EXPLORATION 1.5. Magic Squares. PART 1: Describing magic squares and finding patterns

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Bertram Carter
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1 chapter /27/05 2:13 PM Page 7 CHAPTER 1 Foundations for Learning Mathematics 7 EXPLORATION 1.5 Magic Squares Have you ever seen a magic square? Magic squares have fascinated human beings for many thousands of years. The oldest recorded magic square, the Lo Shu magic square shown in the figure below, dates to 2200 B.C. It is supposed to have been marked on the back of a divine tortoise that appeared before Emperor Yu when he was standing on the bank of the Yellow River. In the Middle Ages, many people considered magic squares to be able to protect them against illness! Even in the twentieth century, people in many countries still use magic squares as amulets. We will spend some time exploring magic squares because they reveal some amazing patterns. As a teacher, you will find that most students love working with magic squares and other magic figures. Many elementary teachers have their students explore magic squares partly because they are a lot of fun, but also because all five of the NCTM process standards emerge nicely from these explorations. PART 1: Describing magic squares and finding patterns 1. Let us begin with the simple magic square shown in the figure. Take a couple of minutes to write down why this is a magic square and to write down all the patterns you see in this square Compare your observations with those of others in your group. Add to your list new observations and patterns that you heard about from other members. 3. Based on your observations and discussion, what makes a magic square magic? Imagine telling this to a friend who has never heard of magic squares. 4. Take a few minutes to think of questions you might have about magic squares. Write down these questions. 5. Listen to the questions that other students suggest. Your instructor may select additional questions to investigate.

2 chapter /27/05 2:13 PM Page 8 8 CHAPTER 1 Foundations for Learning Mathematics PART 2: Patterns in all 3 3 magic squares Use the 3 3 magic square templates on page Make a completely new 3 3 magic square. You are not restricted to consecutive numbers, though I suggest restricting yourself to positive whole numbers, simply to make it easier to see patterns that are true in all the magic squares. 2. Display each magic square. Have each person explain any strategies that made creating the square easier, that is, strategies beyond grope and hope. Note any strategies that you did not discover but that you would like to remember. 3. Write down patterns that seem to be true for all 3 3 magic squares and patterns that seem to be true only for some 3 3 magic squares. (Hint: There are lots of patterns!) Then go around the group and have members share their responses. 4. Select one non-simple pattern that is true for all 3 3 magic squares. a. Describe this pattern, using only words. Imagine that you are talking on the phone to a friend who has seen magic squares but didn t realize there are patterns that are true for all 3 3 magic squares. b. Give this description to some friends who are not in this course. Check to see how well they understand the pattern just from reading your description. If they understand the pattern, fine. If they don t, revise your description and repeat the process. In the latter case, write down what you learned about communicating. PART 3: Using algebra to describe 3 3 magic squares 1. As you may have already discovered, the middle number of the 3 3 magic square is a key number. What if we called the middle number m? Reflect on the patterns you observed in all magic squares and the strategies students used to construct magic squares. Can we represent the other numbers in the magic square in terms of m? With the group working together, take a few minutes to think about this question, and look at relationships between the middle number and other numbers in the sequence. Describe your present thinking and work before moving on. 2. One key to solving this problem is to realize how many more variables are needed. The only complete algebraic representation of 3 3 magic squares that I know of requires two more variables. Let us call them x and y. That is, using three variables, we can state instructions for making any 3 3 magic square. In other words, by using m, x, and y, we can represent the value of each cell in the magic square. Work on this problem in your group. Briefly explain, in writing, how you solved the problem. 3. At this point, you have a solution, generated either from your group or from the whole-class discussion. Think about your description of how to make a generic magic square, and then read the following quote. Does this experience change your attitude toward algebra? Does it help you to see the use of symbols in a new light? Briefly explain your response. Mathematics is often considered a difficult and mysterious science, because of the numerous symbols which it employs....[t]he technical terms of any profession or trade are incomprehensible to those who have never been trained to use them.

3 chapter /27/05 2:13 PM Page 9 CHAPTER 1 Foundations for Learning Mathematics 9 But this is not because they are difficult in themselves. On the contrary they have invariably been introduced to make things easy. So in mathematics, granted that we are giving any serious attention to mathematical ideas, the symbolism is invariably an immense simplification. 1 PART 4: Further explorations for 3 3 magic squares 1. How many different 3 3 magic squares can you make starting with the two numbers shown in the figure? Below are questions about four possible transformations of a 3 3 magic square. In each case, record your initial guess and your reasoning before you test your guess. Then test your guess. If you were correct, refine your justification if needed. If you were wrong, look for a flaw or incompleteness in your reasoning. If you were wrong, can you now justify the correct answer? a. If you doubled each number in a magic square, would it still be a magic square? b. If you added the same number to each number in a magic square, would it still be a magic square? c. If you multiplied each number in the magic square by 3 and then subtracted 2 from that number, would it still be a magic square? d. If you squared each number in the magic square, would it still be a magic square? 3. Look back on your data from different 3 3 magic squares, and answer the following questions: a. Is there a relationship between the magic sum and whether the number in the center is even or odd? b. Divide the set of magic squares into two subsets: those in which the nine numbers are consecutive numbers (for example, 10 18) and those in which the nine numbers are not consecutive numbers. Are there any other differences between these two sets of magic squares? PART 5: 5 5 magic squares 1. Three 5 x 5 magic squares are shown below. What do you see (observations and/or patterns)? Alfred North Whitehead, Introduction to Mathematics, (New York, 1911, pp ), cited in Robert Moritz, On Mathematics (New York: Dover Publications, 1914), p. 199.

4 chapter /27/05 2:13 PM Page CHAPTER 1 Foundations for Learning Mathematics 2. Describe similarities and differences that you see between 3 3 and 5 5 magic squares. 3. Use the 5 5 magic square templates on page 11. Your instructor will walk you through a set of instructions for generating 5 5 magic squares. Your task is to make sense of these rules and then write out a set of rules that you could to a friend who needs them to make 5 5 magic squares successfully.

5 chapter /27/05 2:13 PM Page 11 CHAPTER 1 Foundations for Learning Mathematics Magic Square Templates For EXPLORATION Magic Square Templates For EXPLORATION 1.5

EXTENSION. Magic Sum Formula If a magic square of order n has entries 1, 2, 3,, n 2, then the magic sum MS is given by the formula

EXTENSION. Magic Sum Formula If a magic square of order n has entries 1, 2, 3,, n 2, then the magic sum MS is given by the formula 40 CHAPTER 5 Number Theory EXTENSION FIGURE 9 8 3 4 1 5 9 6 7 FIGURE 10 Magic Squares Legend has it that in about 00 BC the Chinese Emperor Yu discovered on the bank of the Yellow River a tortoise whose