On a clock the three hands - the second, minute and hour hands - are on the same axis. How often in a 24 hour day will the second hand be parallel to either of the two other hands?

Sometime during every hour the minute hand lies directly above the hour hand. At what time between 4 and 5 o'clock does this happen?

Can you explain why every year must contain at least one Friday the thirteenth?

If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?

A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?

Every day at noon a boat leaves Le Havre for New York while another boat leaves New York for Le Havre. The ocean crossing takes seven days. How many boats will each boat cross during their journey?

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

Suppose you are a bellringer. Can you find the changes so that, starting and ending with a round, all the 24 possible permutations are rung once each and only once?

Basic strategy games are particularly suitable as starting points for investigations. Players instinctively try to discover a winning strategy, and usually the best way to do this is to analyse. . . .

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

A brief video looking at how you can sometimes use symmetry to distinguish knots. Can you use this idea to investigate the differences between the granny knot and the reef knot?

The triathlon is a physically gruelling challenge. Can you work out which athlete burnt the most calories?

32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50 x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if possible.

This article for pupils gives an introduction to Celtic knotwork patterns and a feel for how you can draw them.

Christmas trees are planted in a rectangular array. Which is the taller tree, A or B?

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

In a league of 5 football teams which play in a round robin tournament show that it is possible for all five teams to be league leaders.

This article explores ths history of theories about the shape of our planet. It is the first in a series of articles looking at the significance of geometric shapes in the history of astronomy.

This article for students gives some instructions about how to make some different braids.

Fancy a game of cricket? Here is a mathematical version you can play indoors without breaking any windows.

The second in a series of articles on visualising and modelling shapes in the history of astronomy.

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .

The King showed the Princess a map of the maze and the Princess was allowed to decide which room she would wait in. She was not allowed to send a copy to her lover who would have to guess which path. . . .

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . .

Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?

Formulate and investigate a simple mathematical model for the design of a table mat.

How does the time of dawn and dusk vary? What about the Moon, how does that change from night to night? Is the Sun always the same? Gather data to help you explore these questions.

Build a scaffold out of drinking-straws to support a cup of water

Simple models which help us to investigate how epidemics grow and die out.

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .

The third installment in our series on the shape of astronomical systems, this article explores galaxies and the universe beyond our solar system.

A brief video explaining the idea of a mathematical knot.

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and record your findings in truth tables.

This article for students introduces the idea of naming knots using numbers. You'll need some paper and something to write with handy!

In this article for teachers, Alan Parr looks at ways that mathematics teaching and learning can start from the useful and interesting things can we do with the subject, including. . . .

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

PhysNRICH is the area of the StemNRICH site devoted to the mathematics underlying the study of physics