# Chapter 1 - Problem Solving and Critical Thinking - 1.3 Problem Solving - Exercise Set 1.3: 47

a. 5 22 18 28 15 2 12 8 25 The sums are 45. b. 4 9 8 11 7 3 6 5 10 The sums are 21.

#### Work Step by Step

The square looks like this: 5 _ 18 _ 15 _ _ _ 25 A magic square gives the same result for sums of all digits in any row, column or diagonal. a. We have all the digits of one diagonal; therefore we know that the sum of our square is $5+15+25=45$. With that, we can calculate the upper middle space by looking at the first row: $45-18-5=22$ the middle right space looking at the last column: $45-25-18=2$ and the bottom left space looking at it's diagonal: $45-15-18=12$ Now the square looks like this: 5 22 18 _ 15 2 12 _ 25 The two remaining spaces are: $45-15-2=28$ $45-15-22=8$ This is the full magic square: 5 22 18 28 15 2 12 8 25 b. The numbers of the letters in each word for the numbers in the square are as follows: five – 4 twenty-two – 9 eighteen – 8 twenty-eight-11 fifteen – 7 two – 3 twelve - 6 eight – 5 twenty-five – 10 putting these numbers back in the square, we get: 4 9 8 11 7 3 6 5 10 The sums for each row, column and diagonal are: $4+9+8=21$ $11+7+3=21$ $6+5+10=21$ $4+11+6=21$ $9+7+5=21$ $8+3+10=21$ $4+7+10=21$ $6+7+8=21$ This proves this square is a magic square too.

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