Magic Squares Maths Homework For Form

The Problem

Tui realises that there are nine positions in a magic square. Can she make up a magic square using each of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 only once?

Teaching sequence

  1. Show the students a magic square such as the one below, and ask why it is called a magic square.
  1. Have them to check that the rows all have the same sum (of 12); that the columns all have the same sum; and that the diagonals have the same sum.
  2. Pose Tui’s problem.
  3. Have students work in pairs  to create their magic square.
    Do not tell them that the sum of a magic square is three times the centre entry. They will need this information, but the point is to have them work this out themselves.
  4. As some pairs report back, have them justify each step in the production of the magic square. Ask:
    How did you solve the problem? (There is more than one way.) 
    How many different magic squares can you find?
    How many do they think there are?
  5. Have the students record what they have discovered.
  6. Pose the Extension as appropriate.

Extension to the problem

What other magic squares can you make with nine consecutive numbers?

Students may like to investigate consecutive numbers b – 1, b and b + 1 to show that there are only four magic squares that use only these three numbers.
Does the same argument hold with 2b – 2, 2b and 2b + 2? What about b, c and 2c - b? 

Solution

Suppose that the magic square is as in the diagram below.

Let s = the sum of three numbers in a row. Therefore:

s   = A + B + C   =    D + E + F      =    G + H  + I
     = A + D + G   =    B + E + H     =    C + F + I
     = A + E + I    =     C + E + G

We know that:
A + B + C + D + E + F + G + H + I = 1 + 2 +…+ 9 = 45
Can we use this information to establish s?

We can see that the first three equations for s are A + B + C; D + E + F; and G + H  + I.  If we sum these three equations, we get:
3s = A + B +C + D + E + F + G + H + I = 45. So 3s = 45, which means that s = 15.

There are two ways to go from here. Call this point step *. Thus the sum has to be 15.  How many ways can this sum be attained?  Let's see how many ways each number 1 to 9 can be used in a sum of three to get 15.
9 + 5 + 1
9 + 4 + 2
8 + 6 + 1
8 + 5 + 2
8 + 4 + 3
7 + 6 + 2
7 + 5 + 3
6 + 5 + 4

As can be seen, there are only eight possible equations of three numbers using the numbers 1 to 9 that give a sum of 15.  Thus, all these eight equations have to be used together in the solution.  We can see from the eight algebraic equations that each corner square is in three equations (i.e., a, c, g, i), the centre square is in four equations (i.e., e) and the remaining middle row squares are in two equations (i.e., b, d, f, h).

We can see from the eight equations using the numbers 1 to 9, that 5 is the only number which appears in four equations, thus 5 must be the centre square, E:

   
 5 
   


We can also see that 9, 7, 3, and 1 only appear twice each in these eight equations, thus they must be B, D, F, and H. We can also see that 9 and 1 have to be in an equation together, as do 7 and 3. Therefore, they are restricted to the following arrangements (and their rotations about the centre square):

 9 
357
 1 
 9 
753
 1 

The remaining numbers 2, 4, 6, and 8 must therefore be in the corners, i.e., A, C, G and I. Checking the eight algebraic equations, we can see that this is possible, as all of these numbers each appear three times in the eight equations using the numbers 1 to 9. As these numbers are in the corners, they all appear in one equation with each of the other corner numbers. We can also see that 2 is in equations with 9, 7, and 5; 4 is in equations with 9, 5 and 3; 6 is in equations with 7, 5 and 1; and 8 is in equations with 1, 3 and 5. Therefore, the only following arrangements can be formed (and their rotations about the centre square):

This is the same arrangement, reflected through B, E and H. Thus, 15 is the only sum, and this sum can only be attained using the above eight equations of three numbers using the numbers 1 to 9. Due to the symmetry of a 3 by 3 grid, reflecting or rotating the solution can form 8 different arrangements of this solution:

We have therefore proved that there is only one magic square that can be made from the numbers 1 to 9 inclusive (subject to rotations and reflections of the square).

But earlier, at step * we could have used more algebra. Think about all of the rows, columns and diagonals through the centre square. So, if s is the sum, then
A + E + I = s
B + E + H = s
C + E + G = s
D + E + F = s

Add all of these up and you’ll get

(A + D + G) + (B + E + H) + (C + F + I) + 3E = 4s.

Now each term in the brackets equals s as they are the column sums. So
s + s + s + 3E = 4s.

This means that 3E = s. And since s = 15, then e = 5.

Now one way to make 15 is to 1 + 5 + 9, so suppose that 1, 5, 9 are down the main diagonal.

Then B + C = 14 so {B, C} = {5, 9} or {6, 8}. But we’ve already used 5 and 9 so {B, C} = {6, 8}. This presents a problem. If C = 6 then G = 0 and if C = 8, then G = -2. This can only mean that 1, 5, 9 are not on the main diagonal.

Lay off of the morning math worksheets and try out some of these magical math puzzles that show students how math can be enchanting!

1. Calendar Magic 9
Impress your friends with this math multiplication magic trick from Murderous Maths! Kids tell a friend to put a square around 9 numbers on a calendar ( 3 x 3 box). Then, they say they can find the sum of the 9 numbers within the square in a flash! Abracadabra and alakazam! All they do is multiply the number in the center of the square by 9 and presto! They magically have the answer!

2. Shoe Math Magic
Multiply shoe size by 5 (must be a whole number, round up if you have to)
Add 50
Multiply by 20
Add 1015 (change each year / next year 1016)
Subtract the year you were born
First digit = shoe size
Last 2 digits = age

3. Magic Square
A Magic Square is a great tactile, thinking game for kids, that has them rearrange three numerals (horizontal, vertical, and diagonal) so they all equal one sum, a magical number! I was inspired by Love 2 Learn 2 Day’s milk cap magic square, so I made my own! Kids love the use of milk caps because they can slide and glide them around on a flat table top. Magic Squares are also a good way for kids to improve their addition skills using a group of addends, three whole numbers in an equation.

4. Perimeter Magic Triangles
Perimeter Magic Triangles
 help children improve their addition skills using three addends! All you need are 6 milk caps labeled 1-6. Kids slide the milk caps around, forming a triangle. Their goal is to have all 3 sides add up to equal the same sum.

5. Toothpick Math Puzzles
Geometric toothpick puzzles that help develop problem solving and critical thinking skills.
Lists of puzzles to try:
Puzzle Playground
Planet Seed
Toothpick Triangles

6. Art in Numbers
Practice your multiplication tables by creating grid paper designs from Sharynideas! Kids identify patterns in their multiplication tables. When they identify a repeating pattern, they create art! Check out the activity here! Also, visit NRich Math for a slightly different way to create designs from your times tables!

7. Walk Through Paper
Can you walk through a hole in an 8.5×11″ sheet of paper? Pass out a sheet of paper to all of your students and see if they can figure out how to cut a hole large enough for them to fit through. Then, show them this magical trick! Visit Pleacher or The Math Lab for the how to (and a free printable with lines to cut the “perfect” hole)! Afterwards, stretch your paper out and try to find the area and perimeter of your paper! How did it change? For younger students, this project can tie into a basic measuring unit. You can even fit through an index card! Click here for details!

8. Domino Math Puzzles
NRich Math has LOTS of fun math puzzles to help develop kids’ problem solving skills. All they need is a set of dominoes! Click here for the full list! Check out domino magic squares and rectangles here! If you don’t have dominoes, you can find printable versions online.
Featured above:
Domino Multiplication
Domino Magic Square (Rows, columns, and diagonal share the same sum)
Domino Magic Window (All sides equal same sum)

9. Subtraction Squares
Kids choose 4 numbers (anything they’d like) to write on the vertices of the larger square. Then, they just subtract (corner to corner). They write the differences on the next largest square’s corners (where it meets in between on the line) and so on…

The big surprise is their final square! Like magic, all of the corners are the exact same numeral! Inference: See if kids can predict what will be their mystifying number before solving all their squares!

10. X=
Magic Math Trick (x=2)
1. Think of a whole number 1 through 10
2. Double it!
3. Add 4
4. Divide by 2
5. Subtract the original number
Is the numeral 2?!
Visit The Math Lab for the algebra behind this math magic!

Magic Math Trick (x=18)
1. Choose a number, any number!
2. Multiply the number by 100.
3. Subtract the original number from the answer.
4. Add the digits in your answer.
Is your number 18?!

Example:
5
5 x 100 = 500
500 – 5 = 495
4 + 9 + 5 = 18

Now, let’s try a larger number!
1,467
1,467 x 100 = 146,700
146,700 – 1,467 = 145,233
1 + 4 + 5 + 2 + 3 + 3 = 18
It’s magic!

Erin Bittman is a fashion designer turned teacher. Check out her blog E is for Explore!

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