In this unit students develop and use algebraic representations of 3-by-3 magic squares in which the sum of the three numbers in each row, column, and diagonal, add to a constant number known as the Magic Number. Students are posed an array of problems to solve using their algebraic tools.

- devise a rule for ensuring that sets of numbers can be arranged into 3-by-3 magic squares
- represent 3-by-3 magic squares algebraically
- devise rules for determining the Magic Number for magic squares
- represent magic squares using parametric equations
- solve equations that have been formed from magic squares.

Although many students may have some experience with ‘Magic Squares’ it is likely that those experiences will have been limited to work with arithmetic. In this unit, students begin with magic squares involving sets of simple numbers. Once they see how the magic squares work they quickly advance to representing simple magic squares algebraically where letters are used as generalised unknown numbers. They use these simple algebraic magic squares and the patterns they see to form more general 3-by-3 magic squares. A feature of this unit includes using problem-solving strategies: Trial and Improvement, Looking for Patterns, and Generalising. Students complete the Unit by finding a rule for calculating the ‘Magic Number’ for any *n-by*-*n* magic square where *n* is any natural number ≥ 3.

arrays, constant, substitution, simplify, infinite, consecutive, generalised unknowns, natural numbers, parametric equations, algebraic expressions

Each session in this unit comprises a sequence of tasks or challenges. It is suggested that students have an opportunity to engage in each task by themselves before they begin sharing their ideas with other students or with their teacher. The role of the teacher is to first listen carefully to students as they try to explain their thinking to others and then interact with the students only when they or their peers appear to have reached an impasse. Such interactions might then be to help clarify students’ thinking or even redirect thinking where necessary.

Notes have been included at the beginning of each session to provide mathematical background including solutions, as well as on aspects that are likely to pose some difficulty for students. The exact nature of any interactions will depend on the situation and on the thinking progress that has been made.

#### Session 1

A square grid of numbers in which the numbers in each row, column and diagonal add up the same number is often called a *magic square*. The sum of the numbers in each row, column and diagonal is called the *magic number*.

The magic number for each of the magic squares above is 15. Notice that the set of nine numbers 1, 1, 1, 5, 5, 5, 8, 8, 8 cannot be arranged to make a 3-by-3 magic square

Students are likely to use a ‘Trial and Improvement’ strategy to complete the magic squares. Encourage students to share their rules as to which sets numbers can be arranged into a 3-by-3 magic square and which cannot. The 3-by-3 magic square here can only be formed when the three different numbers involved are in a linear sequence, that is the ‘gap’ between the smallest number and the middle number is the same as the gap between the middle number and the largest number.

#### Task 1

Arrange the nine numbers, 4, 4, 4, 5, 5, 5, 6, 6, 6 on the 3-by-3 grid below so that each row, column and diagonal adds up to 15.

The completed square is called a*magic square*. The*magic number*for the square is 15.Now arrange the nine numbers, 3, 3, 3, 5, 5, 5, 7, 7, 7 on a 3-by-3 grid to make a magic square with magic number 15.

#### Task 2

Make up your own 3-by-3 magic square with magic number 15.

Jen tries to make magic squares using first 1, 1, 1, 5, 5, 5, 8, 8, 8, and then –1, –1, –1, 5, 5, 5, 11, 11, 11. Make 3-by-3 magic squares using these two sets of numbers.

Write a rule in words that tells whether a set of nine numbers, made with 3 sets of three identical numbers as above, can be used to make a magic square.

#### Session 2

Students use their rule from Session 1 to make magic squares that involve larger numbers. Question 4 of Task 3 is an *open* question since there is more than one solution. In fact there is an infinite number of solutions. Draw students’ attention to this during a discussion of answers. They should try to explain why this must be so.

In Task 4, students investigate what happens when each number in a 3-by-3 magic square they make is trebled. They will see that the square remains magic and that the magic number is also trebled. Similarly ‘adding 4 then halving’ each number in a magic square maintains the ‘magic’ but changes the magic number so that it is now ‘one half of four more’ than the original magic number. Encourage students to use their own words to explain how the magic numbers change.

#### Task 3

Make magic squares using the following sets of nine numbers.

a. 4, 4, 4, 13, 13, 13, 22, 22, 22 b. 27, 27, 27, 43, 43, 43, 59, 59, 59Make three different 3-by-3 magic squares that have a magic number of 15.

Find the magic number for each square and then complete the magic square.

Complete these magic squares. Use just three different numbers in the nine spaces.

#### Task 4

Find the magic number for each unfinished magic square. Then complete the magic square

.Double the value of each number in your finished magic squares in Question 1 above and check if your new square is magic. Find the magic number.

Suppose each number in a 3-by-3 magic square is trebled. What can you say about the new magic number? Make a magic square and check your prediction.

Suppose 4 is added to each number in a 3-by-3 magic square. Then each new number is halved. What can you say about the new magic number? Make a magic square and check your prediction.

#### Session 3

In this session, literal symbols are used to represent generalised unknown numbers. In Task 6, students will need early support as they for example, ‘multiply each algebraic expression by 4’. Students might try multiplying the first row of algebraic expressions on their own before any discussion occurs. It is likely that some students will multiply *n* – *c* by 4 to produce 4*n* – *c* rather than 4*n* – 4*c*. Where this happens it may be helpful to have the students resolve their misunderstanding by substituting different numerical values into 4 x (n – *c*), 4n – *c* and 4*n* – 4*c* to check which expressions produce the same values. This approach may be helpful for the remaining questions in this task.

#### Task 5

Complete this unfinished magic square and find an algebraic expression for the magic number. The letters,

*n*and*c*stand for*any*numbers.Make three different 3-by-3 magic squares by substituting different values for

*n*and*c*in the completed magic square above. Find the magic number for each magic square.

#### Task 6

- Multiply by 4, each algebraic expression in the completed magic square for Question 1 in Task 5.
- Check that the new square is magic and find the algebraic expression for the magic number.

- Subtract 7 from each algebraic expression in the completed magic square for Question 1 in Task 5.
- Check that the new square is magic and find the algebraic expression for the magic number.

Complete the first magic square below. Then add

*k*to each algebraic expression in the magic square. Check that your new square is magic and find the algebraic expression for the magic number.Multiply by

*p*, each algebraic expression in your new square in Question 3. Check that your new square is magic and find the algebraic expression for the magic number.Complete the first magic square below. Then subtract

*q*from*x*times each algebraic expression in the magic square. Check that your new square is magic and find the algebraic expression for the magic number.

#### Session 4

Completing the magic square in question 1 of Task 7 will be challenging for most students. They must first work out the expression for the magic number, in this case

it is 3(*x* + *y*), which is 3 times the expression for the centre number. The expression that goes in the empty cell in the middle row is then 3(*x* + *y*) – 2*x* – (*x* + *y*) which simplifies to 2*y*. In Question 3, since the magic number is 18 then 3(*x* + *y*) = 18 giving *x* + *y* = 6. In fact, any values for *x* and *y* that satisfy *x* + *y* = 6 will produce a magic square that has 18 for its magic number. As earlier, this is also an *open* question that has an infinite number of solutions. Such open questions can deepen students’ understanding and are also useful for assessing understanding.

The algebraic magic square that is the basis on the questions in Task 8 is a generalised form of a 3-by-3 magic square. Students may see that this magic square has been produced by adding *z* to the expression in each cell in the algebraic magic square in Task 7. Alternatively, they may recall from earlier work that 3(*x* + *y* + *z*) is the expression for the magic number (3 times the centre number). So 3(*x* + *y* + *z*) – (2*x* + *z*) – (*x* + *y* + *z*) or 2*y* + *z* is the expression that goes in the empty cell in the middle row. The algebraic expressions that go in the remaining empty cells can then in turn be determined by subtracting a known expression in a particular row or column from the expression for the magic number, 3(*x* + *y* + *z*). Substituting numerical values for *x*, *y* and *z* produces magic squares. In question 3, the magic square has consecutive whole number values from 1 to 9. It is sometimes known as a ‘normal’ 3-by-3 magic square with magic number 15.

In Question 4b, since 3(*x* + *y* + *z*) is the algebraic expression for the magic number, then 3(*x* + *y* + *z*) = 27 giving *x* + *y* + *z* = 9. This means that there is an infinite number of sets of values for *x*, *y*, and *z* that can be* *chosen to produce a 3-by-3 magic square with magic number 27. Similarly for Question 4c, using any set of values for *x*, *y*, and *z* so that*x* + *y* + *z* = 0 produces a magic square with magic number zero.

#### Task 7

- Find the algebraic expressions to make the following grid into a magic square.
- Write and then simplify the algebraic expression for the magic number in your completed magic square for Question 1a.

Substitute the given sets of values for

*x*and*y*in your completed magic square for Question 1a.

a.*x*= 1,*y*= 2 b.*x*= 2,*y*= – 2 c.*x*= 10,*y*= 43- Make a magic square based on the algebraic magic square in 1a. Make sure the magic number is 18.

Mene decides to make a magic square by putting

*x*= 1,*y*= 5 in the algebraic magic square in 1a. How can he tell that the magic number is 18 before he actually makes the magic square?How many magic squares, based on the algebraic square in Question 1a, can be made with magic number,

*m*? Explain your reasoning.

#### Task 8

- Find the algebraic expressions to make the next grid into a magic square.
- Write and then simplify the algebraic expression for the magic number in your completed magic square for Question 1a in this Task.

Substitute the given values for

*x*,*y*, and*z*below to make magic squares. Check that each completed square is magic.

a.*x*= 2,*y*= 1,*z*= 1 b.*x*= 1,*y*= –1,*z*= 4 c.*x*= 5,*y*= 12,*z*= 16Sadie substitutes

*x*= 3,*y*= 1,*z*= 1 in the algebraic magic square in Question 1 above. She notices a pattern in her magic square. Describe the pattern.Sadie now uses

*x*= 3,*y*= 4,*z*= 5 to make a magic square. She predicts that the magic number is 12. Explain Sadie’s reasoning.Choose values for

*x*,*y*and*z*to give a magic number of 27 for a magic square. Check that you are correct by making the magic square.Choose values for

*x*,*y*and*z*to give a magic number of 0 for a magic square. Check that you are correct by making the magic square.

#### Session 5

In this session, the generalised unknowns *x* , *y*, and *z* are represented in parametric form. So for example, in Question 1, Task 9, using the parametric equations *x* = *a* and*y* = 2*a* + 1 produces the following 3-by-3 magic square in one unknown, *a*.

In Question 2, the parametric equations, *x* = *a*^{2} + 2 and *y* = 4 – 5*a* produce a magic square with magic expression 3*a*^{2} – 15*a* + 18. When the magic number is 0, then

3*a*^{2} – 15*a* + 18 = 0 so that *a*^{2} – 5*a* + 6 = 0. This factorises to (*a* – 2)(*a* – 3) = 0. So either (*a* – 2) = 0 giving *a* = 2, or (*a* – 3) = 0 giving *a* = 3.

In Task 10, when the parametric equations, *x* = *a*, *y* = 3*a* and *z* = 3 – *a* are substituted for *x*, *y*, and *z* then the expression for the magic number is 9*a*_{}+ 9. So when the magic number is 63, then 9*a*_{}+ 9 = 63 giving *a* = 6.

In Task 11, students will need to look closely at the six “normal” magic squares. They may first see that for the *n*-by-*n* magic squares, where *n* is an odd number greater than or equal to 3, the magic number is *n* times the middle number in the magic square. For the *n*-by-*n* magic squares, where *n* is even, there is no ‘middle’ number. A ‘hidden’ middle number can be determined by calculating the mean of the four numbers in the middle of the magic square. So a rule for the magic number of the “normal” *n*-by-*n* magic squares, can be expressed as *n *multiplied by the ‘middle’ number of the magic square. However, this rule seems to be of little value if the magic square cannot be seen. But all is not lost. The middle number can also be calculated by finding the mean of the first, (1), and last, (*n*^{2}), numbers in the sequence of consecutive numbers that make up the magic square. So the rule for the magic number of any “normal” magic square is *n* x . This is more usually expressed as , where *n* is any natural number ≥ 3.

#### Task 9

Substitute

*x*=*a*and*y*= 2*a*+ 1 to make a new magic square.Find an algebraic expression for the magic number in your new magic square.

Find the value for

*a*that makes the magic number zero.

Substitute

*x*=*a*^{2}+ 2 and*y*= 4 – 5*a*to make a new magic square.Find an algebraic expression for the magic number in your new magic square.

Find values for

*a*that make the magic number, 0.

#### Task 10

Substitute

*x*=*a*,*y*= 3*a*and*z*= 3 –*a*to make a new magic square.Find the algebraic expression for the magic number in your new magic square.

Find the value for

*a*when the magic number is 63.

#### Task 11

The following magic squares consist of consecutive whole numbers starting with 1. They are sometimes known as “normal” magic squares.

The magic number for each square is shown under the square. For example, the magic number for the “normal” 3-by-3 magic square is M_{3-by-3} = 15 = 3 x 5.

- Look carefully at the magic squares and then predict the magic number for a 9-by-9 “normal” magic square. Explain your reasoning.
- Now predict the magic number for a 10-by-10 “normal” magic square. Explain your reasoning.
- Write a rule for the magic number of an
*n*-by-*n*“normal” magic square where*n*is any natural number greater than or equal to 3. Explain how the rule works.