Creating and manipulating expressions and formulae

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

Find b where 3723(base 10) = 123(base b).

The square of a number is 12 more than the number itself. The cube of the number is 9 times the number. What is the number?

Quince, quonce and quance are three types of fruit. Can you work out the order of heaviness of the fruits?

Tina has chosen a number and has noticed something about its factors. What number could she have chosen? Are there multiple possibilities?

Can you explain why a sequence of operations always gives you perfect squares?

What is the total number of squares that can be made on a 5 by 5 geoboard?