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?

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

Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?

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.

Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?

Can you use your powers of logic and deduction to work out the missing information in these sporty situations?

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?

Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.

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?

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

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.

This Sudoku, based on differences. Using the one clue number can you find the solution?

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.

Four small numbers give the clue to the contents of the four surrounding cells.

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 put the numbers 1 to 8 into the circles so that the four calculations are correct?

Can you find which shapes you need to put into the grid to make the totals at the end of each row and the bottom of each column?

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

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?

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.

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

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?

How many models can you find which obey these rules?

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?

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

This Sudoku puzzle can be solved with the help of small clue-numbers on the border lines between pairs of neighbouring squares of the grid.

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

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

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

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?

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

A Sudoku that uses transformations as supporting clues.

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

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

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

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?

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?

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

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

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

Investigate the different ways you could split up these rooms so that you have double the number.