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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?
Some diagrammatic 'proofs' of algebraic identities and inequalities.
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. . . .
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
Prove Pythagoras Theorem using enlargements and scale factors.
Mark a point P inside a closed curve. Is it always possible to find two points that lie on the curve, such that P is the mid point of the line joining these two points?
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?
This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .
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. . . .
Can you discover whether this is a fair game?
We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
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. . . .
This is an interactivity in which you have to sort into the correct order the steps in the proof of the formula for the sum of a geometric series.
Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.
Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.
Can you rearrange the cards to make a series of correct mathematical statements?
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Investigate circuits and record your findings in this simple introduction to truth tables and logic.
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.
Here the diagram says it all. Can you find the diagram?
An article about the strategy for playing The Triangle Game which appears on the NRICH site. It contains a simple lemma about labelling a grid of equilateral triangles within a triangular frame.
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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.
Can you invert the logic to prove these statements?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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. . . .
An introduction to some beautiful results of Number Theory
These proofs are wrong. Can you see why?
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.
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.
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.
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .
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?
Can you work through these direct proofs, using our interactive proof sorters?
Have a go at being mathematically negative, by negating these statements.
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.
This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.
Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .
Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.
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.
Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.
Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...
Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum orf two or more cubes.
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...
This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.
An article which gives an account of some properties of magic squares.