Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?
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. . . .
A 3x3x3 cube may be reduced to unit cubes in six saw cuts. If after every cut you can rearrange the pieces before cutting straight through, can you do it in fewer?
Which of the following cubes can be made from these nets?
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.
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
Can you mentally fit the 7 SOMA pieces together to make a cube? Can you do it in more than one way?
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?
These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?
This is the first article in a series which aim to provide some insight into the way spatial thinking develops in children, and draw on a range of reported research. The focus of this article is the. . . .
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?
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?
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?
Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.
A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?
Can you discover whether this is a fair game?
To avoid losing think of another very well known game where the patterns of play are similar.
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube?
Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube so that the surface area of the remaining solid is the same as the surface area of the original?
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?
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.
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?
Can you mark 4 points on a flat surface so that there are only two different distances between them?
How many different symmetrical shapes can you make by shading triangles or squares?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?
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. . . .
A useful visualising exercise which offers opportunities for discussion and generalising, and which could be used for thinking about the formulae needed for generating the results on a spreadsheet.
Can you find a way of representing these arrangements of balls?
A game for 2 players
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.
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.
What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?
How can you make an angle of 60 degrees by folding a sheet of paper twice?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?