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?
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Can you maximise the area available to a grazing goat?
A huge wheel is rolling past your window. What do you see?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable.
Decide which of these diagrams are traversable.
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.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
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?
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?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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?
Draw a square. A second square of the same size slides around the
first always maintaining contact and keeping the same orientation.
How far does the dot travel?
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
ABC is an equilateral triangle and P is a point in the interior of
the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP
must be less than 10 cm.
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
How many different symmetrical shapes can you make by shading triangles or squares?
Can you discover whether this is a fair game?
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
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
A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .
To avoid losing think of another very well known game where the
patterns of play are similar.
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?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
A game for 2 players
Can you describe this route to infinity? Where will the arrows take you next?
Can you find a rule which connects consecutive triangular numbers?
Show that all pentagonal numbers are one third of a triangular number.
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?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .
Four rods, two of length a and two of length b, are linked to form
a kite. The linkage is moveable so that the angles change. What is
the maximum area of the kite?
Seven small rectangular pictures have one inch wide frames. The
frames are removed and the pictures are fitted together like a
jigsaw to make a rectangle of length 12 inches. Find the dimensions
of. . . .
In this problem, we have created a pattern from smaller and smaller
squares. If we carried on the pattern forever, what proportion of
the image would be coloured blue?
You can move the 4 pieces of the jigsaw and fit them into both
outlines. Explain what has happened to the missing one unit of
ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP
: PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED.
What is the area of the triangle PQR?
Have a go at this 3D extension to the Pebbles problem.