During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

A challenging activity focusing on finding all possible ways of stacking rods.

Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.

How many different symmetrical shapes can you make by shading triangles or squares?

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

On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?

Use the clues about the symmetrical properties of these letters to place them on the grid.

Stuart's watch loses two minutes every hour. Adam's watch gains one minute every hour. Use the information to work out what time (the real time) they arrived at the airport.

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

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

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

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.?

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?

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?

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

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?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

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

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

Can you find all the different ways of lining up these Cuisenaire rods?

Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

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?

This activity investigates how you might make squares and pentominoes from Polydron.

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?

What is the date in February 2002 where the 8 digits are palindromic if the date is written in the British way?

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.

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

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

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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

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?

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?

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?

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

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?

In this matching game, you have to decide how long different events take.

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

The pages of my calendar have got mixed up. Can you sort them out?

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

In how many ways can you stack these rods, following the rules?

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

Have a go at balancing this equation. Can you find different ways of doing it?

What happens when you round these three-digit numbers to the nearest 100?

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

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

Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?

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