who are board showing 398712 + 436512 = 447212 ing intuitions about comedy into concrete as happy solving differential equa- – sent shivers down Simon Singh’s spine. jokes is like wrestling mathematical tions as crafting jokes. ‘I was so shocked,’ he writes, ‘I almost hunches into proofs and formulas. Comedy Now, science writer Simon Singh has snapped my slide rule.’ The numbers are and maths, says Cohen, are both explora- revealed The Simpsons’ secret math- a fake exception to a famous mathemati- tions into the unknown. ematical formula in a new book*. He cal rule known as Fermat’s Last Theorem. combed through hundreds of episodes One episode from 1990 features a Mathletes and trawled obscure internet forums to teacher making a maths joke to a class of Can maths really be funny? There are many discover that behind the show’s comic brilliant students in which Bart Simpson who will think comparing jokes to equa- exterior lies a hidden core of advanced has been accidentally included. -
The Observability of the Fibonacci and the Lucas Cubes
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 255 (2002) 55–63 www.elsevier.com/locate/disc The observability of the Fibonacci and the Lucas cubes Ernesto DedÃo∗;1, Damiano Torri1, Norma Zagaglia Salvi1 Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy Received 5 April 1999; received inrevised form 31 July 2000; accepted 8 January2001 Abstract The Fibonacci cube n is the graph whose vertices are binary strings of length n without two consecutive 1’s and two vertices are adjacent when their Hamming distance is exactly 1. If the binary strings do not contain two consecutive 1’s nora1intheÿrst and in the last position, we obtainthe Lucas cube Ln. We prove that the observability of n and Ln is n, where the observability of a graph G is the minimum number of colors to be assigned to the edges of G so that the coloring is proper and the vertices are distinguished by their color sets. c 2002 Elsevier Science B.V. All rights reserved. MSC: 05C15; 05A15 Keywords: Fibonacci cube; Fibonacci number; Lucas number; Observability 1. Introduction A Fibonacci string of order n is a binary string of length n without two con- secutive ones. Let and ÿ be binary strings; then ÿ denotes the string obtained by concatenating and ÿ. More generally, if S is a set of strings, then Sÿ = {ÿ: ∈ S}. If Cn denotes the set of the Fibonacci strings of order n, then Cn+2 =0Cn+1 +10Cn and |Cn| = Fn, where Fn is the nth Fibonacci number. -
Figurate Numbers
Figurate Numbers by George Jelliss June 2008 with additions November 2008 Visualisation of Numbers The visual representation of the number of elements in a set by an array of small counters or other standard tally marks is still seen in the symbols on dominoes or playing cards, and in Roman numerals. The word "calculus" originally meant a small pebble used to calculate. Bear with me while we begin with a few elementary observations. Any number, n greater than 1, can be represented by a linear arrangement of n counters. The cases of 1 or 0 counters can be regarded as trivial or degenerate linear arrangements. The counters that make up a number m can alternatively be grouped in pairs instead of ones, and we find there are two cases, m = 2.n or 2.n + 1 (where the dot denotes multiplication). Numbers of these two forms are of course known as even and odd respectively. An even number is the sum of two equal numbers, n+n = 2.n. An odd number is the sum of two successive numbers 2.n + 1 = n + (n+1). The even and odd numbers alternate. Figure 1. Representation of numbers by rows of counters, and of even and odd numbers by various, mainly symmetric, formations. The right-angled (L-shaped) formation of the odd numbers is known as a gnomon. These do not of course exhaust the possibilities. 1 2 3 4 5 6 7 8 9 n 2 4 6 8 10 12 14 2.n 1 3 5 7 9 11 13 15 2.n + 1 Triples, Quadruples and Other Forms Generalising the divison into even and odd numbers, the counters making up a number can of course also be grouped in threes or fours or indeed any nonzero number k. -
Single Digits
...................................single digits ...................................single digits In Praise of Small Numbers MARC CHAMBERLAND Princeton University Press Princeton & Oxford Copyright c 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved The second epigraph by Paul McCartney on page 111 is taken from The Beatles and is reproduced with permission of Curtis Brown Group Ltd., London on behalf of The Beneficiaries of the Estate of Hunter Davies. Copyright c Hunter Davies 2009. The epigraph on page 170 is taken from Harry Potter and the Half Blood Prince:Copyrightc J.K. Rowling 2005 The epigraphs on page 205 are reprinted wiht the permission of the Free Press, a Division of Simon & Schuster, Inc., from Born on a Blue Day: Inside the Extraordinary Mind of an Austistic Savant by Daniel Tammet. Copyright c 2006 by Daniel Tammet. Originally published in Great Britain in 2006 by Hodder & Stoughton. All rights reserved. Library of Congress Cataloging-in-Publication Data Chamberland, Marc, 1964– Single digits : in praise of small numbers / Marc Chamberland. pages cm Includes bibliographical references and index. ISBN 978-0-691-16114-3 (hardcover : alk. paper) 1. Mathematical analysis. 2. Sequences (Mathematics) 3. Combinatorial analysis. 4. Mathematics–Miscellanea. I. Title. QA300.C4412 2015 510—dc23 2014047680 British Library -
Fibonacci's Perfect Spiral
Middle School Fibonacci's Perfect Spiral This l esson was created to combine math history, math, critical thinking, and art. Students will l earn about Fibonacci, the code he created, and how the Fibonacci sequence relates to real l ife and the perfect spiral. Students will practice drawing perfect spirals and l earn how patterns are a mathematical concept that surrounds us i n real l ife. This l esson i s designed to take 3 to 4 class periods of 45 minutes each, depending on the students’ focus and depth of detail i n the assignments. Common Core CCSS.MATH.CONTENT.HSF.IF.A.3 Recognize that Standards: sequences are functions, sometimes defined recursively, whose domain i s a subset of the i ntegers. For example, the Fibonacci sequence i s defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n1) for n ≥ 1. CCSS.MATH.CONTENT.7.EE.B.4 Use variables to represent quantities i n a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. CCSS.MATH.CONTENT.7.G.A.1 Solve problems involving scale drawings of geometric figures, i ncluding computing actual l engths and areas from a scale drawing and reproducing a scale drawing at a different scale. CCSS.MATH.CONTENT.7.G.A.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. CCSS.MATH.CONTENT.8.G.A.2 Understand that a twodimensional figure i s congruent to another i f the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. -
Number Gossip About 10 Years Ago and at first I Uploaded It on My Personal Website [6]
Number Gossip Tanya Khovanova Department of Mathematics, MIT April 15, 2008 Abstract This article covers my talk at the Gathering for Gardner 2008, with some additions. 1 Introduction My pet project Number Gossip has its own website: http://www.numbergossip.com/, where you can plug in your favorite integer up to 9,999 and learn its properties. A behind-the-scenes program checks your number for 49 regular properties and also checks a database for unique properties I collected. 2 Eight The favorite number of this year’s Gathering is composite, deficient, even, odious, palindromic, powerful, practical and Ulam. It is also very cool as it has the rare properties of being a cake and a narcissistic number, as well as a cube and a Fibonacci number. And it also is a power of two. In addition, eight has the following unique properties: • 8 is the only composite cube in the Fibonacci sequence • 8 is the dimension of the octonions and is the highest possible dimension of a normed division algebra • 8 is the smallest number (except 1) which is equal to the sum of the digits of its cube 3 Properties There are 49 regular properties that I check for: arXiv:0804.2277v1 [math.CO] 14 Apr 2008 abundant evil odious Smith amicable factorial palindrome sociable apocalyptic power Fibonacci palindromic prime square aspiring Google pentagonal square-free automorphic happy perfect tetrahedral cake hungry power of 2 triangular Carmichael lazy caterer powerful twin Catalan lucky practical Ulam composite Mersenne prime undulating compositorial Mersenne prime primorial untouchable cube narcissistic pronic vampire deficient odd repunit weird even 1 I selected regular properties for their importance as well as their funny names, so your favorite number could be lucky and happy at the same time, as is the case for 7. -
Student Task Statements (8.7.B.3)
GRADE 8 MATHEMATICS BY NAME DATE PERIOD Unit 7, Lesson 3: Powers of Powers of 10 Let's look at powers of powers of 10. 3.1: Big Cube What is the volume of a giant cube that measures 10,000 km on each side? 3.2: Taking Powers of Powers of 10 1. a. Complete the table to explore patterns in the exponents when raising a power of 10 to a power. You may skip a single box in the table, but if you do, be prepared to explain why you skipped it. expression expanded single power of 10 b. If you chose to skip one entry in the table, which entry did you skip? Why? 2. Use the patterns you found in the table to rewrite as an equivalent expression with a single exponent, like . GRADE 8 MATHEMATICS BY NAME DATE PERIOD 3. If you took the amount of oil consumed in 2 months in 2013 worldwide, you could make a cube of oil that measures meters on each side. How many cubic meters of oil is this? Do you think this would be enough to fill a pond, a lake, or an ocean? 3.3: How Do the Rules Work? Andre and Elena want to write with a single exponent. • Andre says, “When you multiply powers with the same base, it just means you add the exponents, so .” • Elena says, “ is multiplied by itself 3 times, so .” Do you agree with either of them? Explain your reasoning. Are you ready for more? . How many other whole numbers can you raise to a power and get 4,096? Explain or show your reasoning. -
INDEX to ANSWERS Issue Page Number Symmetryplus 60 2
INDEX TO ANSWERS Issue Page number SYMmetryplus 60 2 SYMmetryplus 61 4 SYMmetryplus 62 5 SYMmetryplus 63 8 SYMmetryplus 64 9 SYMmetryplus 65 10 SYMmetryplus 66 13 SYMmetryplus 67 15 SYMmetryplus 68 17 SYMmetryplus 69 21 SYMmetryplus 70 25 SYMmetryplus 71 29 SYMmetryplus 72 33 SYMmetryplus 73 37 SYMmetryplus 74 44 1 ANSWERS FROM ISSUE 60 SOME TRIANGLE NUMBERS – 2 Many thanks to Andrew Palfreyman who found five, not four solutions! 7 7 7 7 7 1 0 5 3 0 0 4 0 6 9 0 3 9 4 6 3 3 3 3 1 Grid A 6 6 1 3 2 6 2 0 1 6 0 0 Grid B CROSSNUMBER Many thanks again to Andrew Palfreyman who pointed out that 1 Down and 8 Across do not give unique answers so there are four possible solutions. 1 2 1 2 1 2 1 2 1 4 4 8 1 4 4 8 1 4 4 8 1 4 4 8 3 3 3 3 3 9 1 9 8 9 1 9 3 9 1 9 8 9 1 9 4 5 4 5 4 5 4 5 2 3 1 0 2 3 1 0 2 3 1 0 2 3 1 0 6 7 6 7 6 7 6 7 1 0 9 8 1 0 9 8 1 0 9 8 1 0 9 8 8 9 8 9 8 9 8 9 3 6 1 0 3 6 1 0 9 6 1 0 9 6 1 0 10 10 10 10 5 3 4 3 5 3 4 3 5 3 4 3 5 3 4 3 TREASURE HUNTS 12, 13 12 This is a rostral column in St Petersburg, Russia. -
2018 Summer High School Program Placement Exam for Level B NAME
2018 Summer High School Program Placement Exam for Level B NAME: Multiple Choice: Indicate your answer in the box to the right of each question. 1 1. Which of the following is closest to 2 ? 2 7 9 10 23 (A) 3 (B) 13 (C) 19 (D) 21 (E) 48 2. If 42x = 4 and 4−5y = 32, then what is the value of (−4)x−y? 1 1 (A) −4 (B) −2 (C) − 4 (D) 4 (E) 4 3. On an analog clock, how many times do the hour and minute hands coincide between 11:30am on Monday and 11:30am on Tuesday? (A) 12 (B) 13 (C) 22 (D) 23 (E) 24 4. If −1 ≤ a ≤ 2 and −2 ≤ b ≤ 3, find the minimum value of ab. (A) −6 (B) −5 (C) −4 (D) −3 (E) −2 5. A fair coin is tossed 5 times. What is the probability that it will land tails at least half the time? 3 5 1 11 13 (A) 16 (B) 16 (C) 2 (D) 16 (E) 16 6. Let f(x) = j5 − 2xj. If the domain of f(x) is (−3; 3], what is the range of f(x)? (A) [−1; 11) (B) [0; 11) (C) [2; 8) (D) [1; 11) (E) None of these 7. When (a+b)2018 +(a−b)2018 is fully expanded and all the like terms are combined, how many terms are there? (A) 1008 (B) 1009 (C) 1010 (D) 2018 (E) 2019 8. How many factors does 10! have? (A) 64 (B) 80 (C) 120 (D) 240 (E) 270 9. -
Various Arithmetic Functions and Their Applications
University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2016 Various Arithmetic Functions and their Applications Florentin Smarandache University of New Mexico, [email protected] Octavian Cira Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Applied Mathematics Commons, Logic and Foundations Commons, Number Theory Commons, and the Set Theory Commons Recommended Citation Smarandache, Florentin and Octavian Cira. "Various Arithmetic Functions and their Applications." (2016). https://digitalrepository.unm.edu/math_fsp/256 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. Octavian Cira Florentin Smarandache Octavian Cira and Florentin Smarandache Various Arithmetic Functions and their Applications Peer reviewers: Nassim Abbas, Youcef Chibani, Bilal Hadjadji and Zayen Azzouz Omar Communicating and Intelligent System Engineering Laboratory, Faculty of Electronics and Computer Science University of Science and Technology Houari Boumediene 32, El Alia, Bab Ezzouar, 16111, Algiers, Algeria Octavian Cira Florentin Smarandache Various Arithmetic Functions and their Applications PONS asbl Bruxelles, 2016 © 2016 Octavian Cira, Florentin Smarandache & Pons. All rights reserved. This book is protected by copyright. No part of this book may be reproduced in any form or by any means, including photocopying or using any information storage and retrieval system without written permission from the copyright owners Pons asbl Quai du Batelage no. -
Solve the Rubik's Cube Using Proc
Solve the Rubik’s Cube using Proc IML Jeremy Hamm Cancer Surveillance & Outcomes (CSO) Population Oncology BC Cancer Agency Overview Rubik’s Cube basics Translating the cube into linear algebra Steps to solving the cube using proc IML Proc IML examples The Rubik’s Cube 3x3x3 cube invented in 1974 by Hungarian Erno Rubik – Most popular in the 80’s – Still has popularity with speed-cubers 43x1018 permutations of the Rubik’s Cube – Quintillion Interested in discovering moves that lead to permutations of interest – or generalized permutations Generalized permutations can help solve the puzzle The Rubik’s Cube Made up of Edges and corners Pieces can permute – Squares called facets Edges can flip Corners can rotate Rubik’s Cube Basics Each move can be defined as a combination of Basic Movement Generators – each face rotated a quarter turn clockwise Eg. – Movement of front face ¼ turn clockwise is labeled as move F – ¼ turn counter clockwise is F-1 – 2 quarter turns would be F*F or F2 Relation to Linear Algebra The Rubik’s cube can represented by a Vector, numbering each facet 1-48 (excl centres). Permutations occur through matrix algebra where the basic movements are represented by Matrices Ax=b Relation to Linear Algebra Certain facets are connected and will always move together Facets will always move in a predictable fashion – Can be written as: F= (17,19,24,22) (18,21,23,20) (6,25,43,16) (7,28,42,13) (8,30,41,11) Relation to Linear Algebra Therefore, if you can keep track of which numbers are edges and which are corners, you can use a program such as SAS to mathematically determine moves which are useful in solving the puzzle – Moves that only permute or flip a few pieces at a time such that it is easy to predict what will happen Useful Group Theory Notes: – All moves of the Rubik’s cube are cyclical where the order is the number of moves needed to return to the original Eg.