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M at h e m at i c a l Problem Solving david kennedy AMT P u b l i s h i n g Contents Prefacev Introduction1 Making An Organised List 4 Looking For A Pattern 11 Drawing A Picture Or Diagram 19 Making The Problem Simpler 26 Acting Out The Problem 31 Using Models 34 Trial And Improvement 40 Making A Table (Part 1) 53 Making A Table (Part 2) 63 Working Backwards 73 Using Logic 84 Using Algebra 89 Taking A Different Viewpoint 95 Introduction1 Introduction In order to become an efficient problem solver, we must first appreciate the different strategies available. Regular practice of these strategies will greatly strengthen our effectiveness as problem solvers and should lead us to become more efficient ones. Alternatives Many problems can be solved using different strategies. Example: Shaking hands If ten people meet in a room and each person shakes hands with each of the others, how many handshakes take place? Alternative Solution Strategies 1. Act It Out: Let ten people shake hands with each other and keep a tally. 2. Draw a Diagram: Count the lines. 3. Use Logic: Ten people each shake hands with nine others, so 10 lots of 9 = 90 but each handshake has been counted by both shakers so you need to halve this number, that is, 45. 4. Make an Organised List: Suppose their names are A, B, C, D, E, F, G, H, I and J, then the results could be recorded as follows (where AB represents A shaking hands with B). AB AC BC AD BD CD AE BE CE DE AF BF CF DF EF AG BG CG DG EG FG AH BH CH DH EH FH GH AI BI CI DI EI FI GI HI AJ BJ CJ DJ EJ FJ GJ HJ IJ 2 Mathematical Problem Solving 5. Make the Problem Simpler and Look for a Pattern: With 2 people there is 1 handshake. With 3 people there are 3 handshakes. With 4 people there are 6 handshakes. With 5 people there are 10 handshakes.......and so on. Students may notice that these are the triangular numbers. Combinations Many solutions involve a combination of different strategies. Example: The Magic Square Place the digits 1 to 9 in the square below so that each row, column and diagonal has the same sum. Solution 1. Make the Problem Simpler: Sum of all numbers = 45 so sum of each row = 15. 2. Make an Organised List: Find combinations of three digits that total 15. 951 951 861 852 843 762 753 654 3. Use Logic: The 5 occurs in four combinations so must go in the centre square. 5 Introduction3 The 8 occurs in three combinations so must go in a corner square. 8 5 So it follows that the 2 must be in the opposite corner. 8 5 2 The 6 also occurs in three combinations so must go in another corner. 8 6 5 2 So it follows that the 4 must be in the opposite corner. 8 6 5 4 Now, the rest are obvious. 2 8 1 6 3 5 7 4 9 2 4 Mathematical Problem Solving Making an Organised List A list helps us account for all possibilities and avoid repetitions. It is important to find and use a system. Often it is simply a case of listing results in numerical or alphabetical order. However, we need to be rigorous in the application of our system. Example 1 How many triangles can you find in the shape below? Solution First label each small section with a letter. a e b c d Now list all the possibilities in alphabetical order. a b c d e abcbc cd de abcdebe abe So there are 12 different triangles in the figure. Example 2 Domino tiles are all different. Each tile consists of two halves. Each half can be either blank or a number from 1 to 6. How many dominoes are there in a full set? Making an Organised List 5 Solution The possible numbers on each half are 0, 1, 2, 3, 4, 5 and 6. List the possible combinations as ordered pairs in numerical order. You could start at 6,6 or 0,0 but be consistent! (Note: 6,5 is the same as 5,6 in this problem!) 6,6 6,5 5,5 6,4 5,4 4,4 6,3 5,3 4,3 3,3 6,2 5,2 4,2 3,2 2,2 6,1 5,1 4,1 3,1 2,1 1,1 6,0 5,0 4,0 3,0 2,0 1,0 So there are 28 dominoes in a full set. 0,0