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?

Peaches in General

Stage: 4 Challenge Level:

There's a problem which goes like this :

A monkey had some peaches.

He ate half of them plus one more.

On the second day, he ate half of the rest plus one more.

On the third day, he ate half of the rest plus one more again.

On the fourth day, he found there was only one left.

How many did he have at the beginning?

By the time you reach Stage 4 you will start to feel that problems like this are examples or instances of something more general.

Try this problem as it is and then generalise your solution as far as you think you can.

Using a spreadsheet may be a help, but you can judge that for yourself.