Conjecturing and generalising

problem
Odd squares

Age

7 to 11

Challenge level

Think of a number, square it and subtract your starting number. Is the number you're left with odd or even? How do the images help to explain this?
problem
Pair products

Age

14 to 16

Challenge level

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
problem
Picturing square numbers

Age

11 to 14

Challenge level

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
problem
Picturing triangular numbers

Age

11 to 14

Challenge level

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
problem
Generally geometric

Age

16 to 18

Challenge level

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
problem
Dotty triangles

Age

11 to 14

Challenge level

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
problem
Squares, squares and more squares

Age

11 to 14

Challenge level

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?
problem
Taking steps

Age

7 to 11

Challenge level

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
problem
Triangles and petals

Age

14 to 16

Challenge level

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?
problem
Farey sequences

Age

11 to 14

Challenge level

There are lots of ideas to explore in these sequences of ordered fractions.