Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

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?

How far have these students walked by the time the teacher's car reaches them after their bus broke down?

Can you work out how long it will take for John to walk to and from school?

A rectangular plank fits neatly inside a square frame when placed diagonally. What is the length of the plank?

Can you work out how long Aimee would take to get up the escalator if she walked?

A walk is made up of diagonal steps from left to right, starting at the origin and ending on the x-axis. How many paths are there for 4 steps, for 6 steps, for 8 steps?

Heidi and Peter pass two signs which say how far their destination is. How long will it take them to get there?

This ladybird is taking a walk round a triangle. Can you see how much he has turned when he gets back to where he started?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Weekly Problem 44 - 2006

How long does it take an athelete to walk, cycle and run three miles if this is 10 minutes slower than cycling the whole distance?

These pieces of wallpaper need to be ordered from smallest to largest. Can you find a way to do it?

A Sudoku that uses transformations as supporting clues.

Plane 1 contains points A, B and C and plane 2 contains points A and B. Find all the points on plane 2 such that the two planes are perpendicular.

Resources to accompany Charlie's workshop at Walthamstow School for Girls.

This is a simple version of an ancient game played all over the world. It is also called Mancala. What tactics will increase your chances of winning?

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

How long will it take Mary and Nigel to wash an elephant if they work together?

What fraction of customers buy Kleenz after the advertising campaign?

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?

Have you ever noticed the patterns in car wheel trims? These questions will make you look at car wheels in a different way!

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

Use this information to work out whether the front or back wheel of this bicycle gets more wear and tear.

There are some water lilies in a lake. The area that they cover doubles in size every day. After 17 days the whole lake is covered. How long did it take them to cover half the lake?

With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?

Which bottles will hold the most lemonade for our picnic? How could we find out?

These watermelons have been entered into a competition. Use the information to work out the number of points each one was awarded.

The problems in this feature give you chance to consider different ways of showing (or representing) your thinking.

Sanjay Joshi, age 17, The Perse Boys School, Cambridge followed up the Madrass College class 2YP article with more thoughts on the problem of the number of ways of expressing an integer as the sum. . . .

How many different rhythms can you make by putting two drums on the wheel?

Two right-angled triangles are connected together as part of a structure. An object is dropped from the top of the green triangle where does it pass the base of the blue triangle?

The NRICH Stage 5 weekly challenges are shorter problems aimed at Post-16 students or enthusiastic younger students. There are 52 of them.

Weekly challenges are here for NRICH! To celebrate this event, we've collected a set of 20 essential problems for you to try.

Mr Ross tells truths or lies depending on the day of the week. Can you catch him out?

This article for teachers recounts the history of measurement, encouraging it to be used as a spring board for cross-curricular activity.

Can you use this information to estimate how much the different fruit selections weigh in kilos and pounds?

Can you work out how many spheres will balance a single pyramid?

Weighing the baby at the clinic was a problem. Can you work out the total weight of the baby, the nurse and me from the information given?

Use the number weights to find different ways of balancing the equaliser.

Different combinations of the weights available allow you to make different totals. Which totals can you make?

The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .

Consider these weird universes and ways in which the stick man can shoot the robot in the back.