Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

In how many different ways can I colour the five edges of a pentagon red, blue and green so that no two adjacent edges are the same colour?

In how many ways can you fit all three pieces together to make shapes with line symmetry?

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

How many different symmetrical shapes can you make by shading triangles or squares?

An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?

A game for 2 people. Take turns joining two dots, until your opponent is unable to move.

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.

There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you. . . .

Can you make a tetrahedron whose faces all have the same perimeter?

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.

If you move the tiles around, can you make squares with different coloured edges?

I found these clocks in the Arts Centre at the University of Warwick intriguing - do they really need four clocks and what times would be ambiguous with only two or three of them?

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?

Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet?

How can visual patterns be used to prove sums of series?

Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . .

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.

A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

To avoid losing think of another very well known game where the patterns of play are similar.

This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo!

The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.

Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible?

A cube is made from smaller cubes, 5 by 5 by 5, then some of those cubes are removed. Can you make the specified shapes, and what is the most and least number of cubes required ?

What can you see? What do you notice? What questions can you ask?

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

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.

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the. . . .

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

How many moves does it take to swap over some red and blue frogs? Do you have a method?

This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . .