How many moves does it take to swap over some red and blue frogs? Do you have a method?
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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?
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?
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?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
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?
A game for 2 players. Can be played online. One player has 1 red
counter, the other has 4 blue. The red counter needs to reach the
other side, and the blue needs to trap the red.
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.
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?
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?
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.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
How many different symmetrical shapes can you make by shading triangles or squares?
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?
In how many ways can you fit all three pieces together to make
shapes with line symmetry?
A huge wheel is rolling past your window. What do you see?
Can you mentally fit the 7 SOMA pieces together to make a cube? Can
you do it in more than one way?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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.
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
If you move the tiles around, can you make squares with different coloured edges?
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?
Can you describe this route to infinity? Where will the arrows take you next?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
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 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?
Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Have a go at this 3D extension to the Pebbles problem.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Is it possible to rearrange the numbers 1,2......12 around a clock
face in such a way that every two numbers in adjacent positions
differ by any of 3, 4 or 5 hours?
Mathematics is the study of patterns. Studying pattern is an
opportunity to observe, hypothesise, experiment, discover and
Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?
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?
Imagine you are suspending a cube from one vertex (corner) and
allowing it to hang freely. Now imagine you are lowering it into
water until it is exactly half submerged. What shape does the
surface. . . .
The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?
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?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
What is the shape of wrapping paper that you would need to completely wrap this model?
Can you find a way of representing these arrangements of balls?
When dice land edge-up, we usually roll again. But what if we
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?
This article for teachers discusses examples of problems in which
there is no obvious method but in which children can be encouraged
to think deeply about the context and extend their ability to. . . .
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Can you find a rule which connects consecutive triangular numbers?