Problem Solving and Critical Thinking

Transcription

1 Problem Solving and Critical Thinking The ability to solve problems using thought and reasoning is indispensable in our everyday lives. It can also provide entertainment as you exercise those gray cells in your brain. Recall Polya's Four Steps in Problem Solving as you solve these puzzles: 1. Understand the problem. 2. Devise a plan. 3. Carry out the plan and solve the problem. 4. Look back and check the answer. Sudoku A recent craze is Sudoku (sue-doe-koo), which was preceded by magic squares. (See Exercise Set 1.3 in the textbook.) Here is a Sudoku puzzle with directions: Here is how to solve a sudoku puzzle (no math): 1. You'll be given some numbers to start each puzzle. These cannot be moved or changed. 2. Each row, column, and 3-by-3 grid must contain one and only one instance of each number (1-9). You can never have the same number in any row, column or 3-by-3 grid, or "region." 3. A properly constructed sudoku puzzle has only one correct answer. 4. There can be different levels of difficulty, usually so noted in or around the puzzle.

2 USA Weekend Magazine at for March 4, Apply good problem solving techniques. Step 1: Do you understand the rules? Step 2: Did you devise a plan of attack? Step 3: Did you fill in all of the boxes? Step 4: Does your solution meet the requirements? You can purchase puzzle books devoted entirely to Sudoku puzzles and many newspapers offer a daily puzzle. There's even a website that offers free sodoku solving and teaching tools: Try this site for the solution to the puzzle. Note: this is an "outside" site that CTC has no control over. Puzzles Everyone likes puzzles, but how often do you try to find a mathematical or logical explanation for how they work? Phone number puzzle 1. Grab a calculator. (You won't be able to do this one in your head) 2. Key in the first three digits of your phone number (NOT the area code)

3 3. Multiply by Add 1 5. Multiply by Add the last 4 digits of your phone number 7. Add the last 4 digits of your phone number again. 8. Subtract Divide number by 2 And the result is.... Can you explain how this works? Don't peek until you have tried. Explanation: Let's say your phone number is Let x represent the prefix 987 and y present the last four digits Multiply the prefix by 80: 80x. Add 1: 80x + 1 Multiply by 250: 250(80x + 1) = 20000x Add the last four digits and then add them again: 20000x y + y = 20000x y Subtract 250: 20000x y = 20000x + 2y Divide by 2: 20000x/2 + 2y/2 = 10000x + y. What does multiplying a number by do? It puts four zeroes at the end of the number x + y = 10000(987) = = ISBN Puzzle "A modern though little realized example of undecimal counting is seen in the ISBN of published books. Any ISBN comprises ten digits. If you multiply the first by ten, the second by nine, the third by eight, and so on, summing the results as you go along, the result will always be divisible by eleven." Source: William Hartston; What Are The Chances Of That?: Fabulous Facts About Figures; Metro Books; 2004.

4 Questions: 1. Is this little known fact true? 2. Does it work only with ISBNs? 3. Can you "prove" it mathematically? Solution: Have you strained your brain enough trying to figure this one out? Well, here is one explanation. (Don't peek until you have tried this on your own!) Let the ten digits of the ISBN be represented by the first ten letters of the alphabet: Place in ISBN Represented by A B C D E F G H I J Multiply by Product 10(A) 9(B) 8(C) 7(D) 6(E) 5(F) 4(G) 3(H) 2(I) 1(J) Add the items in the last row of the table: ( )(A + B + C + D + E + F + G + H + I + J) = 55(A + B + C + D + E + F + G + H + I + J) Since 55 is divisible by 11, 55(A + B + C + D + E + F + G + H + I + J) is also divisible by 11. Logic puzzles Numbers aren't always involved, as illustrated in these logic puzzles. Can you solve these word games? Think literally. Each communicates a famous person, place, or thing or saying. USA Weekend Magazine at for March 4, 2007.

5 Answers: Pull over to the curb ("PULL" over two "THE CURB") No one to blame (no 1 two "BLAME") Kevin Federline ("KEV" in "FEDERLINE") Graph Theory Puzzles are not a recent development. The Konigsberg bridge problem has been pondered since the early 1700s. The river Pregel divides the town of Konigsberg, Germany, into four separate land masses, A, B, C, and D. Seven bridges connect the various parts of town, and some of the town's curious citizens wondered if it were possible to take a journey across all seven bridges without having to cross any bridge more than once. All who tried ended up in failure, including the Swiss mathematician, Leonhard Euler (pronounced "Oiler"), a notable genius of the eighteenth-century. This is a diagram of the city along with the bridges that connect its parts. Euler did succeed in explaining why such a journey was impossible, not only for the Konigsberg bridges, but whether such a journey was possible or not for any network of bridges anywhere. Euler reasoned that for such a journey to be possible that each land mass should have an even number of bridges connected to it, or if the journey would begin at one land mass and end at another, then exactly those two land masses could have an odd number of connecting bridges while all other land masses must have an even number of connecting bridges. Euler realized that all problems of this form could be represented by replacing areas of land by points (he called them vertices), and the bridges to and from them by arcs. For Konigsberg, let us represent land with red dots and bridges with black curves:

6 Simplified farther, we get a diagram like this: The problem now becomes one of drawing this picture without retracing any line and without picking your pencil up off the paper. Consider this: all four of the vertices in the above picture have an odd number of arcs connected to them. Take one of these vertices, say one of the ones with three arcs connected to it. Say you're going along, trying to trace the above figure out without picking up your pencil. The first time you get to this vertex, you can leave by another arc. But the next time you arrive, you can't. So you'd better be through drawing the figure when you get there! Alternatively, you could start at that vertex, and then arrive and leave later. But then you can't come back. Thus every vertex with an odd number of arcs attached to it has to be either the beginning or the end of your pencil-path. So you can only have up to two 'odd' vertices! Thus it is impossible to draw the above picture in one pencil stroke without retracing and the graph is not traversable. You will study these concepts in detail in Chapter 10. In Summary Puzzles can be fun, challenging, or frustrating. The more you practice and apply sound, logical techniques to their solution, the better you will become at finding answers to puzzles and hopefully, at troubleshooting and identifying solutions to daily events.