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?

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.

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.

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?

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.?

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

How long does it take to brush your teeth? Can you find the matching length of time?

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

Can you find all the different triangles on these peg boards, and find their angles?

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

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

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

The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . .

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

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

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

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

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

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

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?

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

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. . . .

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?

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

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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

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.

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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

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?

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

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....?

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?

Try out the lottery that is played in a far-away land. What is the chance of winning?

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

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

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?

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

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?

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

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

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

This is a variation of sudoku which contains a set of special clue-numbers. Each set of 4 small digits stands for the numbers in the four cells of the grid adjacent to this set.

An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.

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?

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?

When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?

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