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Always Perfect

Age 14 to 18
Challenge Level


Why do this problem?

This problem offers an excellent context for observing, conjecturing and thinking about proof. It also offers an opportunity to discuss the relationship between geometrical and algebraic methods for representing how numbers behave - an engaging introduction to number theory.

Along the way, students have the opportunity of practising routine algebraic procedures. The first problem involves factorising a quadratic expression to get a perfect square, and the second problem involves factorising a quartic expression.

Possible approach

This activity featured in an NRICH Secondary webinar in March 2021.
Start by asking:
"Take two numbers that differ by 2, multiply them together and add 1. What were your answers?"
Write a selection of students' responses on the board.
"What do you notice?" The answer is always a square number. The answer is always the square of the number between the two chosen ones.
"Will this always happen? Can you explain why?"

Give students some time to discuss with their partner why the answers are always square numbers. Circulate and listen out for interesting insights.

Bring the class together and share any explanations they have found. Perhaps share Charlie's and Claire's representations from the problem if they haven't emerged.

"We've worked out what happens when you find the product of two numbers that differ by 2, and then add 1.

I'd like you to explain what happens when you:

  • find the product of two numbers that differ by 4, and then add 4
  • find the product of two numbers that differ by 6, and then add 9
  • find the product of two numbers that differ by ...

"Start by testing some numerical examples before trying to generalise using the representations we've looked at. Can you prove your results?"

Bring the class together to discuss how they would predict what they will have to add to the product of two numbers that differ by 8, 10, 12,..., 2k in order to get a square number.
The second part of the problem may only be suitable for 16-18 year olds, but there is a video in Getting Started which may help support younger students.  The problem can be tackled in a very similar way to the previous part, by discussing part (a) together before letting the class explore the rest of the questions.  

Key question

Is there a way to represent the product of the two numbers that will explain the patterns you noticed?

Possible support

Pair Products may be a suitable problem to attempt before this one.

Possible extension

Part 2 of the problem could be used as an extension activity.
Otherwise students could take a look at Common Divisor