The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.
A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.
An article which gives an account of some properties of magic squares.
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. . . .
Take a number, add its digits then multiply the digits together, then multiply these two results. If you get the same number it is an SP number.
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. . . .
If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.
Relate these algebraic expressions to geometrical diagrams.
Can you use the diagram to prove the AM-GM inequality?
Solve this famous unsolved problem and win a prize. Take a positive integer N. If even, divide by 2; if odd, multiply by 3 and add 1. Iterate. Prove that the sequence always goes to 4,2,1,4,2,1...
In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.
Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Follow the hints and prove Pick's Theorem.
The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!
A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
Here the diagram says it all. Can you find the diagram?
Four jewellers share their stock. Can you work out the relative values of their gems?
In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.
This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Some diagrammatic 'proofs' of algebraic identities and inequalities.
Explore a number pattern which has the same symmetries in different bases.
You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?
Can you make sense of these three proofs of Pythagoras' Theorem?
Have a go at being mathematically negative, by negating these statements.
These proofs are wrong. Can you see why?
This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.
An introduction to some beautiful results of Number Theory
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?
Can you find the value of this function involving algebraic fractions for x=2000?
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.
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. . . .
With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.
A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?