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. . . .

You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. How should you distribute the balls between the two. . . .

To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize?

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

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?

Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.

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.

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

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

A manager of a forestry company has to decide which trees to plant. What strategy for planting and felling would you recommend to the manager in order to maximise the profit?

A car is travelling along a dual carriageway at constant speed. Every 3 minutes a bus passes going in the opposite direction, while every 6 minutes a bus passes the car travelling in the same. . . .

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

A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . .

Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever. . . .

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

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.

It is possible to identify a particular card out of a pack of 15 with the use of some mathematical reasoning. What is this reasoning and can it be applied to other numbers of cards?

bioNRICH is the area of the stemNRICH site devoted to the mathematics underlying the study of the biological sciences, designed to help develop the mathematics required to get the most from your. . . .

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

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

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

Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?

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

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. . . .

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

How do you write a computer program that creates the illusion of stretching elastic bands between pegs of a Geoboard? The answer contains some surprising mathematics.

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

Christmas trees are planted in a rectangular array of 10 rows and 12 columns. The farmer chooses the shortest tree in each of the columns... the tallest tree from each of the rows ... Which is. . . .

At Holborn underground station there is a very long escalator. Two people are in a hurry and so climb the escalator as it is moving upwards, thus adding their speed to that of the moving steps. . . .

The builders have dug a hole in the ground to be filled with concrete for the foundations of our garage. How many cubic metres of ready-mix concrete should the builders order to fill this hole to. . . .

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. . . .

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

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

Your school has been left a million pounds in the will of an ex- pupil. What model of investment and spending would you use in order to ensure the best return on the money?

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

A brief video explaining the idea of a mathematical knot.

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

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?

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

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

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. . . .

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.

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. . . .

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. . . .

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.

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.

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?

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?