Conjecturing and generalising

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.
problem
Polite numbers

Age

16 to 18

Challenge level

A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?
problem
Fibonacci factors

Age

16 to 18

Challenge level

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
problem
Chocolate 2010

Age

14 to 16

Challenge level

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
problem
How many miles to go?

Age

11 to 14

Challenge level

How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?
problem
Mind reading

Age

11 to 14

Challenge level

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I know?