Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .
Can you discover whether this is a fair game?
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
A blue coin rolls round two yellow coins which touch. The coins are
the same size. How many revolutions does the blue coin make when it
rolls all the way round the yellow coins? Investigate for a. . . .
Some puzzles requiring no knowledge of knot theory, just a careful
inspection of the patterns. A glimpse of the classification of
knots and a little about prime knots, crossing numbers and. . . .
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?
We are given a regular icosahedron having three red vertices. Show
that it has a vertex that has at least two red neighbours.
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
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?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
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 find a rule which relates triangular numbers to square numbers?
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. . . .
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Can you describe this route to infinity? Where will the arrows take you next?
Simple additions can lead to intriguing results...
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
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?
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
A game for 2 players
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
Show that all pentagonal numbers are one third of a triangular number.
Can you find a rule which connects consecutive triangular numbers?
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?
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?
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.
If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.
A huge wheel is rolling past your window. What do you see?
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?
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.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
To avoid losing think of another very well known game where the
patterns of play are similar.
Have a go at this 3D extension to the Pebbles problem.
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
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?
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. . . .
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?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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. . . .
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?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
Can you maximise the area available to a grazing goat?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?