Creating and manipulating expressions and formulae

Can you find a rule which relates triangular numbers to square numbers?

Look at the times that Harry, Christine and Betty take to pack boxes when working in pairs, to find how fast Christine can pack boxes by herself.

Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.

Can you use the diagram to prove the AM-GM inequality?

Choose any four consecutive even numbers. Multiply the two middle numbers together. Multiply the first and last numbers. Now subtract your second answer from the first. Try it with your own numbers. Why is the answer always 8?

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

Show that all pentagonal numbers are one third of a triangular number.

What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?