Relate these algebraic expressions to geometrical diagrams.
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. . . .
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
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.
Can you make sense of these three proofs of Pythagoras' Theorem?
Freddie Manners, of Packwood Haugh School in Shropshire solved an alphanumeric without using the extra information supplied and this article explains his reasoning.
Kyle and his teacher disagree about his test score - who is right?
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
Can you use the diagram to prove the AM-GM inequality?
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?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.
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.
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.
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.
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.
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. . . .
This is the second article on right-angled triangles whose edge lengths are whole numbers.
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.
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?
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.
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?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
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?
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?
What can you say about the common difference of an AP where every term is prime?
Can you rearrange the cards to make a series of correct mathematical 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.
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.
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a 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!
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
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.
Follow the hints and prove Pick's Theorem.
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.
Three frogs started jumping randomly over any adjacent frog. Is it possible for them to finish up in the same order they started?
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?
Here the diagram says it all. Can you find the diagram?
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
Have a go at being mathematically negative, by negating these statements.
What fractions can you divide the diagonal of a square into by simple folding?
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. . . .