Exploring and noticing

problem
Sponge sections

Age

7 to 11

Challenge level

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.
problem
Special numbers

Age

11 to 14

Challenge level

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
problem
Fractional triangles

Age

7 to 11

Challenge level

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
problem
Farey sequences

Age

11 to 14

Challenge level

There are lots of ideas to explore in these sequences of ordered fractions.
problem
Flipping twisty matrices

Age

14 to 18

Challenge level

Investigate the transformations of the plane given by the 2 by 2 matrices with entries taking all combinations of values 0, -1 and +1.
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
Marvellous matrix

Age

7 to 11

Challenge level

Follow the directions for circling numbers in the matrix. Add all the circled numbers together. Note your answer. Try again with a different starting number. What do you notice?
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
Beads

Age

7 to 11

Challenge level

Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?
problem
Trebling

Age

7 to 11

Challenge level

Can you replace the letters with numbers? Is there only one solution in each case?