Problem Solving Skills (14021601-3 ) Lecture 1 Puzzle-Based Learning • Puzzles represent “ unstructured ” problems; they teach how to think “ out of the box. ” • Puzzles are not attached to any chapter of any text. • Puzzles illustrate many general and powerful problem-solving techniques. • Puzzles illustrate importance of science. • Puzzles are fun and easy to remember! 2 1
Puzzle-Based Learning • Puzzles introduce a variety of topics, like: probability, statistics, modeling, simulation, induction, data mining, forecasting, optimization, and many others, in a very entertaining way … • Puzzles give some perspective on science in general and mathematics in particular. 3 A puzzle A man has to take a wolf, a goat, and some cabbage across a river. His rowboat has enough room for the man plus either the wolf or the goat or the cabbage. If he takes the cabbage with him, the wolf will eat the goat. If he takes the wolf, the goat will eat the cabbage. Only when the man is present are the goat and the cabbage safe from their enemies. How should the man carry the wolf, goat, and cabbage across the river? 4 2
A puzzle, cont. A solution: 5 Supporting evidence • English scholar Alcuin (born in 732) published his main work, Problems to Sharpen the Young (one of the puzzles there was the famous river-crossing puzzle – used 1,200 years later in all Artificial Intelligence courses all over the world!). • The first known puzzles are from Neolithic Age (3,000 – 2,500 BC). • Statements by famous mathematicians (Polya, Gardner, Steinhauss, Smullyan, etc.). 6 3
Learning objectives • After completing this unit, you should have a better understanding of: – The nature of unstructured problems. – The various elements that impact real-world problems (probability, constraints, etc.). – How to approach and solve unstructured problems. – Various problem-solving techniques (e.g. simulation) and disciplines (e.g. statistics). • After completing this unit, you should be a better problem solver! 7 Learning objectives Solution Problem 8 4
Main goal After completing this unit, you will look at problems in a different way … 9 The first “ problem ” A farmer has: 20 pigs, 40 cows, and 60 horses. How many horses does he have, if he calls the cows horses? 10 5
The first “ problem ” A farmer has: 20 pigs, 40 cows, and 60 horses. How many horses does he have , if he calls the cows horses? Answer: 11 Puzzle 1.2 You drive a car at a constant speed of 40 km/h from A to B , and on arrival at B you return immediately to A , but at a higher speed of 60 km/h. What was your average speed for the whole trip? A B 12 6
Puzzle 1.2 You drive a car at a constant speed of 40 km/h from A to B , and on arrival at B you return immediately to A , but at a higher speed of 60 km/h. What was your average speed for the whole trip? A B 13 Model of the problem (Puzzle 1) 14 7
The model solution 15 Consequences • Boat on the river (boat: 50 km/h, the current: 10 km/h). • Plane and wind. • Regular trips, when we drive different segments with different speeds … 16 8
Some experiments • Puzzle-Based Learning web-site: www.PuzzleBasedLearning.edu.au Experiment with Puzzle 1.2 – set different speeds for the boat and the current of the river. Observe the average speed. 17 Rule #1 Rule #1: Be sure you understand the problem, and all the basic terms and expressions used to define it. 18 9
Two groups of students Two groups of students are attending a college. One day, one of the students from group A approached another from group B , and said “ We are taller than you! ” 19 What does it mean? Does this statement mean that: 1. Each a is taller than each b ? 2. The tallest a is taller than the tallest b ? 3. Each a is taller than some b ’ s? 4. Each b is smaller that some a ’ s? 5. Each a has a corresponding b (and each of them a different one) whom he surpasses in height? 6. Each b has a corresponding a (and each of them a different one) by whom he is surpassed? 7. The shortest b is shorter than the shortest a ? 8. The shortest a exceeds more b 's than the tallest b exceeds a 's? 20 10
What does it mean? Does this statement mean that: 9. The sum of heights of a 's is greater than the sum of heights of b 's? 10.The average height of a 's is greater than the average height of b 's? 11.There are more a 's who exceed some b than there are b 's who exceed some a ? 12.There are more a 's with height greater than the average height of b 's, than there are b 's with height greater than the average height of a 's? 13.The median height of a 's is greater than that of b 's? 21 Check some implications • 1 → 2, 1 → 3, 1 → 4, 1 → 7, 1 → 8, 1 → 10, 1 → 11, 1 → 12, 1 → 13, • 2 → 4, • 3 → 7, • 4 → 2, • 5 → 7, • 6 → 2, 6 → 4, 6 → 9, • 7 → 3, • 8 → 3, 8 → 7. 22 11
Puzzle 1.4 Three cats catch three rats in three minutes. How long it will take them to catch 100 rats? 23 Puzzle 1.4 Three cats catch three rats in three minutes. How long it will take them to catch 100 rats? 24 12
Lessons learned Rule #1: Be sure you understand the problem, and all the basic terms and expressions used to define it. 25 Critical thinking “ Arguments for banning guns are mostly myths, and what we need now is not more laws, but more law enforcement. One myth is that most murderers are ordinary, law-abiding citizens who kill a relative or acquaintance in a moment of anger only because a gun was available. In fact, every study of homicide shows the overwhelming majority of murderers are career criminals, people with lifelong histories of violence. The typical murderer has a prior criminal history averaging at least six years, with four major felony arrests. ” 26 13
Critical thinking “ Arguments for banning guns are mostly myths, and what we need now is not more laws, but more law enforcement. One myth is that most murderers are ordinary, law-abiding citizens who kill a relative or acquaintance in a moment of anger only because a gun was available. In fact, every study of homicide shows the overwhelming majority of murderers are career criminals, people with lifelong histories of violence. The typical murderer has a prior criminal history averaging at least six years, with four major felony arrests. ” 27 Key questions • Which words or phrases are ambiguous? • What are the assumptions? • What significant information is omitted? 28 14
Another example “ Late afternoon was partially cloudy, yet there were many people on the streets of this large city. ” Late afternoon? Partially cloudy? Many people? Large city? 29 Classic Logic vs. Fuzzy Logic a X or a a is a member of X X in some degree … reasonable salary 1 1 0 0 45K 70K 35K 50K 65K 80K 30 15
Exercise 1 • Find the simplest solution for the following: With a 7-minute hourglass and an 11-minute hourglass, find the simplest way to time the boiling of an egg for15 minutes. • Discuss the solution. Provide an argument that your solution is not only correct, but also the simplest … 31 Exercise 2 • Find the solution for the following: Three backpackers cooked rice for dinner. The first one gave 400g of rice and the second – 200g of rice. The third one did not have any rice so he gave $6 to the other two. How should they divide the $6 in a fair way (assume they equally shared the dinner)? • Discuss the solution. Provide an argument that your solution is fair. 32 16
Exercise 3 • Find the solution for the following: Mr. White, Mr. Green, and Mr. Brown have in total six children: five boys and one girl. Bill has one child more than Mr. Green. Greg has as many children as Walter and Bill. Mr. Brown has as many boys as Mr. Green. What is the first name of Mr. White and many boys does he have? • Justify your the solution. 33 Any Question ??? 17
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