The prompt designed by **Ann Macdonald **(a maths teacher in Brighton, UK) is made up of two equations. The picture shows the initial questions and observations from year 7 students. They have tried to identify, extend and generalise a pattern and have wondered about other cases (such as, negative numbers and cubes). When the square is in the form (10*n* + 5)^{2} students can verify the equation is true for different types of values of *n*. From the general form of the equation (10*n* + 5)^{2} = *n* x (*n* + 1) x 100 + 25

they can show that both sides equal 100*n*^{2} + 100*n* + 25.

A more open prompt uses only the second equation:

45^{2} = 4 x 5 x 100 + 25

This might lead to a longer period of exploration and different conjectures. One conjecture, which turns out to be false, is that the form of the equation works for all double-digit numbers (for example, 24^{2} = 2 x 4 x 100 + 16 with the addend being the square of the '4').

In his book *Getting the Buggers to Add Up*, **Mike Ollerton** discusses the prompt in the following form:

(115)^{2} = 11 x 12[25]

He explains that the '25' in the square brackets emerges from 5^{2}^{ }and this forms the tens and units digits when combined with the product of 11 and 12. Thus the answer becomes 13 225. In fact, placing the '25' at the end is equivalent to multiplying the product of 11 and 12 by 100 in the original prompt. Mike explored the prompt with a fellow participant at a conference session. They asked* why* the prompt worked and whether it would work in a similar way when squaring numbers that end in six, seven or any other digit.

You can follow Ann Macdonald on twitter **@****M****ckyntyre**. Mike Ollerton writes widely about ideas for teaching maths and can be contacted through his website. He is on twitter @MichaelOllerton.