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?
We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4
These gnomons appear to have more than a passing connection with
the Fibonacci sequence. This problem ask you to investigate some of
List any 3 numbers. It is always possible to find a subset of
adjacent numbers that add up to a multiple of 3. Can you explain
why and prove it?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
Can you use the diagram to prove the AM-GM inequality?
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
This article for teachers describes several games, found on the
site, all of which have a related structure that can be used to
develop the skills of strategic planning.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Can you describe this route to infinity? Where will the arrows take you next?
Find some examples of pairs of numbers such that their sum is a
factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and
16 is a factor of 48.
What is the volume of the solid formed by rotating this right
angled triangle about the hypotenuse?
Choose any 3 digits and make a 6 digit number by repeating the 3
digits in the same order (e.g. 594594). Explain why whatever digits
you choose the number will always be divisible by 7, 11 and 13.
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
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?
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nĀ² Use the diagram to show that any odd number is the difference of two squares.
A game for 2 players
A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
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
Can all unit fractions be written as the sum of two unit fractions?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
A collection of games on the NIM theme
Start with any number of counters in any number of piles. 2 players
take it in turns to remove any number of counters from a single
pile. The winner is the player to take the last counter.
Jo has three numbers which she adds together in pairs. When she
does this she has three different totals: 11, 17 and 22 What are
the three numbers Jo had to start with?”
Can you find the area of a parallelogram defined by two vectors?
It would be nice to have a strategy for disentangling any tangled
Can you tangle yourself up and reach any fraction?
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?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
An account of some magic squares and their properties and and how to construct them for yourself.
Janine noticed, while studying some cube numbers, that if you take
three consecutive whole numbers and multiply them together and then
add the middle number of the three, you get the middle number. . . .
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
To avoid losing think of another very well known game where the
patterns of play are similar.