# TAKE-AWAY GAMES. ALLEN J. SCHWENK California Institute of Technology, Pasadena, California INTRODUCTION

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 TAKE-AWAY GAMES ALLEN J. SCHWENK California Institute of Technology, Pasadena, California L INTRODUCTION Several games of Tf take-away?f have become popular. The purpose of this paper is to determine the winning strategy of a general class of takeaway games, in which the number of markers which maybe removed each turn is a function of the number removed on the preceding turn. By-products of this investigation are a new generalization of Zeckendorfs Theorem [ 3 ], and an affirmative answer to a conjecture of Gaskell and Whinihan [2]. Definitions: (1-1) Let a take-away game be defined as a two-person game in which the players alternately diminish an original stock of markers subject to various restrictions, with the player who removes the last marker being the winner** (1-2) A turn or move shall consist of removing a number of these markers. (1-3) Let the original number of markers in the stock be N(0). (1-4) After the k move there will be N(k) markers remaining. (1-5) The player who takes the first turn shall be called player A. The other player shall be called player B. (1-6) Let T(k) = N(k- 1) - N(k). That is, T(k) is the number of markers removed in the k move. (1-7) The winning strategy sought will always be a forced win for Player A. All games considered in this paper are further restricted by the following rules: (a) T(k) > 1 for all k = 1,2,- 8 '. (b) T(l) < N(0) (Thus, N(0) > 1.) (c) For all k = 2, 3,» ", T(k) < m,, where m, is some function of T ( k - 1). *This definition is essentially taken from Golomb [1]. 225

2 226 TAKE-AWAY GAMES [April Rule (a) guarantees that the game will terminate after a finite number of moves since the number of markers in the stock is strictly decreasing, and hence, must reach zero. Rule (b) dispenses with the uninteresting case of immediate victory. Rule (c) is the source of the distinguishing characteristics of the various games which shall be considered. H. MOTIVATION Example (n-1) A simple game occurs when m, is defined to be constant, m, and we require f (1) m. The well known strategy is: If N(0) ^ 0 mod (m + 1), remove N(0) mod (m + 1) markers. On subsequent moves, Player A selects T(2j + 1) to be equal to m T(2j). If N(0) = Omod (m + 1), Player B can win by applying Player A T s strategy above. A simple way to express this result is to write the integer N(k) in a base m + 1 number system. Thus, N(k) = a 0 + a^m + 1) + a 2 (m + I) a.(m+l) 3, where this representation is unique. Player A f s strategy is to remove a 0 markers, provided a 0 ^ 0. If a 0 = 0, Player A is faced with a losing position. systems. This result suggests a connection between winning strategies and number Example (IE-2) Consider the game defined by the rule m, = T(k - 1), the number of markers removed on the preceding move. In other words, T(k) <T(k - 1). To find a winning strategy, express N(0) as a binary number, e.g., 12 = 1100 B. Define /N(0)/ as follows: If N = ( V n - l ' " a i a o ) B ' in the binary system, then

3 1970] TAKE-AWAY GAMES 227 / N / = a n + a n _ a i + a 0. If /N(0)/ = k > 1,. Player A removes a number corresponding to the last "one" in the binary expansion. (Thus, for N(0) = 12 = 1100, Player A removes 4 = 100.) Now /N(l)/ = k - 1 > 0. Player B now has no move which reduces /N(l)/; to do so, he would have to remove twice as many as the rules permit. In addition, any move Player B does make produces an N(2) such that Player A can again remove the last " 1 " in the expansion of N(2) 8 To see this, note that N(l) can be rewritten as V n - l ' " V ' ) B + 1 B ' Now, since N(k) is strictly decreasing, it must reach zero. However, / 0 / = 0 and / N / > 0 for all positive integers N. Since Player B never decreases /N(k)/, Player B cannot produce zero; hence, Player A must win. If, on the other hand, /N(0)/ = 1, it is clear that Player A cannot win because N(0) = = 11' 1_ Any move by Player A permits Player B to remove the last " 1 " in the expansion* thus applying the strategy formerly used by A above. Again we see a connection with number systems. A generalization of this method now suggests itself: Find a way to express every positive integer as a unique sum of losing positions. Then a losing position has norm 1. For any other position, the norm reducing strategy described above will work if, given Player A T s move, (i) Player B cannot reduce /N(k)/, and (ii) any move Player B does make permits Player A to reduce /N(k+1)/. m. THE GENERAL GAME Now consider any game in which m, is a function of the number of markers removed on the preceding move; i. e., let m = f(t(k - 1)). Suppose f(n) ^ n and f(n) ^ f(n - 1) for all positive integers, n. Note that example II-2 satisfied this hypothesis. We want f to be a monotonic nondecreasing

4 228 TAKE-AWAY GAMES [April function so that if Player B removes more markers he cannot limit Player A to removing fewer markers and thus foil the norm reducing strategy. In addition, we want f(n) ^ n to guarantee the existence of a legal move at all times, and to permit the following definition: Definition (ni-1) Define a sequence (H.) by: Hj = 1 and H. + 1 = H. + H. where j is the smallest index such that f(h.) H,. Clearly this is well defined because if the above inequality holds for no smaller j, at least we know it holds for j = k. Theorem (3II-2) Every positive integer can be represented as a unique sum of H. f s, such that n N = ^2 H. and f(h. ) < H. for i = 1, 2,,n ~ 1.. = 1 +l The theorem is trivially true when N = 1, for Hi = 1 Assume that the theorem holds for all N < H, ; and let H, ^ N < H, -. By induction, n N = Hk + E H J. x i=l where f(h. ) < H. for i = 1,2,*,n - 1. Thus, for the existence of a +l representation, we need only show that f(h. ) < H,. Suppose f(h. ) H,. Then recall that H, H = H, + H^ B.g where 4 is the minimal coefficient for which f (Hg ^ H,. Hence j ^ i and so H k + i = H k + H - H k + H J n - N contradicting the choice of N. Thus we have existence.

5 1970] TAKE-AWAY GAMES 229 F o r uniqueness, note that: f(h. ) < H. Ji 32 implies E H. i = l < H.,- Jl 32 + l f(h. ) < H. J2 33 implies 3 i=l f(h. ) < H. implies J n - 1 'n n Z H. i=i < H. J_ 1 l. I +1 i n T h u s, for H, - > N > H,, the l a r g e s t t e r m in any sum for N m u s t be H,. If N has two r e p r e s e n t a t i o n s, so does N - H,, but this violates the induction hypo the si s. T h u s, the representation i s unique. Definition (m-3) / N / i s the number of t e r m s in the "H sum" for N. L e m m a (IH-4) If / N ( k ) / = 1 and the player cannot move N(k) m a r k e r s, then any move he does make p e r m i t s his opponent to reduce /N(k + 1)/* F o r simplicity, let us a s s u m e that k is odd. T h u s, we will prove that P l a y e r A can remove an appropriate number of m a r k e r s so that /N(k + 2 ) / /N(k + 1)/. Rewrite N(k) = H. 3o = H., + H. : 30-1 Jl = H J o _ 1 + H J i _ H J n _ 1 + l <

6 230 TAKE-AWAY GAMES [April for some n where f(h. ) ^ H., > f(h., ) J i+1 J i * 3 x i+l for i = 0, 1,, n - 1 Note that this is equivalent to H. = H. - + H. k' 1 +l Now P l a y e r B removes T(k + 1), with H. < T(k + 1) < H. for some i Ji+1 Ji between 0 and n, where H. = 1. P l a y e r A may remove up to J n+1 f(t(k + 1)) >f(hl ) +l Hence, P l a y e r A m a y elect to remove and H. - T(k + 1 ) < H. - H. = H. _, J i+1 J i N(k + 2) = H H. - J i - l " 1 JO" 1 Since f(h. _ 1 ) ^ H. 1 for i = l, 2, - - -, n, we have /N(k + 2 ) / = i. T Ji J i-1 Let / H. - T(k + 1)/ = a > 0. h Now N(k + 1) = N(k + 2) + H. - T(k + 1). i Let H# be the l a r g e s t t e r m in the H sum of H. - T(k + 1). Clearly Ji H, < H. T whence f(h*) < f(h. - ) < H. -. Thus J i - 1 /N(k + 1)/ = I + a > i = /N(k + 2 ) /, and this completes the proof of the l e m m a.

7 1970] TAKE-AWAY GAMES 231 T h e o r e m (III-5) Let us consider a game defined by f satisfying the p r o p e r t i e s stated above. Also let (H.) and the n o r m be defined a s above, If / N ( 0 ) / > 1, P l a y e r A can force a win. If / N ( 0 ) / = 1, P l a y e r B can force a win. If / N ( 0 ) / > 1, let N(0) = H. + + H j n with f(hj.) < H - j. + r P l a y e r A r e - moves H.. Since P l a y e r B can remove a t m o s t f(h. ) < H. it J2 Jl 32 is clear that Player B cannot reduce /N(l)/ or affect any of the last n - 2 terms in the sum, so we may just as well consider n «2. Now we invoke Lemma (HI-4)* so Player A can reduce /JN(2)/. Thus, Player A can force a win. If /N(0)/ = 1, Since Player A cannot remove N(0) markers, Lemma (ni-4) tells us that Player B will be able to reduce /N(l)/. If /N(l)/ = 1, this means that he can remove N(l) and win immediately. /N(l)/ > 1, Player B can apply Player A! s strategy from the first part of this proof. Thus, Player B can force a win. If IV. BY-PRODUCTS In the case when f(t(k - 1)) = 2T(k - 1), the foregoing results produce the conclusions of Whinihan and Gaskell [2] regarding "Fibonacci Nim," We note that in this case: Hj = 1 H2 = Hj + Hj = 2 H3 = H2 + Hj = 3 and in general, if H. = H. - + H. 0 n-i n - i - 1 n - i - 2 for i = 0 9 1, and 2, then

8 232 TAKE-AWAY GAMES [April 2 H n - 3 ^ H n - 2 > 2 H n H n - 2 * H n - 1 > 2 H n - 3 So 2H 1 ^ H > 2H 0 n - 1 n n-2 by adding the inequalities above. Hence H + - = H + H -. This p r o c e s s continues by induction so that the sequence Fibonacci n u m b e r s. (H.) is indeed the sequence of Also in this c a s e, T h e o r e m (III-2) becomes "Zeckendorf's theorem" [3], which states that every positive integer can be uniquely expressed a s a Fibonacci sum with no two consecutive subscripts appearing. Another interesting fact, conjectured by Whinihan and Gaskell [ 2 ], is that for the game m, = ct(k - 1), where c i s any real number ^ 1, (H.) m u s t become a simple r e c u r s i o n sequence for sufficiently l a r g e subscripts; i. e., there exist integers & k and n 0 u such that H,- = H + H, for all n+1 n n-k n ;> n 0. Let us now consider how to prove the conjecture, and how to calculate k and n 0 a s a function of c. L e m m a (IV-1) If ch., < I - I H. < ch., j ~ r then c H. ^ ^ H.^,. I + I j+1 Since ch. < H. <i.ch., we m u s t have H. ^ = H. + H.. Also, H., n = i - I j I 3+1 J i i+l H. + H, where ch, ^ H.. Now I k k I c H ch. + ch k > c H. + H. - l l > H. + H. = H. J _ 1. - J i J+1 T h e o r e m (IV-2) T h e r e exists an integer k such that ch, < H for all n > k. n Ac n Since E.^ = H. + H. where ch. ^ H., it follows that J+1 J i i J

9 1970] TAKE-AWAY GAMES 233 If we choose k such that H., ^H- then H) k > c, VJ lt A k >. H j-k+l HJ T h u s, ch.. < H. for all j > k. This completes the proof of the theorem. Corollary (IV-3) (H ) must become a simple recursion sequence for sufficiently large n. L e m m a (IV-1) says that the difference between successive indices a s described before is monotonically nondecreasing. the sequence of differences is bounded. T h e o r e m (IV-2) says that Thus the difference m u s t be constant for all l a r g e n. This is equivalent to saying that (H ) is a r e c u r s i o n sequence for n ^ r n 0. Q. E. D. T h e o r e m (IV-4) If H = H H. + i _ k for some j, and for i = 0, l, - - -, k + 1, then this equation holds for every positive integer i. By induction, we need only show that H. + k + 3 = H. + k H By definition, H.,, l 0 = H.,,,- + H., - implies ch. < H.,,,- < c H., -. H., - = H. + p 3+k+2 j+k+1 j+1 j j+k+1 ^ j+1 j+1 3 H., implies F 3-k whence ch., < H.^- < ch., _,_-, 3-k k+l c(h. + H.. ) < H. M _,, + H.^- < c O L ^ + H.. ^ ), J j - k ' 3 + k+l J-k+1

10 234 TAKE-AWAY GAMES [April or c H j+l < H j+k+2 * ch j+2 ' SO H j + k + 3 = H j + k H j + 2- Q - E - D - This theorem tells us that k has reached the recursion value when k has been the difference for k + 2 successive indices. V. CONCLUSION We have discovered some interesting properties of take-away games and their winning strategies. The subject, however, is by no means exhausted. For example, in Theorem (IV-4) we showed that for every _c > 1 there exists a k such that.... By inspection, I have found: If c = 1 then k = 0 = 2 = 1 = 3 = 3 = 4 = 5 = 5 = 7 = 6 = 9 = 7 = 12 = 8 = 1 4 It is not clear whether or not a simple relation exists between c and k. In Section IV, we found that f(x) = ex gives rise to a recursion relation for (H.). Other special cases of f can be studied, to learn about the corresponding sequence (H.); or one might try to reverse the approach by proceeding from (H.) to f, as opposed to the approach taken in this paper. It is also possible to generalize in other ways. For example, if f(n) and g(n) satisfy the hypotheses of Section EI, then (f+ g)(n) = f(n) + g(n) and (fg)(n) = f(n)g(n) also satisfy the hypotheses. Can the corresponding strategies and sequences be related? Can the procedure be generalized for functions which are not monotonic? These problems are suggested for those interested in pursuing the subject further. [Continued on page 241. ]

### Take-Away Games MICHAEL ZIEVE

Games of No Chance MSRI Publications Volume 29, 1996 Take-Away Games MICHAEL ZIEVE Abstract. Several authors have considered take-away games where the players alternately remove a positive number of counters

### Math 55: Discrete Mathematics

Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What

### MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

### Mathematical Induction

Mathematical S 0 S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S 10 Like dominoes! Mathematical S 0 S 1 S 2 S 3 S4 S 5 S 6 S 7 S 8 S 9 S 10 Like dominoes! S 4 Mathematical S 0 S 1 S 2 S 3 S5 S 6 S 7 S 8 S 9 S 10

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS

CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS JEREMY BOOHER Continued fractions usually get short-changed at PROMYS, but they are interesting in their own right and useful in other areas

### Induction and Recursion

Induction and Recursion Winter 2010 Mathematical Induction Mathematical Induction Principle (of Mathematical Induction) Suppose you want to prove that a statement about an integer n is true for every positive

### OPTIMAL SELECTION BASED ON RELATIVE RANK* (the "Secretary Problem")

OPTIMAL SELECTION BASED ON RELATIVE RANK* (the "Secretary Problem") BY Y. S. CHOW, S. MORIGUTI, H. ROBBINS AND S. M. SAMUELS ABSTRACT n rankable persons appear sequentially in random order. At the ith

### Math 317 HW #5 Solutions

Math 317 HW #5 Solutions 1. Exercise 2.4.2. (a) Prove that the sequence defined by x 1 = 3 and converges. x n+1 = 1 4 x n Proof. I intend to use the Monotone Convergence Theorem, so my goal is to show

### p 2 1 (mod 6) Adding 2 to both sides gives p (mod 6)

.9. Problems P10 Try small prime numbers first. p p + 6 3 11 5 7 7 51 11 13 Among the primes in this table, only the prime 3 has the property that (p + ) is also a prime. We try to prove that no other

### NIM Games: Handout 1

NIM Games: Handout 1 Based on notes by William Gasarch 1 One-Pile NIM Games Consider the following two-person game in which players alternate making moves. There are initially n stones on the board. During

### SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION

CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION Alessandro Artale UniBZ - http://www.inf.unibz.it/ artale/ SECTION 5.2 Mathematical Induction I Copyright Cengage Learning. All rights reserved.

### Mathematical Induction

Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers

### 3.1. Sequences and Their Limits Definition (3.1.1). A sequence of real numbers (or a sequence in R) is a function from N into R.

CHAPTER 3 Sequences and Series 3.. Sequences and Their Limits Definition (3..). A sequence of real numbers (or a sequence in R) is a function from N into R. Notation. () The values of X : N! R are denoted

### Subsets of Euclidean domains possessing a unique division algorithm

Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness

### PYTHAGOREAN NUMBERS. Bowling Green State University, Bowling Green, OH (Submitted February 1988)

S u p r i y a Mohanty a n d S. P. Mohanty Bowling Green State University, Bowling Green, OH 43403-0221 (Submitted February 1988) Let M be a right angled triangle with legs x and y and hypotenuse z. Then

### N E W S A N D L E T T E R S

N E W S A N D L E T T E R S 73rd Annual William Lowell Putnam Mathematical Competition Editor s Note: Additional solutions will be printed in the Monthly later in the year. PROBLEMS A1. Let d 1, d,...,

### Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition

Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing

### 15-251: Great Theoretical Ideas in Computer Science Anupam Gupta Notes on Combinatorial Games (draft!!) January 29, 2012

15-251: Great Theoretical Ideas in Computer Science Anupam Gupta Notes on Combinatorial Games (draft!!) January 29, 2012 1 A Take-Away Game Consider the following game: there are 21 chips on the table.

### Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

### Further linear algebra. Chapter I. Integers.

Further linear algebra. Chapter I. Integers. Andrei Yafaev Number theory is the theory of Z = {0, ±1, ±2,...}. 1 Euclid s algorithm, Bézout s identity and the greatest common divisor. We say that a Z divides

### Solutions for Practice problems on proofs

Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg

### MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS

1 CHAPTER 6. MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS 1 Introduction Recurrence

### Gambling Systems and Multiplication-Invariant Measures

Gambling Systems and Multiplication-Invariant Measures by Jeffrey S. Rosenthal* and Peter O. Schwartz** (May 28, 997.. Introduction. This short paper describes a surprising connection between two previously

### Every Positive Integer is the Sum of Four Squares! (and other exciting problems)

Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer

### CHAPTER 3. Sequences. 1. Basic Properties

CHAPTER 3 Sequences We begin our study of analysis with sequences. There are several reasons for starting here. First, sequences are the simplest way to introduce limits, the central idea of calculus.

### Taylor and Maclaurin Series

Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

### Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.

Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,

### SECTION 10-2 Mathematical Induction

73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms

### arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu

Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing

### A Curious Case of a Continued Fraction

A Curious Case of a Continued Fraction Abigail Faron Advisor: Dr. David Garth May 30, 204 Introduction The primary purpose of this paper is to combine the studies of continued fractions, the Fibonacci

### Principle of (Weak) Mathematical Induction. P(1) ( n 1)(P(n) P(n + 1)) ( n 1)(P(n))

Outline We will cover (over the next few weeks) Mathematical Induction (or Weak Induction) Strong (Mathematical) Induction Constructive Induction Structural Induction Principle of (Weak) Mathematical Induction

### CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

### Basic building blocks for a triple-double intermediate format

Basic building blocks for a triple-double intermediate format corrected version) Christoph Quirin Lauter February 26, 2009 Abstract The implementation of correctly rounded elementary functions needs high

### Generalized golden ratios of ternary alphabets

Generalized golden ratios of ternary alphabets V. Komornik Strasbourg Numération University of Liège June 8, 2011 V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, 2011 1 / 19 Abstract

### CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

### Mathematical Induction

Chapter 2 Mathematical Induction 2.1 First Examples Suppose we want to find a simple formula for the sum of the first n odd numbers: 1 + 3 + 5 +... + (2n 1) = n (2k 1). How might we proceed? The most natural

### Nim Games. There are initially n stones on the table.

Nim Games 1 One Pile Nim Games Consider the following two-person game. There are initially n stones on the table. During a move a player can remove either one, two, or three stones. If a player cannot

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.

The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,

### Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

### THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### 2 When is a 2-Digit Number the Sum of the Squares of its Digits?

When Does a Number Equal the Sum of the Squares or Cubes of its Digits? An Exposition and a Call for a More elegant Proof 1 Introduction We will look at theorems of the following form: by William Gasarch

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,

### Section 6-2 Mathematical Induction

6- Mathematical Induction 457 In calculus, it can be shown that e x k0 x k k! x x x3!! 3!... xn n! where the larger n is, the better the approximation. Problems 6 and 6 refer to this series. Note that

### The Alternating Series Test

The Alternating Series Test So far we have considered mostly series all of whose terms are positive. If the signs of the terms alternate, then testing convergence is a much simpler matter. On the other

### Chapter 2 Sequences and Series

Chapter 2 Sequences and Series 2. Sequences A sequence is a function from the positive integers (possibly including 0) to the reals. A typical example is a n = /n defined for all integers n. The notation

### Find-The-Number. 1 Find-The-Number With Comps

Find-The-Number 1 Find-The-Number With Comps Consider the following two-person game, which we call Find-The-Number with Comps. Player A (for answerer) has a number x between 1 and 1000. Player Q (for questioner)

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### SCORE SETS IN ORIENTED GRAPHS

Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in

### MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 4: Fourier Series and L 2 ([ π, π], µ) ( 1 π

MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION Chapter 4: Fourier Series and L ([, π], µ) Square Integrable Functions Definition. Let f : [, π] R be measurable. We say that f

### Math 55: Discrete Mathematics

Math 55: Discrete Mathematics UC Berkeley, Spring 2012 Homework # 9, due Wednesday, April 11 8.1.5 How many ways are there to pay a bill of 17 pesos using a currency with coins of values of 1 peso, 2 pesos,

### Continued fractions and good approximations.

Continued fractions and good approximations We will study how to find good approximations for important real life constants A good approximation must be both accurate and easy to use For instance, our

### Mathematical induction & Recursion

CS 441 Discrete Mathematics for CS Lecture 15 Mathematical induction & Recursion Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Proofs Basic proof methods: Direct, Indirect, Contradiction, By Cases,

### The Pigeonhole Principle

The Pigeonhole Principle 1 Pigeonhole Principle: Simple form Theorem 1.1. If n + 1 objects are put into n boxes, then at least one box contains two or more objects. Proof. Trivial. Example 1.1. Among 13

### THE 2-ADIC, BINARY AND DECIMAL PERIODS OF 1/3 k APPROACH FULL COMPLEXITY FOR INCREASING k

#A28 INTEGERS 12 (2012) THE 2-ADIC BINARY AND DECIMAL PERIODS OF 1/ k APPROACH FULL COMPLEXITY FOR INCREASING k Josefina López Villa Aecia Sud Cinti Chuquisaca Bolivia josefinapedro@hotmailcom Peter Stoll

### MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform

MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish

### CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

### FACTORING LARGE NUMBERS, A GREAT WAY TO SPEND A BIRTHDAY

FACTORING LARGE NUMBERS, A GREAT WAY TO SPEND A BIRTHDAY LINDSEY R. BOSKO I would like to acknowledge the assistance of Dr. Michael Singer. His guidance and feedback were instrumental in completing this

### Lecture 13: Martingales

Lecture 13: Martingales 1. Definition of a Martingale 1.1 Filtrations 1.2 Definition of a martingale and its basic properties 1.3 Sums of independent random variables and related models 1.4 Products of

### Wald s Identity. by Jeffery Hein. Dartmouth College, Math 100

Wald s Identity by Jeffery Hein Dartmouth College, Math 100 1. Introduction Given random variables X 1, X 2, X 3,... with common finite mean and a stopping rule τ which may depend upon the given sequence,

### 10. Graph Matrices Incidence Matrix

10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

### POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS

POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).

### WORST-CASE PERFORMANCE ANALYSIS OF SOME APPROXIMATION ALGORITHMS FOR MINIMIZING MAKESPAN AND FLOWTIME

WORST-CASE PERFORMANCE ANALYSIS OF SOME APPROXIMATION ALGORITHMS FOR MINIMIZING MAKESPAN AND FLOWTIME PERUVEMBA SUNDARAM RAVI, LEVENT TUNÇEL, MICHAEL HUANG Abstract. In 1976, Coffman and Sethi conjectured

### An example of a computable

An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

### MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins

MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins The RSA encryption scheme works as follows. In order to establish the necessary public

### Mathematical Induction

MCS-236: Graph Theory Handout #A5 San Skulrattanakulchai Gustavus Adolphus College Sep 15, 2010 Mathematical Induction The following three principles governing N are equivalent. Ordinary Induction Principle.

### ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

### Homework until Test #2

MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

### CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen

CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

### Lecture 3. Mathematical Induction

Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

### CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### 1. Please write your name in the blank above, and sign & date below. 2. Please use the space provided to write your solution.

Name : Instructor: Marius Ionescu Instructions: 1. Please write your name in the blank above, and sign & date below. 2. Please use the space provided to write your solution. 3. If you need extra pages

### Generating Elementary Combinatorial Objects

Fall 2009 Combinatorial Generation Combinatorial Generation: an old subject Excerpt from: D. Knuth, History of Combinatorial Generation, in pre-fascicle 4B, The Art of Computer Programming Vol 4. Combinatorial

### Math 2602 Finite and Linear Math Fall 14. Homework 9: Core solutions

Math 2602 Finite and Linear Math Fall 14 Homework 9: Core solutions Section 8.2 on page 264 problems 13b, 27a-27b. Section 8.3 on page 275 problems 1b, 8, 10a-10b, 14. Section 8.4 on page 279 problems

### Reading 13 : Finite State Automata and Regular Expressions

CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model

### MATH 289 PROBLEM SET 4: NUMBER THEORY

MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides

### Mathematical Induction. Lecture 10-11

Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### Collatz Sequence. Fibbonacci Sequence. n is even; Recurrence Relation: a n+1 = a n + a n 1.

Fibonacci Roulette In this game you will be constructing a recurrence relation, that is, a sequence of numbers where you find the next number by looking at the previous numbers in the sequence. Your job

### Cartesian Products and Relations

Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special

### CS 4310 HOMEWORK SET 1

CS 4310 HOMEWORK SET 1 PAUL MILLER Section 2.1 Exercises Exercise 2.1-1. Using Figure 2.2 as a model, illustrate the operation of INSERTION-SORT on the array A = 31, 41, 59, 26, 41, 58. Solution: Assume

### Computer Algorithms. Merge Sort. Analysis of Merge Sort. Recurrences. Recurrence Relations. Iteration Method. Lecture 6

Computer Algorithms Lecture 6 Merge Sort Sorting Problem: Sort a sequence of n elements into non-decreasing order. Divide: Divide the n-element sequence to be sorted into two subsequences of n/ elements

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly