Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
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?
What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?
Can you work through these direct proofs, using our interactive proof sorters?
Here the diagram says it all. Can you find the diagram?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
Can you discover whether this is a fair game?
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.
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.
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...
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
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. . . .
Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.
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.
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
Tom writes about expressing numbers as the sums of three squares.
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.
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.
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.
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?
The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .
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.
Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.
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.
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?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
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?
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!
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.
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.
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.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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 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.
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. . . .
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.
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. . . .
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
Have a go at being mathematically negative, by negating these statements.
Can you make sense of these three proofs of Pythagoras' Theorem?
Can you invert the logic to prove these statements?
Can the pdfs and cdfs of an exponential distribution intersect?