Euler's Squares

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

Odd Differences

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

Peeling the Apple or the Cone That Lost Its Head

How much peel does an apple have?

Funnel

Stage: 4 Challenge Level:

See the Hint section for detailed support for students, but one of the main aims of this problem is to use a spreadsheet to open up a problem that is otherwise rather tedious to pursue by calculation. Students may need help thinking through the calculation required and the spreadsheet commands (formulae) necessary to achieve that. This is a challenging problem.

The relationship between surface area and volume enclosed is an important theme for able students at Stage 4 and using IT effectively should be part of a good student's repertoire of problem solving approaches.

With able students it is good to draw out that the ratio between the volume and the surface area will change with scale, but the optimum shape for the cone will not.
Perhaps reasoning in terms of units : measurements in a different unit of length would change all the numbers (how ?) but not the height-radius ratio (or cone angle) where the surface area takes its minimum value.