Classic (and not so classic) Puzzles

Transcription

1 Classic (and not so classic) Puzzles OLLI Fall, 2018 Mary Jane Sterling 1

2 2

3 Mathematical puzzles vary from the simple to deep problems which are still unsolved. The whole history of mathematics is interwoven with mathematical games which have led to the study of many areas of mathematics. Number games, geometrical puzzles, network problems and combinatorial problems are among the best known types of puzzles. 3

4 The Rhind papyrus shows that early Egyptian mathematics was largely based on puzzle-type problems. Written in around 1850 BC, it contains a rather familiar type of puzzle. Seven houses contain seven cats. Each cat kills seven mice. Each mouse had eaten seven ears of grain. Each ear of grain would have produced seven hekats of wheat. What is the total of all of these? 4

5 5

6 Greek mathematics produced many classic puzzles. Perhaps the most famous are from Archimedes in his book The Sandreckoner where he gives the Cattle Problem. If thou art diligent and wise, O Stranger, compute the number of cattle of the Sun... 6

7 Tangrams are of Chinese origin and require little mathematical skill. It is interesting however to see how many convex figures you can make from the 7 tangram pieces. 7

8 Fibonacci, is famed for his invention of the sequence: 1, 1, 2, 3, 5, 8, 13,... where each number is the sum of the previous two. A problem related to the sequence is the Rabbit Problem. A man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begins a new pair which from the second month on becomes productive? 8

9 9

10 10

11 One of the earliest mentions of chess in puzzles is by the Arabic mathematician Ibn Kallikan who, in 1256, poses the problem of the grains of wheat, 1 on the first square of the chess board, 2 on the second, 4 on the third, 8 on the fourth, etc. 11

12 Magic squares involve using all the numbers 1, 2, 3,..., n 2 to fill the squares of an n n board so that each row, each column and both main diagonals sum to the same number. They are claimed to go back as far as 2200 BC when the Chinese called them lo-shu. In the early 16 th Century Cornelius Agrippa constructed squares for n = 3, 4, 5, 6, 7, 8, 9 which he associated with the seven planets then known (including the Sun and the Moon). 12

13 Dürer s Melancholia 13

14 Bachet is famed as a collector of mathematical puzzles which he published in It contains many problems including: river crossing problems, weighing problems, number tricks, magic squares etc. Here is an example of one of his weighing problems: What is the least number of weights that can be used on a scale pan to weigh any integral number of pounds from 1 to 40 inclusive, if the weights can be placed in either of the scale pans? 14

15 To balance against: 1, 2, 3, 4, 5, 6, 7, 8,, 39, 40 Use weights: 1, 2, 4, 8, 16, 32 15

16 Euler is perhaps the mathematician whose puzzles have led to the most deep mathematical disciplines. In addition to magic square problems and number problems he considered the Knight's Tour of the chess board, the Thirty Six Officers problem and the Seven Bridges of Königsberg. The Seven Bridges of Königsberg heralds the beginning of graph theory and topology. 16

17 The Thirty Six Officers Problem, posed in 1779, asks if it is possible to arrange 6 regiments consisting of 6 officers each of different ranks in a 6 6 square so that no rank or regiment will be repeated in any row or column. The problem is insoluble but it has led to important work in combinatorics. 17

18 Seven Bridges of Königsberg 18

19 We re in good company! 19

20 Pop Ups 20

21 A ship is docked in the harbor. Over the side hangs a rope ladder with rungs a foot apart. The tide rises at the rate of 9 inches per hour. At the end of 6 hours, how much of the rope ladder will still remain above water, assuming that 9 feet were above water when the tide began to rise? 21

22 A Petri dish hosts a healthy colony of bacteria. Once a minute, every bacterium divides into two. The colony was founded by a single cell at noon. At exactly 12:30 (30 minutes later), the Petri dish was completely full. At what time was it half-full? 12:00 noon 1 bacterium 12:01 pm 2 bacteria 12:02 pm 4 bacteria 12:30 pm Full 22

23 Find the hidden country in the following sentence: Tom s cold was made worse by his swollen glands; perhaps a tonic would help. 23

24 What value of * makes this equation correct? 24

25 What is the parking spot number where this car is parked?

26

27 A man buys a string 25,000 miles long and sets out to stretch it around the circumference of the earth. When he reaches his starting point, he discovers that the string is, in fact, 25,000 miles and one yard long. Rather than cut the string, he decides to tie the ends together and distribute the extra 36 inches evenly around the entire circumference. How far does the string stand out from the earth because of the extra yard? (Choose the closest answer.) 27

28 Circumference: 25,000 miles A inch B inch C. 0.1 inch D. 1 inch E. 2 inches F. 3 inches G. 4 inches H. 5 inches I. 6 inches J. 7 inches K. 8 inches L. 9 inches M.10 inches 25,000 miles + 1 yard 28

29 During World War II, the mathematician Abraham Wald was asked to help with determining which parts of the allied forces' planes must be armored better. After examining the surviving American planes, he noticed that there were many holes in the fuselage, and very few in the engines. After careful thinking, he suggested that the armor on the engines must be improved. Why? 29

30 What do the following have in common: deft, first, calmness, canopy, laughing, stupid, hijack. 30

31 What is it that happens once in a second, once in a month, once in a century, but not at all in a year or week? 31

32 One person went to the store and bought groceries for $13.59 total. He paid with a $100 bill, took his change, and left the store. There was something special about this transaction. What is it? 32

33 If your doctor gave you three pills and told you to take one every half hour, how long would they last? 33

34 34

35 Johnny s mother had three children. The first child was named April. The second child was named May. What was the third child s name?? 35

36 Use the numbers on the tree and arrange them to first create the smallest sum and then the largest sum. 36

37 I start with the letter E. I end with the letter E. I usually contain only one letter Yet I am not the letter E! What am I? 37

38 My current age, is the age of my brother (who is 14), plus one third of my age. How old will I be when my brother is twice his current age? 38

39 Find four consecutive letters in the alphabet which can be rearranged to spell a common word. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 39

40 40

41 = =

42 What makes this number unique: 8,549,176,320? 42

43 A clerk at a butcher shop stands five feet ten inches tall and wears size 13 sneakers. What does he weigh? 43

44 Four years ago, Meg put a nail on a tree in order to mark her height. If the tree grows 10 inches per year, and currently the nail is 5 inches lower than Meg, how much has Meg grown over these four years? 44

45 What is the missing number? 45

46 Billie was born on December 28th, yet her birthday always falls in the summer. How is this possible? 46

47 47

48 What is orange and sounds like a parrot? 48

49 When I was two years old, my sister was half my age. Now I m 100 years old. How old is my sister? 49

50 Job Polish Herb What is special about the words: job, polish, herb? 50

51 Insert three arithmetic operations on the dashes to make the equation true = 19 51

52 52

53 14 of the kids in the class are girls. 8 of the kids wear blue shirts. 2 of the kids are neither girls or wear a blue shirt. If 5 of the kids are girls who wear blue shirts, how many kids are in the class? 53

54 Find the 8 numbers hidden in this paragraph. 54

55 Find the 8 numbers hidden in this paragraph. 55

56 Your parents have six sons including you and each son has one sister. How many people are in the family? 56

57 Emily loves cats, and she keeps some as pets. All but two of them are completely black. All but two of them are completely white. All but two of them are completely ginger. How many cats does she have in total? 57

58 What is your answer? 58

59 What phrase is represented here? 59

60 60

61 What phrase is represented here? 61

62 What is Spooky s favorite number? 62

63 63

64 64

65 A man wanted to encrypt his password but he needed to do it in a way so that he could remember it. He had to use seven characters consisting of letters and numbers only (no symbols like! or <). In order to remember it, he wrote down You force heaven to be empty. What is his password? 65

66 66

67 What comes next? 67

68 A family of five people drove in a car for 300 miles at an average speed of 50 miles per hour. For the whole journey, nobody notices that the car had a flat tire. How come nobody noticed? 68

69 A bat and a ball cost $1.10. The bat costs one dollar more than the ball. How much does the ball cost? 69

70 70

71 71

72 72

73 73

74 74

75 75

76 In British Columbia, you cannot take a picture of a man with a wooden leg. Why not? 76

77 Find a number less than 100 that is increased by one-fifth of its value when its digits are reversed. 77

78 You have a 3-gallon jug and a 5-gallon jug. You need to measure out exactly 7 gallons of water. How can you do it? 78

79 Fill 5 gallon jug. Pour 3 gallons into smaller jug. 3 gallons 2 gallons Empty the 3 gallon jug. Pour the 2 gallons into small jug. 2 gallons Empty Fill 5 gallon jug. 2 gallons 5 gallons 79