NRICH: Square Number Surprises

There are unexpected discoveries to be made about square numbers…

You’re probably very familiar with the sequence of Square Numbers:


Below are some ideas to explore with square numbers.
For each one, try a few examples and see what you notice.

  • Add two consecutive square numbers and then subtract 1
  • Square any odd number, then subtract 1
  • Multiply two consecutive odd numbers and then add 1
  • Multiply two consecutive even numbers and then add 1

Can you explain what you’ve noticed?
Can you prove that it will always happen?

Can you come up with any Square Number Surprises of your own?

Age 14 to 16

Algebra & Pre-Algebra, Quadratic Equations
High School, Educator

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NRICH (University of Cambridge)

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