Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
Here the diagram says it all. Can you find the diagram?
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.
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.
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
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 serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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.
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
Tom writes about expressing numbers as the sums of three squares.
Follow the hints and prove Pick's Theorem.
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?
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. . . .
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
These proofs are wrong. Can you see why?
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?
Have a go at being mathematically negative, by negating these statements.
Can you invert the logic to prove these statements?
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.
Can you rearrange the cards to make a series of correct mathematical statements?
Sort these mathematical propositions into a series of 8 correct statements.
Can you work through these direct proofs, using our interactive proof sorters?
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.
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
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.
There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?
Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.
Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?
Which of these roads will satisfy a Munchkin builder?
An introduction to some beautiful results of Number Theory
A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .
We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.
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.
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 of two or more cubes.
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
Explore a number pattern which has the same symmetries in different bases.
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.
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
Some diagrammatic 'proofs' of algebraic identities and inequalities.
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.
Can you discover whether this is a fair game?
Can you explain why a sequence of operations always gives you perfect squares?
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.
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!
An article which gives an account of some properties of magic squares.
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?
If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.
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.