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Square Number Surprises

Age 14 to 16 Challenge Level:

You're probably very familiar with the sequence of Square Numbers: $$1, 4, 9, 16, 25...$$
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?

With thanks to Don Steward, whose ideas formed the basis of this problem.