Easy, Hard, Impossible! A Lecture in CE Freshman Seminar Series:

Slides:

Advertisements

Similar presentations

22C:19 Discrete Math Sequence and Sums Fall 2011 Sukumar Ghosh.

Advertisements

Recursion Lecture 18: Nov 18.

Lecture 1Easy, Hard, Impossible!Handout Easy: Euclid Sequences Form a sequence of number pairs (integers) as follows: Begin with any two positive numbers.

Mar. 2015Easy, Hard, Impossible!Slide 1 Easy, Hard, Impossible! A Lecture in CE Freshman Seminar Series: Puzzling Problems in Computer Engineering.

HandoutMay 2007Binary Search A Lecture in CE Freshman Seminar Series: Ten Puzzling Problems in Computer Engineering.

Apr. 2007Easy, Hard, Impossible!Slide 1 Easy, Hard, Impossible! A Lecture in CE Freshman Seminar Series: Ten Puzzling Problems in Computer Engineering.

Revision Using Recursion1 Recursion. Revision Using Recursion2 Recall the Recursion Pattern Recursion: when a method calls itself Classic example--the.

HandoutMay 2007Sorting Networks A Lecture in CE Freshman Seminar Series: Ten Puzzling Problems in Computer Engineering.

Discrete Mathematics Recursion and Sequences

CSC 2300 Data Structures & Algorithms January 30, 2007 Chapter 2. Algorithm Analysis.

HandoutApr. 2007Satisfiability A Lecture in CE Freshman Seminar Series: Ten Puzzling Problems in Computer Engineering.

Arithmetic and Geometric Series (11.5) Short cuts.

MATH 224 – Discrete Mathematics

2, 4, 6, 8, … a1, a2, a3, a4, … Arithmetic Sequences

Sequences Suppose that $5,000 is borrowed at 6%, compounded annually. The value of the loan at the start of the years 1, 2, 3, 4, and so on is $5000,

Factorial Notation For any positive integer n, n! means: n (n – 1) (n – 2)... (3) (2) (1) 0! will be defined as equal to one. Examples: 4! = =

1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 11 Further Topics in Algebra.

13.2 Recursive Definitions Objective 1) Provide the recursive definition for sequences; 2) Identify the type of a sequence from a recursive definition.

1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 10 Further Topics in Algebra.

Lecture#16 Discrete Mathematics. Recursion Now, 1 is an odd positive integer by the definition base. With k = 1, = 3, so 3 is an odd positive integer.

Fall 2015 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1.

HandoutMay 2007Task Scheduling A Lecture in CE Freshman Seminar Series: Ten Puzzling Problems in Computer Engineering.

Dr Zhang Fall 2014 Fordham University

Program Design. The design process How do you go about writing a program? –It’s like many other things in life Understand the problem to be solved Develop.

Pythagorean Theorem – Prove It! Converse of Pythagorean Theorem 1.) Is a triangle with side measures of 30, 40, and 50 units a right triangle? Prove it!

Infinite Geometric Series Recursion & Special Sequences Definitions & Equations Writing & Solving Geometric Series Practice Problems.

Arithmetic Sequences Sequence is a list of numbers typically with a pattern. 2, 4, 6, 8, … The first term in a sequence is denoted as a 1, the second term.

Discrete Mathematics Lecture # 22 Recursion.  First of all instead of giving the definition of Recursion we give you an example, you already know the.

Algebra II Honors Problem of the Day Homework: p odds Find the first 6 terms of the sequence defined as: Fibonacci!

Mathematical Induction Thinking Skill: Develop Confidence in Reason Warm Up: Find the k+1 term (P k+1 ) 1) 2)

13.3 Arithmetic and Geometric Series and Their Sums Finite Series.

Recursion1 © 2013 Goodrich, Tamassia, Goldwasser.

Section 11.2 Arithmetic Sequences and Series Copyright ©2013, 2009, 2006, 2005 Pearson Education, Inc.

22C:19 Discrete Structures Sequence and Sums Fall 2014 Sukumar Ghosh.

Warm Up DNE Sums of Infinite Series.

SEQUENCES AND SERIES 11.1 NOTES. WARM-UP Describe the pattern below & list the next 3 numbers in the pattern: 1, 1, 2, 3, 5, 8, 13, 21…

CMSC201 Computer Science I for Majors Lecture 19 – Recursion (Continued) Prof. Katherine Gibson Prof. Jeremy Dixon Based on slides from UPenn’s.

Fibonacci’s rabbits Fibonacci posed the following problem:

Algebra II Honors Problem of the Day

CMSC201 Computer Science I for Majors Lecture 20 – Recursion (Continued) Prof. Katherine Gibson Based on slides from UPenn’s CIS 110, and from previous.

Recursion 5/22/2018 Presentation for use with the textbook Data Structures and Algorithms in Java, 6th edition, by M. T. Goodrich, R. Tamassia, and M.

Sequences and Series 9.1.

Mathematical Induction II

Sequences and Series.

Ch. 8 – Sequences, Series, and Probability

Lecture Outline for Recurrences

Copyright © Cengage Learning. All rights reserved.

Recursion © 2013 Goodrich, Tamassia, Goldwasser Recursion Sequences

Finite Geometric Series

75 Has 7 in the tens place Odd Number Less than 100

Pre Calculus 11 Section 1.2 Arithmetic Series

Copyright © Cengage Learning. All rights reserved.

CS 250, Discrete Structures, Fall 2013

Section 2.1 Arithmetic Sequences and Series

Placement and Routing A Lecture in CE Freshman Seminar Series:

Easy, Hard, Impossible! A Lecture in CE Freshman Seminar Series:

Discrete Mathematics and its Applications

Induction Rosen 5 Lecture 8: Oct 29, 30.

Chapter 3 :Recursion © 2011 Pearson Addison-Wesley. All rights reserved.

12 Has 1 in the tens place Even Number Less than 20

Addition and Subtraction within 1,000

LT: I can evaluate Arithmetic and Geometric Series.

Year 2 Summer Term Week 11 Lesson 3

Year 2 Summer Term Week 11 Lesson 3

Software Development Techniques

Repeating Decimals

Presentation transcript:

Easy, Hard, Impossible! A Lecture in CE Freshman Seminar Series: Ten Puzzling Problems in Computer Engineering Apr. 2007 Easy, Hard, Impossible!

Easy: Euclid Sequences Form a sequence of number pairs (integers) as follows: Begin with any two positive numbers as the first pair In each step, the next number pair consists of (1) the smaller of the current pair of values, and (2) their difference Stop when the two numbers in the pair become equal (10, 15) (10, 5) (5, 5) (22, 6) (6, 16) (6, 10) (6, 4) (4, 2) (2, 2) (9, 23) (9, 14) (9, 5) (5, 4) (4, 1) (1, 3) (1, 2) (1, 1) Why is the process outlined above guaranteed to end? Challenge: Repeat this process for a few more starting number pairs and see if you can discover something about how the final number pair is related to the starting values Apr. 2007 Easy, Hard, Impossible!

Not So Easy: Fibonacci Sequences Form a sequence of numbers (integers) as follows: Begin with any two numbers as the first two elements In each step, the next number is the sum of the last two numbers already in the sequence Stop when you have generated the j th number (j is given) 5 16 21 37 5 16 j = 4 2 0 2 0 2 2 4 6 10 16 26 j = 9 1 1 1 1 2 3 5 8 13 21 34 55 89 144 j = 12 Challenge: See if you can find a formula that yields the j th number directly (i.e., without following the sequence) when we begin with 1 1 Apr. 2007 Easy, Hard, Impossible!

Very Hard: Collatz Sequences Form a sequence of numbers (integers) as follows: Begin with a given number To find the next number in each step, halve the current number if it is even or triple it and add 1 if it is odd The pattern 4 2 1 repeats (5 steps to reach the end) 5 16 8 4 2 1 22 11 34 17 52 26 13 40 20 10 5 . . . 1 (15 steps) (19 steps) 9 28 14 7 22 11 . . . 1 Challenge: Repeat this process for 27 and some other starting values. See if you can discover something about how various sequences end; i.e., do all sequences end in the same way, fall into several categories, or do not show any overall pattern at all? Reference: http://en.wikipedia.org/wiki/Collatz_conjecture Apr. 2007 Easy, Hard, Impossible!