What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

If you had 36 cubes, what different cuboids could you make?

How many models can you find which obey these rules?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

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

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?

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

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.

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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

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.

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

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?

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

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?

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

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

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

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

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?

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

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

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

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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

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?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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

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

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

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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?

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

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