BLUEBIRD MATH CIRCLE �

17 May 2023

Mark Saul, Ph.D.

“Think deeply of simple things.”

-- Arnold Ross

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MAGIC SQUARES

LAST TIME WE MET:

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We constructed 3x3 magic squares with the entries {1,2,3,4,5,6,7,8,9}.

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Here is one result:

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All the rows add up to 15.

All the columns add up to 15.

Each of the diagonals adds up to 15.

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The row sums, column sums, and diagonal sums are all equal.

This is a magic square.

All the rows add up to 15.

All the columns add up to 15.

Each of the diagonals adds up to 15.

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The row sums, column sums, and diagonal sums are all equal.

This is a magic square.

TASK 1: This magic square contains the numbers 1 through 9.

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Can you construct a magic square containing the numbers 2 through 10?

Here are four magic squares.

You can check that in fact they are magic.

How are they different?

How are they the same?

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The Magic Constant Must be 15

Why? Well, if we add up the sums of the three rows, we must get:

1+2+3+4+5+6+7+8+9 = 45 (except the addends will be in a different order) .

And 45/3 = 15.

If the magic constant (sum of each row) is S,

then:

a+b+c = S

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d+e+f = S

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g+h+I = S

So S+S+s = 3S = a+b+c+d+e+f+g+h+I,

which is 1+2+3+4+5+6+7+8+9, = 45

(with the addends in some order.

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Thus 3S = 45, and S = 15.

The middle number must be 5

Why?

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Well, try to put some other number in the middle, say 6.

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Then the numbers in the two opposite corners, p and q must add up to 15-6 = 9

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The middle number must be 5

Why?

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Well, try to put some other number in the middle, say 6.

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Then the numbers in the two opposite corners, p and q must add up to 15-6 = 9

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But so must r + s and t+u and v+w. Each pair must add up to 9.

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That means that all the nine numbers will add up to 9+9+9+9+6 = 42.

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But they don’t! They add up to 45. So 6 in the middle must be wrong.

The middle number must be 5

In algebra;

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The sums along the arrows below are all 15, and the arrows cover each number once EXCEPT the middle number, which is counted FOUR times (once for each arrow).

a + e + f = 15

b + e + h = 15

d + e + f = 15

g + e + c = 15

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So

a + b + c + d + 4e + f + g +h +i = 60

But a + b + c + d +e + f + g +h +i = 45

So 3e = 15 and e = 5

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Challenge (for later): Can you prove that the numbers in the corners must all be even?

Hints: What can you say about the parity of the sum of two even numbers?

Of two odd numbers?

Of an odd number and an even number?

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If three numbers add up to 15, what can you say about the parity of these numbers?

The property of being odd or being even is called the parity of a number.

The numbers in the corners must all be even

Why?

(This is not so simple)

Suppose we put a 6 next to the five.

Then we must put a 4 opposite it.

(Remember, the row sum must be 15.)

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What can go in the upper left?

If it’s an even number, then the lower right must also be even.

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The numbers in the corners must all be even

Why?

(This is not so simple)

Suppose we put a 6 next to the five.

Then we must put a 4 opposite it.

(Remember, the row sum must be 15.)

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What can go in the upper left?

If it’s an even number, then the lower right must also be even.

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For example, if the upper left is 2, then the lower right must be 8 (to make the diagonal 15).

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The numbers in the corners must all be even

Why?

(This is not so simple)

Suppose we put a 6 next to the five.

Then we must put a 4 opposite it.

(Remember, the row sum must be 15.)

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What can go in the upper left?

If it’s an even number, then the lower right must also be even.

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For example, if the upper left is 2, then the lower right must be 8 (to make the diagonal 15).

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This is trouble!

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The numbers in the corners must all be even

Why?

(This is not so simple)

What can go in the upper left?

If it’s an even number, then the lower right must also be even.

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For example, if the upper left is 2, then the lower right must be 8 (to make the diagonal 15).

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This is trouble!

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Look at the first row. It has two odd numbers and one even number. But the sum of two odds and one even is going to be even!

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The number 15 is NOT even!

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The numbers in the corners must all be even

Why?

(This is not so simple)

What can go in the upper left?

If it’s an even number, then the lower right must also be even.

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For example, if the upper left is 2, then the lower right must be 8 (to make the diagonal 15).

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This is trouble!

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Look at the first row. It has two odd numbers and one even number. But the sum of two odds and one even is going to be even!

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The number 15 is NOT even!

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The numbers in the corners must all be even

Why?

(This is not so simple)

It’s even worse if we put an odd number (say 1) above the 6.

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Then the lower right must be odd (it’s 9).

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Now look at the rightmost column. It needs two even numbers. So the top right must be even.

The numbers in the corners must all be even

Why?

(This is not so simple)

It’s even worse if we put an odd number (say 1) above the 6.

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Then the lower right must be odd (it’s 9).

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Now look at the rightmost column. It needs two even numbers. So the top right must be even.

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Now the top row needs another even number.

The numbers in the corners must all be even

Why?

(This is not so simple)

It’s even worse if we put an odd number (say 1) above the 6.

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Then the lower right must be odd (it’s 9).

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Now look at the rightmost column. It needs two even numbers. So the top right must be even.

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Now the top row needs another even number. And the middle column also needs another even number.

The numbers in the corners must all be even

Why?

(This is not so simple)

It’s even worse if we put an odd number (say 1) above the 6.

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Then the lower right must be odd (it’s 9).

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Now look at the rightmost column. It needs two even numbers. So the top right must be even.

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Now the top row needs another even number. And the middle column also needs another even number.

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That makes FIVE even numbers, but we only have FOUR: 2,4,6,8.

The numbers in the corners must all be even

Why?

(This is not so simple)

Placing just one even number on a ‘side’ square makes it impossible for the square to be magic.

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So all the even numbers must be in the corners.

Eight Magic Squares

(a) (b) (c) (d)

(e) (f) (g) (h)

Eight Magic Squares

One of the ways to describe the relationships among these squares is to say that they have the same rows and columns:

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For example, 6-1-8 always appears as a row or column. The 6-1-8 may appear backwards or upside down, but it never becomes 6-1-4

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They squares also have the same diagonals, but in different positions.

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(e) (f) (g) (h)

(a) (b) (c) (d)

Eight Magic Squares

(a) (b)

In fact, if you look at squares (a) and (b), you will see that you can obtain (b) from (a) by rotating (“twisting”) around the center by 90 degrees.

Eight Magic Squares

(a) (b) (c) (d)

(e) (f) (g) (h)

Which other squares can be obtained from (a) by rotating (twisting) about the center?

Eight Magic Squares

(a) (b) (c) (d)

(e) (f) (g) (h)

And what about the other squares? How can you obtain them from (a)?

Try this yourself!

Eight Magic Squares

(a) (b) (c) (d)

(e) (f) (g) (h)

Which other squares can be obtained from (a) by rotating (twisting) about the center?

Eight Magic Squares

(a) (b) (c) (d)

(e) (f) (g) (h)

And what about the other squares? How can you obtain them from (a)?

BACK FROM BREAKOUT

Eight Magic Squares

(a) (b) (c) (d)

(e) (f) (g) (h)

And what about the other squares? How can you obtain them from (a)?

Eight Magic Squares

(a) (b) (c) (d)

(e) (f) (g) (h)

Rotate (a) by 90 degrees to get (b).

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Rotate (a) by 180 degrees to get (c).

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Rotate (a) by 270 degrees to get (d).

Eight Magic Squares

(a) (b) (c) (d)

(e) (f) (g) (h)

But to get from (a) to ( e) we cannot just rotate.

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We must reflect (‘flip’) along the diagonal.

Eight Magic Squares

(a) (b) (c) (d)

(e) (f) (g) (h)

How can we get from (a) to (f)?

From (a) to (g)?

From (a) to (h)?

BREAKOUT

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Eight Magic Squares

(a) (b) (c) (d)

(e) (f) (g) (h)

How can we get from (a) to (f)?

From (a) to (g)?

From (a) to (h)?

BACK FROM BREAKOUT

Eight Magic Squares

(a) (b) (c) (d)

(e) (f) (g) (h)

How can we get from (a) to (f)?

From (a) to (g)?

From (a) to (h)?

Eight Magic Squares

(a) (b) (c) (d)

(e) (f) (g) (h)

A reflection (flip) in the middle column takes (a) onto (f).

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A reflection (flip) in the ‘other’ diagonal takes (a) onto (g).

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A reflection (flip) in the middle row takes (a) onto (h).

Eight Magic Squares

(a) (b) (c) (d)

(e) (f) (g) (h)

Notice that when we reflect, the line of reflection does not change.

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But when we rotate, every line changes.

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That is one way we can tell a rotation from a reflection.

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Are there other ways?

Thank you for your kind attention.

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ENJOY MATHEMATICS!

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Mark Saul, Ph.D.

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