Tribute to Martin Gardner: Combinatorial Card Problems
|
|
- Mark Flynn
- 5 years ago
- Views:
Transcription
1 Tribute to Martin Gardner: Combinatorial Card Problems Doug Ensley, SU Math Department October 7, 2010
2 Combinatorial Card Problems The column originally appeared in Scientific American magazine.
3 Combinatorial Card Problems The column originally appeared in Scientific American magazine. Reprinted in the book, Time Travel and Other Mathematical Bewilderments, published in 1988.
4 Combinatorial Card Problems The column originally appeared in Scientific American magazine. Reprinted in the book, Time Travel and Other Mathematical Bewilderments, published in Released in the complete collection, Martin Gardners Mathematical Games, by MAA on DVD.
5 Combinatorial Card Problems The column originally appeared in Scientific American magazine. Reprinted in the book, Time Travel and Other Mathematical Bewilderments, published in Released in the complete collection, Martin Gardners Mathematical Games, by MAA on DVD. Applets to accompany some of the puzzles can be found at
6 Dramatizing an important number theorem... [Place your cards] face down in a row with the ace at the left. The following turning procedure is now applied, starting at the left at each step and proceeding to the right: Turn over every card. Turn over every second card. (Cards 2, 4, 6, 8,10, and Q are turned face down.) Turn over every third card. Continue in this manner, turning every fourth card, every fifth card, and so on until you turn over only the last card.
7 Dramatizing an important number theorem... A good classroom exercise is to prepare 100 small cards bearing numbers 1 through 100, stand them with their backs out in serial order on a blackboard ledge and apply the turning procedure. Sure enough, at the finish the only visible numbers will be the squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. That is too large a sampling to be coincidental. The next step is to prove that no matter how large the deck, only squares survive the turning procedure.
8 Dramatizing an important number theorem... A simple roof introduces one of the oldest and most fundamental of number theorems: A positive integer has an odd number of divisors (the divisors include 1 and the number itself), if and only if the number is a square. This is easy to see. Most divisors of a number come in pairs. Consider 72. The smallest divisor, 1, goes into the number 72 times, giving the pair 1 and 72. The next-larger divisor, 2, goes into the number 36 times, giving the pair 2 and 36. Similarly, 72 = 3 24 = 4 18 = 6 12 = 8 9. The only divisor of a number that is not paired with a different number is a divisor that is a square root. Consequently, all non-squares have an even number of divisors, and all squares have an odd number of divisors.
9 Lehmers Motel Manager Problem Mr. Smith manages a motel. It consists of n rooms in a straight row. There is no vacancy. Smith is a psychologist who plans to study the effects of rearranging his guests in all possible ways. Every morning he gives them a new permutation. The weather is miserable, raining almost daily. To minimize his guests discomfort, each daily rearrangement is made by exchanging the occupants of two adjoining rooms. Is there a simple algorithm that will run through all possible arrangements by switching adjacent occupants at each step?
10 Lehmer s Motel Manager Problem Model the problem for 4 guests using cards A, 2, 3, 4 from your packet. Here is a list of all permutations. Try to do only adjacent swaps while checking off all of them. A234 2A34 3A24 4A23 A243 2A43 3A42 4A32 A324 23A4 32A4 42A3 A A 324A 423A A423 24A3 34A2 43A2 A A 342A 432A
11 Lehmer s Motel Manager Problem [An algorithmic solution] has important applications in computer science. Many problems require a computer in order to run through all permutations of n elements, and if this can be done by exchanging adjacent pairs, there is a significant reduction in computer time.... Hugo Steinhaus, a Polish mathematician, was the first to discover [the algorithm]. It provides a solution for the abacus problem on page 49 of his One Hundred Problems in Elementary Mathematics, first published in Poland in In the early 1960 s the procedure was independently rediscovered at almost the same time by H. F. Trotter and Selmer M. Johnson, each of whom published it separately.
12 Lehmer s Motel Manager Problem Solution. Think about it recursively. First solve the problem for n = 3 cards. A23 A32 3A2 32A 23A 2A3
13 Lehmer s Motel Manager Problem Solution. Think about it recursively. First solve the problem for n = 3 cards, and follow the 3... A23 A32 3A2 32A 23A 2A3
14 Lehmer s Motel Manager Problem Algorithm : Weave the 4 through the solution to the puzzle when n = 3, and you will have a solution to the puzzle when n = 4. A234 A A 243A A243 A A 423A A423 3A24 324A 42A3 4A23 3A42 32A4 24A3 4A32 34A2 23A4 2A43 A432 43A2 234A 2A34
15 John Conway s game of TopSwops Hold packet of 13 cards face up in your hand. Value of top card tells you how many to deal onto the table.
16 John Conway s game of TopSwops Hold packet of 13 cards face up in your hand. Value of top card tells you how many to deal onto the table. After dealing, collect the cards from the table and replace these on top of the packet in your hand.
17 John Conway s game of TopSwops Hold packet of 13 cards face up in your hand. Value of top card tells you how many to deal onto the table. After dealing, collect the cards from the table and replace these on top of the packet in your hand. Applet: topswots/topswots.html
18 John Conway s game of TopSwops
19 John Conway s game of TopSwops The process definitely gets into a loop when an Ace is on top of the packet. Are there longer cycles that repeat, or do we always end up stuck with an Ace on top?
20 John Conway s game of TopSwops Theorem (Wilf) The game of TopSwops always ends with an Ace on top. Proof. For a given arrangement of the cards c 1 c 2... c 13, a card is considered to be in its natural position if card c i has value i. For example, in the following arrangement cards 2 and 7 are in their natural positions: 6, 2, Q, 4, 8, T, 7, 9, 5, J, A, K, 3
21 John Conway s game of TopSwops Theorem (Wilf) The game of TopSwops always ends with an Ace on top. Proof continued... Define the function F as follows: F (c 1, c 2,..., c 13 ) = 2 j c j is natural If the top card has value K, then after one move in the game, that card is in its natural position. Moreover, any cards that were previously in natural positions have values less than K.
22 John Conway s game of TopSwops Theorem (Wilf) The game of TopSwops always ends with an Ace on top. Proof continued... For example, F (6, 2, Q, 4, 8, T, 7, 9, 5, J, A, K, 3) = = 132 and after the TopSwops move, we have F (T, 8, 4, Q, 2, 6, 7, 9, 5, J, A, K, 3) = = 192 Since I gained 2 6 = 64 and risked losing at most , the value of F definitely goes up with each move. The only exception is when A is first. Since F goes up with every other move, the process must terminate with an A on top.
23 John Conway s game of TopSwops Wilf writes, Since the numbers increase steadily but cannot exceed 16382, it follows that the game must halt after at most that many moves. A slightly more careful study shows, in fact, that for a game with n cards, no more than 2 n 1 moves can take place. This raises an interesting unsolved question: What arrangement of the thirteen cards provides the longest possible game of TopSwops?
24 John Conway s other games BotDrops (call the bottom card, count, and put the reversed set on the bottom) is more interesting. If you play it for a while, Conway writes, you might convince yourself that it always loops in a KQKQKQ... sequence, but that is not always the case. On rare occasions other loops are possible. (Can you find one?) Applet: botdrops/botdrops.html
25 John Conway s other games When the game is extended to two or more players, each with a packet, it becomes much harder to analyze. For instance, suppose two players have packets of thirteen cards each. One has spades, the other hearts. They play two player TopSwops as follows. Each shuffles his packet. Player A calls his top card, then B counts that number off his packet and replaces the reversed cards on top of his packet. B now calls his top card, A counts and replaces the reversed cards on top of his packet. This continues with players alternating calls. Applet: topswots/topswotsfor2.html
26 John Conway s other games It is a curious fact, reports Conway, that as soon as an ace is called, the calls go into a loop that starts with an ace, then a sequence, then an ace again (either the same ace or the other one), then the same sequence is repeated in reverse. For example, the first called ace might generate the following loop: Note that the sequence between the first two ace calls is the reverse of the sequence between the second and third ace calls. It is an unproved conjecture (or was when I last heard from Conway) that in two-player TopSwops an ace is always called. It is not known if the game can conclude in a loop without an ace, although it is known that if a loop includes an ace, it includes it just twice.
27 Langford Puzzle Remove from a deck all the cards of three suits that bear values of ace through 9. Try to arrange these twenty-seven cards in a single row to meet the following proviso. Between the first two cards of every value k there are exactly k cards, and between the second and third cards of every value k there also are exactly k cards. For instance, between the first and second 7 s there must be just seven cards, not counting the two 7 s. Similarly, seven cards separate the second and third 7 s. The rule applies to each value from 1 through 9.
28 Silverman Card Puzzle Use two complete suits, say spades and diamonds, and match cards in pairs so that the sum of each pair is a perfect square. Ace = 1, Jack = 11, Queen = 12, King = 13 Applet: combinatorialcards/card_comb.html
29 Silverman Card Puzzle
30 Silverman Card Puzzle General question: For what values of n can the numbers in {1, 2, 3,..., n} be paired with the numbers in {1, 2, 3,..., n} so that the pairs each sum to perfect a square?
31 Silverman Card Puzzle General question: For what values of n can the numbers in {1, 2, 3,..., n} be paired with the numbers in {1, 2, 3,..., n} so that the pairs each sum to perfect a square? Example solution when n = 5: (1,3), (2,2), (3,1), (4,5), (5,4)
32 Silverman Card Puzzle General question: For what values of n can the numbers in {1, 2, 3,..., n} be paired with the numbers in {1, 2, 3,..., n} so that the pairs each sum to perfect a square? Example solution when n = 5: (1,3), (2,2), (3,1), (4,5), (5,4) Card puzzle addresses n = 13.
33 Silverman Card Puzzle Alan Hadsell and Stoddard Vandersteel together used a computer to generalize Silverman s problem. When you use a packet of cards from a single suit, solutions exist only for n = 3, 5, 8, 9, 10, 12, and 13, and each solution is unique.
34 Silverman Card Puzzle Alan Hadsell and Stoddard Vandersteel together used a computer to generalize Silverman s problem. When you use a packet of cards from a single suit, solutions exist only for n = 3, 5, 8, 9, 10, 12, and 13, and each solution is unique. From 14 through 31 all values of n have multiple solutions.
35 Silverman Card Puzzle Alan Hadsell and Stoddard Vandersteel together used a computer to generalize Silverman s problem. When you use a packet of cards from a single suit, solutions exist only for n = 3, 5, 8, 9, 10, 12, and 13, and each solution is unique. From 14 through 31 all values of n have multiple solutions. They report that the number of solutions, beginning with n = 14, are 2, 4, 3, 2, 5, 15, 21, 66, 37, 51, 144, 263, 601, 333, 2119, 2154, 2189, 3280,...
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
More informationDealing with some maths
Dealing with some maths Hayden Tronnolone School of Mathematical Sciences University of Adelaide August 20th, 2012 To call a spade a spade First, some dealing... Hayden Tronnolone (University of Adelaide)
More informationTable 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
More informationMath 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
More informationMath 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
More informationCounting integral solutions
Thought exercise 2.2 20 Counting integral solutions Question: How many non-negative integer solutions are there of x 1 +x 2 +x 3 +x 4 = 10? Thought exercise 2.2 20 Counting integral solutions Question:
More informationThink 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
More informationGough, John , Doing it with dominoes, Australian primary mathematics classroom, vol. 7, no. 3, pp
Deakin Research Online Deakin University s institutional research repository DDeakin Research Online Research Online This is the published version (version of record) of: Gough, John 2002-08, Doing it
More informationLecture 18 - Counting
Lecture 18 - Counting 6.0 - April, 003 One of the most common mathematical problems in computer science is counting the number of elements in a set. This is often the core difficulty in determining a program
More informationNOTES ON SEPT 13-18, 2012
NOTES ON SEPT 13-18, 01 MIKE ZABROCKI Last time I gave a name to S(n, k := number of set partitions of [n] into k parts. This only makes sense for n 1 and 1 k n. For other values we need to choose a convention
More informationThree 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.
More informationTeaching the TERNARY BASE
Features Teaching the TERNARY BASE Using a Card Trick SUHAS SAHA Any sufficiently advanced technology is indistinguishable from magic. Arthur C. Clarke, Profiles of the Future: An Inquiry Into the Limits
More informationAn Optimal Algorithm for a Strategy Game
International Conference on Materials Engineering and Information Technology Applications (MEITA 2015) An Optimal Algorithm for a Strategy Game Daxin Zhu 1, a and Xiaodong Wang 2,b* 1 Quanzhou Normal University,
More informationSMT 2014 Advanced Topics Test Solutions February 15, 2014
1. David flips a fair coin five times. Compute the probability that the fourth coin flip is the first coin flip that lands heads. 1 Answer: 16 ( ) 1 4 Solution: David must flip three tails, then heads.
More informationCounting integral solutions
Thought exercise 2.2 25 Counting integral solutions Question: How many non-negative integer solutions are there of x 1 + x 2 + x 3 + x 4 =10? Give some examples of solutions. Characterize what solutions
More informationChapter 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
More information6) A) both; happy B) neither; not happy C) one; happy D) one; not happy
MATH 00 -- PRACTICE TEST 2 Millersville University, Spring 202 Ron Umble, Instr. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find all natural
More informationCALCULATING 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
More informationECS 20 (Spring 2013) Phillip Rogaway Lecture 1
ECS 20 (Spring 2013) Phillip Rogaway Lecture 1 Today: Introductory comments Some example problems Announcements course information sheet online (from my personal homepage: Rogaway ) first HW due Wednesday
More informationFreeCell Puzzle Protocol Document
AI Puzzle Framework FreeCell Puzzle Protocol Document Brian Shaver April 11, 2005 FreeCell Puzzle Protocol Document Page 2 of 7 Table of Contents Table of Contents...2 Introduction...3 Puzzle Description...
More information1 of 5 7/16/2009 6:57 AM Virtual Laboratories > 13. Games of Chance > 1 2 3 4 5 6 7 8 9 10 11 3. Simple Dice Games In this section, we will analyze several simple games played with dice--poker dice, chuck-a-luck,
More informationA complete set of dominoes containing the numbers 0, 1, 2, 3, 4, 5 and 6, part of which is shown, has a total of 28 dominoes.
Station 1 A domino has two parts, each containing one number. A complete set of dominoes containing the numbers 0, 1, 2, 3, 4, 5 and 6, part of which is shown, has a total of 28 dominoes. Part A How many
More informationChapter 1. Probability
Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.
More informationEXPLAINING THE SHAPE OF RSK
EXPLAINING THE SHAPE OF RSK SIMON RUBINSTEIN-SALZEDO 1. Introduction There is an algorithm, due to Robinson, Schensted, and Knuth (henceforth RSK), that gives a bijection between permutations σ S n and
More informationStaircase Rook Polynomials and Cayley s Game of Mousetrap
Staircase Rook Polynomials and Cayley s Game of Mousetrap Michael Z. Spivey Department of Mathematics and Computer Science University of Puget Sound Tacoma, Washington 98416-1043 USA mspivey@ups.edu Phone:
More informationPOKER (AN INTRODUCTION TO COUNTING)
POKER (AN INTRODUCTION TO COUNTING) LAMC INTERMEDIATE GROUP - 10/27/13 If you want to be a succesful poker player the first thing you need to do is learn combinatorics! Today we are going to count poker
More informationFree Cell Solver. Copyright 2001 Kevin Atkinson Shari Holstege December 11, 2001
Free Cell Solver Copyright 2001 Kevin Atkinson Shari Holstege December 11, 2001 Abstract We created an agent that plays the Free Cell version of Solitaire by searching through the space of possible sequences
More informationYou ve seen them played in coffee shops, on planes, and
Every Sudoku variation you can think of comes with its own set of interesting open questions There is math to be had here. So get working! Taking Sudoku Seriously Laura Taalman James Madison University
More informationLatin 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
More informationAnalyzing Games: Solutions
Writing Proofs Misha Lavrov Analyzing Games: olutions Western PA ARML Practice March 13, 2016 Here are some key ideas that show up in these problems. You may gain some understanding of them by reading
More informationPoker Hands. Christopher Hayes
Poker Hands Christopher Hayes Poker Hands The normal playing card deck of 52 cards is called the French deck. The French deck actually came from Egypt in the 1300 s and was already present in the Middle
More informationCOUNTING AND PROBABILITY
CHAPTER 9 COUNTING AND PROBABILITY It s as easy as 1 2 3. That s the saying. And in certain ways, counting is easy. But other aspects of counting aren t so simple. Have you ever agreed to meet a friend
More informationElementary Combinatorics
184 DISCRETE MATHEMATICAL STRUCTURES 7 Elementary Combinatorics 7.1 INTRODUCTION Combinatorics deals with counting and enumeration of specified objects, patterns or designs. Techniques of counting are
More informationPRIMES 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
More information5.8 Problems (last update 30 May 2018)
5.8 Problems (last update 30 May 2018) 1.The lineup or batting order for a baseball team is a list of the nine players on the team indicating the order in which they will bat during the game. a) How many
More informationLaunchpad 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
More informationMaths Is Fun! Activity Pack Year 4
Maths Is Fun! Activity Pack Year 4 1. Spot the Difference Draw a horizontal line on a piece of paper. Write a 3 digit number at the left hand end and a higher one at the right hand end. Ask your child
More informationEuropean Journal of Combinatorics. Staircase rook polynomials and Cayley s game of Mousetrap
European Journal of Combinatorics 30 (2009) 532 539 Contents lists available at ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc Staircase rook polynomials
More informationProbability. The MEnTe Program Math Enrichment through Technology. Title V East Los Angeles College
Probability The MEnTe Program Math Enrichment through Technology Title V East Los Angeles College 2003 East Los Angeles College. All rights reserved. Topics Introduction Empirical Probability Theoretical
More informationTHE NUMBER WAR GAMES
THE NUMBER WAR GAMES Teaching Mathematics Facts Using Games and Cards Mahesh C. Sharma President Center for Teaching/Learning Mathematics 47A River St. Wellesley, MA 02141 info@mathematicsforall.org @2008
More informationGame, Set, and Match Carl W. Lee September 2016
Game, Set, and Match Carl W. Lee September 2016 Note: Some of the text below comes from Martin Gardner s articles in Scientific American and some from Mathematical Circles by Fomin, Genkin, and Itenberg.
More informationThree Pile Nim with Move Blocking. Arthur Holshouser. Harold Reiter.
Three Pile Nim with Move Blocking Arthur Holshouser 3600 Bullard St Charlotte, NC, USA Harold Reiter Department of Mathematics, University of North Carolina Charlotte, Charlotte, NC 28223, USA hbreiter@emailunccedu
More informationChapter 5: Probability: What are the Chances? Section 5.2 Probability Rules
+ Chapter 5: Probability: What are the Chances? Section 5.2 + Two-Way Tables and Probability When finding probabilities involving two events, a two-way table can display the sample space in a way that
More informationPoker Rules Friday Night Poker Club
Poker Rules Friday Night Poker Club Last edited: 2 April 2004 General Rules... 2 Basic Terms... 2 Basic Game Mechanics... 2 Order of Hands... 3 The Three Basic Games... 4 Five Card Draw... 4 Seven Card
More informationSTRATEGY AND COMPLEXITY OF THE GAME OF SQUARES
STRATEGY AND COMPLEXITY OF THE GAME OF SQUARES FLORIAN BREUER and JOHN MICHAEL ROBSON Abstract We introduce a game called Squares where the single player is presented with a pattern of black and white
More informationTwenty-sixth Annual UNC Math Contest First Round Fall, 2017
Twenty-sixth Annual UNC Math Contest First Round Fall, 07 Rules: 90 minutes; no electronic devices. The positive integers are,,,,.... Find the largest integer n that satisfies both 6 < 5n and n < 99..
More informationCounting in Algorithms
Counting Counting in Algorithms How many comparisons are needed to sort n numbers? How many steps to compute the GCD of two numbers? How many steps to factor an integer? Counting in Games How many different
More informationThe mathematics of the flip and horseshoe shuffles
The mathematics of the flip and horseshoe shuffles Steve Butler Persi Diaconis Ron Graham Abstract We consider new types of perfect shuffles wherein a deck is split in half, one half of the deck is reversed,
More informationCSE 1400 Applied Discrete Mathematics Permutations
CSE 1400 Applied Discrete Mathematics Department of Computer Sciences College of Engineering Florida Tech Fall 2011 1 Cyclic Notation 2 Re-Order a Sequence 2 Stirling Numbers of the First Kind 2 Problems
More informationTaking Sudoku Seriously
Taking Sudoku Seriously Laura Taalman, James Madison University You ve seen them played in coffee shops, on planes, and maybe even in the back of the room during class. These days it seems that everyone
More informationPuzzles to Play With
Puzzles to Play With Attached are some puzzles to occupy your mind. They are not arranged in order of difficulty. Some at the back are easier than some at the front. If you think you have a solution but
More informationSorting Squares. (Martin Gardner)
Sorting Squares (Martin Gardner) A student is given the large square below. They are asked to the paper forwards or backwards along any horizontal or vertical line. They are then asked to keep doing this
More information{ a, b }, { a, c }, { b, c }
12 d.) 0(5.5) c.) 0(5,0) h.) 0(7,1) a.) 0(6,3) 3.) Simplify the following combinations. PROBLEMS: C(n,k)= the number of combinations of n distinct objects taken k at a time is COMBINATION RULE It can easily
More informationUK 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
More informationActivity 6: Playing Elevens
Activity 6: Playing Elevens Introduction: In this activity, the game Elevens will be explained, and you will play an interactive version of the game. Exploration: The solitaire game of Elevens uses a deck
More informationGrade 7/8 Math Circles Game Theory October 27/28, 2015
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Game Theory October 27/28, 2015 Chomp Chomp is a simple 2-player game. There is
More informationAcing Math (One Deck At A Time!): A Collection of Math Games. Table of Contents
Table of Contents Introduction to Acing Math page 5 Card Sort (Grades K - 3) page 8 Greater or Less Than (Grades K - 3) page 9 Number Battle (Grades K - 3) page 10 Place Value Number Battle (Grades 1-6)
More informationHOMEWORK ASSIGNMENT 5
HOMEWORK ASSIGNMENT 5 MATH 251, WILLIAMS COLLEGE, FALL 2006 Abstract. These are the instructor s solutions. 1. Big Brother The social security number of a person is a sequence of nine digits that are not
More informationCSE 312: Foundations of Computing II Quiz Section #2: Inclusion-Exclusion, Pigeonhole, Introduction to Probability
CSE 312: Foundations of Computing II Quiz Section #2: Inclusion-Exclusion, Pigeonhole, Introduction to Probability Review: Main Theorems and Concepts Binomial Theorem: Principle of Inclusion-Exclusion
More informationGEOGRAPHY PLAYED ON AN N-CYCLE TIMES A 4-CYCLE
GEOGRAPHY PLAYED ON AN N-CYCLE TIMES A 4-CYCLE M. S. Hogan 1 Department of Mathematics and Computer Science, University of Prince Edward Island, Charlottetown, PE C1A 4P3, Canada D. G. Horrocks 2 Department
More informationMath 166: Topics in Contemporary Mathematics II
Math 166: Topics in Contemporary Mathematics II Xin Ma Texas A&M University September 30, 2017 Xin Ma (TAMU) Math 166 September 30, 2017 1 / 11 Last Time Factorials For any natural number n, we define
More informationPuzzling Math, Part 2: The Tower of Hanoi & the End of the World!
Puzzling Math, Part 2: The Tower of Hanoi & the End of the World! by Jeremy Knight, Grants Pass High School, jeremy@knightmath.com The Oregon Mathematics Teacher, Jan./Feb. 2014 Grade Level: 6-12+ Objectives:
More information2-1 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
More informationPrinciple of Inclusion-Exclusion Notes
Principle of Inclusion-Exclusion Notes The Principle of Inclusion-Exclusion (often abbreviated PIE is the following general formula used for finding the cardinality of a union of finite sets. Theorem 0.1.
More informationGale s Vingt-et-en. Ng P.T. 1 and Tay T.S. 2. Department of Mathematics, National University of Singapore 2, Science Drive 2, Singapore (117543)
ABSTRACT Gale s Vingt-et-en Ng P.T. 1 and Tay T.S. 2 Department of Mathematics, National University of Singapore 2, Science Drive 2, Singapore (117543) David Gale is a professor emeritus of mathematics
More informationPROBLEM SET 2 Due: Friday, September 28. Reading: CLRS Chapter 5 & Appendix C; CLR Sections 6.1, 6.2, 6.3, & 6.6;
CS231 Algorithms Handout #8 Prof Lyn Turbak September 21, 2001 Wellesley College PROBLEM SET 2 Due: Friday, September 28 Reading: CLRS Chapter 5 & Appendix C; CLR Sections 6.1, 6.2, 6.3, & 6.6; Suggested
More informationThe mathematics of the flip and horseshoe shuffles
The mathematics of the flip and horseshoe shuffles Steve Butler Persi Diaconis Ron Graham Abstract We consider new types of perfect shuffles wherein a deck is split in half, one half of the deck is reversed,
More informationImpartial Combinatorial Games Berkeley Math Circle Intermediate II Ted Alper Evans Hall, room 740 Sept 1, 2015
Impartial Combinatorial Games Berkeley Math Circle Intermediate II Ted Alper Evans Hall, room 740 Sept 1, 2015 tmalper@stanford.edu 1 Warmups 1.1 (Kozepiskolai Matematikai Lapok, 1980) Contestants B and
More information1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?
1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building? 2. A particular brand of shirt comes in 12 colors, has a male version and a female version,
More informationBlock 1 - Sets and Basic Combinatorics. Main Topics in Block 1:
Block 1 - Sets and Basic Combinatorics Main Topics in Block 1: A short revision of some set theory Sets and subsets. Venn diagrams to represent sets. Describing sets using rules of inclusion. Set operations.
More informationTHE 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
More informationCS 3233 Discrete Mathematical Structure Midterm 2 Exam Solution Tuesday, April 17, :30 1:45 pm. Last Name: First Name: Student ID:
CS Discrete Mathematical Structure Midterm Exam Solution Tuesday, April 17, 007 1:0 1:4 pm Last Name: First Name: Student ID: Problem No. Points Score 1 10 10 10 4 1 10 6 10 7 1 Total 80 1 This is a closed
More informationMATH 351 Fall 2009 Homework 1 Due: Wednesday, September 30
MATH 51 Fall 2009 Homework 1 Due: Wednesday, September 0 Problem 1. How many different letter arrangements can be made from the letters BOOKKEEPER. This is analogous to one of the problems presented in
More informationLESSON 2. Objectives. General Concepts. General Introduction. Group Activities. Sample Deals
LESSON 2 Objectives General Concepts General Introduction Group Activities Sample Deals 38 Bidding in the 21st Century GENERAL CONCEPTS Bidding The purpose of opener s bid Opener is the describer and tries
More informationMultiple Choice Questions for Review
Review Questions Multiple Choice Questions for Review 1. Suppose there are 12 students, among whom are three students, M, B, C (a Math Major, a Biology Major, a Computer Science Major. We want to send
More informationCS 202, section 2 Final Exam 13 December Pledge: Signature:
CS 22, section 2 Final Exam 3 December 24 Name: KEY E-mail ID: @virginia.edu Pledge: Signature: There are 8 minutes (3 hours) for this exam and 8 points on the test; don t spend too long on any one question!
More informationHonors Precalculus Chapter 9 Summary Basic Combinatorics
Honors Precalculus Chapter 9 Summary Basic Combinatorics A. Factorial: n! means 0! = Why? B. Counting principle: 1. How many different ways can a license plate be formed a) if 7 letters are used and each
More informationOn Variants of Nim and Chomp
The Minnesota Journal of Undergraduate Mathematics On Variants of Nim and Chomp June Ahn 1, Benjamin Chen 2, Richard Chen 3, Ezra Erives 4, Jeremy Fleming 3, Michael Gerovitch 5, Tejas Gopalakrishna 6,
More informationFrustration solitaire
arxiv:math/0703900v2 [math.pr] 2 Apr 2009 Frustration solitaire Peter G. Doyle Charles M. Grinstead J. Laurie Snell Version dated 2 April 2009 GNU FDL Abstract In this expository article, we discuss the
More informationChapter 2 Integers. Math 20 Activity Packet Page 1
Chapter 2 Integers Contents Chapter 2 Integers... 1 Introduction to Integers... 3 Adding Integers with Context... 5 Adding Integers Practice Game... 7 Subtracting Integers with Context... 9 Mixed Addition
More informationGrades 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
More informationSolutions 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
More informationName: Checked: jack queen king ace
Lab 11 Name: Checked: Objectives: More practice using arrays: 1. Arrays of Strings and shuffling an array 2. Arrays as parameters 3. Collections Preparation 1) An array to store a deck of cards: DeckOfCards.java
More informationKenken For Teachers. Tom Davis January 8, Abstract
Kenken For Teachers Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles January 8, 00 Abstract Kenken is a puzzle whose solution requires a combination of logic and simple arithmetic
More informationDiscrete Finite Probability Probability 1
Discrete Finite Probability Probability 1 In these notes, I will consider only the finite discrete case. That is, in every situation the possible outcomes are all distinct cases, which can be modeled by
More informationFinal Exam, Math 6105
Final Exam, Math 6105 SWIM, June 29, 2006 Your name Throughout this test you must show your work. 1. Base 5 arithmetic (a) Construct the addition and multiplication table for the base five digits. (b)
More informationMath 3338: Probability (Fall 2006)
Math 3338: Probability (Fall 2006) Jiwen He Section Number: 10853 http://math.uh.edu/ jiwenhe/math3338fall06.html Probability p.1/7 2.3 Counting Techniques (III) - Partitions Probability p.2/7 Partitioned
More informationHow to Become a Mathemagician: Mental Calculations and Math Magic
How to Become a Mathemagician: Mental Calculations and Math Magic Adam Gleitman (amgleit@mit.edu) Splash 2012 A mathematician is a conjurer who gives away his secrets. John H. Conway This document describes
More informationThe Pigeonhole Principle
The Pigeonhole Principle Some Questions Does there have to be two trees on Earth with the same number of leaves? How large of a set of distinct integers between 1 and 200 is needed to assure that two numbers
More informationNew Sliding Puzzle with Neighbors Swap Motion
Prihardono AriyantoA,B Kenichi KawagoeC Graduate School of Natural Science and Technology, Kanazawa UniversityA Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Email: prihardono.ari@s.itb.ac.id
More information2009 Philippine Elementary Mathematics International Contest Page 1
2009 Philippine Elementary Mathematics International Contest Page 1 Individual Contest 1. Find the smallest positive integer whose product after multiplication by 543 ends in 2009. It is obvious that the
More informationCSE 312: Foundations of Computing II Quiz Section #2: Inclusion-Exclusion, Pigeonhole, Introduction to Probability (solutions)
CSE 31: Foundations of Computing II Quiz Section #: Inclusion-Exclusion, Pigeonhole, Introduction to Probability (solutions) Review: Main Theorems and Concepts Binomial Theorem: x, y R, n N: (x + y) n
More informationLecture 2.3: Symmetric and alternating groups
Lecture 2.3: Symmetric and alternating groups Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson)
More informationProblem 2A Consider 101 natural numbers not exceeding 200. Prove that at least one of them is divisible by another one.
1. Problems from 2007 contest Problem 1A Do there exist 10 natural numbers such that none one of them is divisible by another one, and the square of any one of them is divisible by any other of the original
More informationProblem Set 8 Solutions R Y G R R G
6.04/18.06J Mathematics for Computer Science April 5, 005 Srini Devadas and Eric Lehman Problem Set 8 Solutions Due: Monday, April 11 at 9 PM in Room 3-044 Problem 1. An electronic toy displays a 4 4 grid
More informationThe mathematics of Septoku
The mathematics of Septoku arxiv:080.397v4 [math.co] Dec 203 George I. Bell gibell@comcast.net, http://home.comcast.net/~gibell/ Mathematics Subject Classifications: 00A08, 97A20 Abstract Septoku is a
More informationLive Casino game rules. 1. Live Baccarat. 2. Live Blackjack. 3. Casino Hold'em. 4. Generic Rulette. 5. Three card Poker
Live Casino game rules 1. Live Baccarat 2. Live Blackjack 3. Casino Hold'em 4. Generic Rulette 5. Three card Poker 1. LIVE BACCARAT 1.1. GAME OBJECTIVE The objective in LIVE BACCARAT is to predict whose
More informationMATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG
MATH DISCRETE MATHEMATICS INSTRUCTOR: P. WENG Counting and Probability Suggested Problems Basic Counting Skills, Inclusion-Exclusion, and Complement. (a An office building contains 7 floors and has 7 offices
More informationJong C. Park Computer Science Division, KAIST
Jong C. Park Computer Science Division, KAIST Today s Topics Basic Principles Permutations and Combinations Algorithms for Generating Permutations Generalized Permutations and Combinations Binomial Coefficients
More informationFundamentals of Probability
Fundamentals of Probability Introduction Probability is the likelihood that an event will occur under a set of given conditions. The probability of an event occurring has a value between 0 and 1. An impossible
More information