Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

How many trapeziums, of various sizes, are hidden in this picture?

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

What is the best way to shunt these carriages so that each train can continue its journey?

Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?

The Zargoes use almost the same alphabet as English. What does this birthday message say?

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

This article for primary teachers suggests ways in which to help children become better at working systematically.

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

How will you go about finding all the jigsaw pieces that have one peg and one hole?

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

My cousin was 24 years old on Friday April 5th in 1974. On what day of the week was she born?

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

An activity making various patterns with 2 x 1 rectangular tiles.

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

These practical challenges are all about making a 'tray' and covering it with paper.

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?