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?”
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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. . . .
What is the value of the integers a and b where sqrt(8-4sqrt3) =
sqrt a - sqrt b?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Can you use the diagram to prove the AM-GM inequality?
For which values of n is the Fibonacci number fn even? Which
Fibonnaci numbers are divisible by 3?
Generalise this inequality involving integrals.
The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles
have radii 1 and 2 units respectively. What about triangles with an
inradius of 3, 4 or 5 or ...?
Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Can you find the area of a parallelogram defined by two vectors?
First of all, pick the number of times a week that you would like
to eat chocolate. Multiply this number by 2...
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?
Can you show that you can share a square pizza equally between two
people by cutting it four times using vertical, horizontal and
diagonal cuts through any point inside the square?
Beautiful mathematics. Two 18 year old students gave eight
different proofs of one result then generalised it from the 3 by 1
case to the n by 1 case and proved the general result.
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
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.
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. . . .
Generalise the sum of a GP by using derivatives to make the
coefficients into powers of the natural numbers.
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0
what can you say about the triangle?
Given that a, b and c are natural numbers show that if sqrt a+sqrt
b is rational then it is a natural number. Extend this to 3
An article which gives an account of some properties of magic squares.
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
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?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Find the vertices of a pentagon given the midpoints of its sides.
Show that for any triangle it is always possible to construct 3
touching circles with centres at the vertices. Is it possible to
construct touching circles centred at the vertices of any polygon?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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 you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
Make and prove a conjecture about the cyclic quadrilateral
inscribed in a circle of radius r that has the maximum perimeter and the maximum area.
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.
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 loser is the player who takes the last counter.
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 describe this route to infinity? Where will the arrows take you next?
A game for 2 players
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
To avoid losing think of another very well known game where the
patterns of play are similar.
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Equal touching circles have centres on a line. From a point of this
line on a circle, a tangent is drawn to the farthest circle. Find
the lengths of chords where the line cuts the other circles.
If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.
Your data is a set of positive numbers. What is the maximum value
that the standard deviation can take?
These gnomons appear to have more than a passing connection with
the Fibonacci sequence. This problem ask you to investigate some of
Here explore some ideas of how the definitions and methods of calculus change if you integrate or differentiate n times when n is not a whole number.