### Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

### Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

### Areas and Ratios

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

# Square Number Surprises

##### Stage: 4 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.