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 ?

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.

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

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.

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?

In this problem we see how many pieces we can cut a cube of cheese into using a limited number of slices. How many pieces will you be able to make?

A visualisation problem in which you search for vectors which sum to zero from a jumble of arrows. Will your eyes be quicker than algebra?

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?

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. . . .

Small circles nestle under touching parent circles when they sit on the axis at neighbouring points in a Farey sequence.

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

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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?

Imagine a rectangular tray lying flat on a table. Suppose that a plate lies on the tray and rolls around, in contact with the sides as it rolls. What can we say about the motion?

Takes you through the systematic way in which you can begin to solve a mixed up Cubic Net. How close will you come to a solution?

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?

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

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

Show that all pentagonal numbers are one third of a triangular number.

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

Can you find a rule which connects consecutive triangular numbers?

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

A circular plate rolls inside a rectangular tray making five circuits and rotating about its centre seven times. Find the dimensions of the tray.

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

A bicycle passes along a path and leaves some tracks. Is it possible to say which track was made by the front wheel and which by the back wheel?

A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.

Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?

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?

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.

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

Use the diagram to investigate the classical Pythagorean means.

See if you can anticipate successive 'generations' of the two animals shown here.

Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.

Can you find a rule which relates triangular numbers to square numbers?

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

On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

What 3D shapes occur in nature. How efficiently can you pack these shapes together?

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. . . .

A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?

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?

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!