28 CHAPTER 1 Critical Thinking Skills The solution to Example 7 is not unique. Other arrangements of the nine numbers in the cells will produce a magic square. Also, other techniques of arriving at a solution for a magic square may be used. In fact, the process described will not work if the number of squares is even, for example, 16 instead of 9. Magic squares are not limited to the operation of addition or to the set of counting numbers. Figure 1.2 (a) (c) (b) (d) 9 4 9 2 8 1 6 3 5 7 4 9 2 8 1 6 5 9 2 8 5 1 5 1 Note that the middle number is 5 and the smallest and largest numbers are 1 and 9, respectively. The sum of 1, 5, and 9 is 15. If the sum of 2 and 8 is added to 5, the sum is 15. Likewise 3, 5, 7, and 4, 5, 6 have sums of 15. We see that in each group of three numbers the sum is 15 and 5 is a member of the group. Now try Exercise 51 Because 5 is the middle number in the list of numbers, place 5 in the center square. Place 9 and 1 to the left and right of 5, as in Fig. 1.2(a). Now we place the 2 and the 8. The 8 cannot be placed next to 9 because 8 9 17, + = which is greater than 15. Place the smaller number 2 next to the larger number 9. We elected to place the 2 in the lower left-hand cell and the 8 in the upper right-hand cell, as in Fig. 1.2(b). The sum of 8 and 1 is 9. To arrive at a sum of 15, we place 6 in the lower right-hand cell, as in Fig. 1.2(c). The sum of 9 and 2 is 11. To arrive at a sum of 15, we place 4 in the upper left-hand cell as in Fig. 1.2(c). Now the diagonals 2, 5, 8, and 4, 5, 6 have sums of 15. The numbers that remain to be placed in the empty cells are 3 and 7. Using arithmetic, we can see that 3 goes in the top middle cell and 7 goes in the bottom middle cell, as in Fig. 1.2(d). A check shows that the sum of the numbers in all the rows, columns, and diagonals is 15. 7 Instructor Resources for Section 1.3 in MyLab Math • Objective-Level Videos 1.3 • Animation: Magic Squares • PowerPoint Lecture Slides 1.3 • MyLab Exercises and Assignments 1.3 • Chapter 1 Projects Magic Squares Exercises Practice the Skills/Problem Solving Throughout this Exercise set, when necessary, round your answer to the nearest hundredth. 1. Reading a Map The scale on a map is 1 inch 12 miles. = How long a distance is a route on the map if it measures 4.25 in.? 51 mi 2. Blueprints Tony, an architect, is designing a new enclosure for the giraffes at a zoo. The scale of his plan is 1 in. 2.5 yd. = He draws a 22.4-in. line on the blueprint to represent the northern boundary of the enclosure. What actual distance does this boundary line represent? 56 yd 3. Height of a Tree At a given time of day, the ratio of the height of an object to the length of its shadow is the same for all objects. If a 3-ft stick in the ground casts a shadow of 1.2 ft, determine the height of a tree that casts a shadow that is 15.36 ft. 38.4 ft 4. Height of a Wind Turbine See Exercise 3. A 6-foot-tall person casts a 2-foot shadow while a wind turbine casts a 95-foot shadow. Determine the height of the wind turbine. 285 ft SECTION 1.3 Jamie Marshall (/ Jon Marshall)-Tribaleye Images/ Alamy Stock Photo
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