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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.
Which of these triangular jigsaws are impossible to finish?
Use this interactivity to sort out the steps of the proof of the formula for the sum of an arithmetic series. The 'thermometer' will tell you how you are doing
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. . . .
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. . . .
A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Prove that if n is a triangular number then 8n+1 is a square number. Prove, conversely, that if 8n+1 is a square number then n is a triangular number.
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?
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!
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 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?
By proving these particular identities, prove the existence of general cases.
Show that if three prime numbers, all greater than 3, form an arithmetic progression then the common difference is divisible by 6. What if one of the terms is 3?
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 variables.
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Can you work out where the blue-and-red brick roads end?
Some diagrammatic 'proofs' of algebraic identities and inequalities.
Can you discover whether this is a fair game?
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.
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.
Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.
We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
An article which gives an account of some properties of magic squares.
Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.
Tom writes about expressing numbers as the sums of three squares.
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.
Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Follow the hints and prove Pick's Theorem.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
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 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?
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
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?
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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.
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. . . .
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...
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 complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.
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.
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 looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
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?