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Euler's Squares

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

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DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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2-digit Square

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

What's Possible?

Age 14 to 16 Challenge Level:

Why do this problem?

This problem starts by asking students to find which numbers can be expressed as the difference of two square numbers, and then suggests some possible avenues for exploration. This can then be used as a springboard to generalisations and the use of algebra for justifications and proof. Along the way, students have the opportunity to make use of the important identity $a^2 - b^2 = (a + b)(a - b)$.

An alternative to this problem which some students may find more accessible is Hollow Squares.

Possible approach

This printable resource may be useful: What's Possible? worksheet.

You may wish to introduce the problem like this:
Ask students for a two-digit number under 30. Write it on the board and express it as the difference between two squares. Repeat several times.

Challenge students to find all numbers between 1 and 30 that can be expressed as the difference of two squares. Encourage them to find more than one solution where possible.

Once students have had time to find most of the answers, ask them to share what they found and list their answers on the board.
‘Have a look at the results so far. What do you notice?’

Give students a couple of minutes to talk to their partners before bringing the class back together.
Here are some of the conjectures and questions that may emerge:
• The difference between squares of consecutive numbers is always odd.
• The difference between squares of consecutive numbers is equal to the sum of the consecutive numbers.
• What happens to the difference when I square two numbers that differ by 2?
• Numbers that can be made in more than one way have lots of factors.
• When is the difference between two square numbers odd?
• And when is it even?
• Why are some numbers impossible to make?
• Can we predict which numbers are impossible to make? 

Alternatively, you could introduce the problem like this:
Arrange the class in groups of two or three, and hand each group one set from these cards. Ask them to work in their groups for the first few minutes, looking at what they notice about their three cards, whether they could create other cards that would belong in the same set, and what questions are prompted by them.
 
Next, bring the class together to share what was on their cards, and what questions occurred to them. Do any of their questions resolve themselves once they see someone else's cards? Make sure conjectures and questions are noted on the board.
 
One way to begin to resolve these conjectures and provoke new ones is to challenge students to find numbers between 1 and 30 which can be expressed as the difference between two square numbers. Encourage them to find more than one solution where possible, and draw attention to systematic ways of working and recording. Once everyone has had some time to find most of the answers, ask students to share what they have found and tabulate the answers on the board. Note down any new conjectures that emerge at this point.
 
Invite students to choose a line of enquiry, working on their own or in small groups. While they are working, circulate and look out for students who are beginning to justify the results they have noticed.

Bring the class back together and invite those students who have useful insights to share them with the class. You may wish to introduce an algebraic and a geometric approach to proving one particular conjecture: for example, that the difference between consecutive squares is always odd. A diagrammatic method for calculating the difference of two squares is explored in the problem Plus Minus.

Now give the class some time to work on proving their conjectures.

You may also wish to set the following challenge:
"In a while, I'm going to give you a number and ask you to quickly find one or more ways to write it as the difference of two squares, or to convince me that it can't be done. Can you develop a strategy that will help you do this?"

Your plenary could involve students presenting their findings to the rest of the class. Expect students to be clear and rigorous in their justifications. Encourage them to challenge any proofs that lack clarity and rigour, and suggest ways of improving them.

Key questions

What do you notice about the difference between squares of consecutive numbers?
What about the difference when I square two numbers which differ by 2? By 3? By 4 ...?
When is the difference between two square numbers odd?
And when is it even?
What do you notice about the numbers you CANNOT make?
Can you prove any of your findings?
 

Possible extension

Every odd prime number can be written as the difference of two squares. Prove that there is only one way to write an odd prime number as the difference of two squares, and that this is the difference of two successive squares.

120 can be written as the difference of the squares of whole numbers in the following ways:
$$31^2-29^2$$
$$17^2-13^2$$
$$13^2-7^2$$
$$11^2-1^2$$
Can you find all the possible ways to write 924 as the difference of the squares of two whole numbers? Can you prove that you have found them all?

*REALLY CHALLENGING*
Can you describe a way for working out how many ways there are to write any number as the difference of two squares?
 

Possible support

Encourage students to list their results in a systematic way, gathering together in separate columns numbers made from squaring two numbers which differ by $1$, $2$, $3$... as started below:

  1 2 3
1 $1^2-0^2$    
2      
3 $2^2-1^2$    
4   $2^2-0^2$  
5 $3^2-2^2$    
6      
7 $4^2-3^2$    
8   $3^2-1^2$  
9 $5^2-4^2$   $3^2-0^2$