Sprouts Explained
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 significant food for thought.
Breaking the Equation 'empirical Argument = Proof '
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.
Air Nets
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Geometry and Gravity 2
This is the second of two articles and discusses problems relating to the curvature of space, shortest distances on surfaces, triangulations of surfaces and representation by graphs.
Impossible Sandwiches
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.
Yih or Luk Tsut K'i or Three Men's Morris
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 knot arithmetic.
The Bridges of Konigsberg
Investigate how networks can be used to solve a problem for the 18th Century inhabitants of Konigsberg.
What Does It All Add Up To?
If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?
To Prove or Not to Prove
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
Some Circuits in Graph or Network Theory
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
An Introduction to Proof by Contradiction
An introduction to proof by contradiction, a powerful method of mathematical proof.
Proof: A Brief Historical Survey
If you think that mathematical proof is really clearcut and universal then you should read this article.
Binomial Coefficients
An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.
A Knight's Journey
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
Picturing Pythagorean Triples
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.
Iffy Logic
Can you rearrange the cards to make a series of correct mathematical statements?
Summing Geometric Progressions
Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?
Network Trees
Explore some of the different types of network, and prove a result about network trees.
Perception Versus Reality
Infographics are a powerful way of communicating statistical information. Can you come up with your own?
Mega Quadratic Equations
What do you get when you raise a quadratic to the power of a quadratic?
Unit Interval
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
Dalmatians
Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.
A Powerful Matrix
What happens when you find the powers of this matrix?
IFFY Triangles
Can you prove these triangle theorems both ways?
Common Divisor
Can you find out what numbers divide these expressions? Can you prove that they are always divisors?
Road Maker
Which of these roads will satisfy a Munchkin builder?
Which Spinners?
Can you work out which spinners were used to generate the frequency charts?
There's a Limit
Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?
Sixational
The nth term of a sequence is given by the formula n^3 + 11n. Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.
Kite in a Square
Can you make sense of the three methods to work out what fraction of the total area is shaded?
Always Two
Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.
Quad in Quad
Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
Impossible Sums
Which numbers cannot be written as the sum of two or more consecutive numbers?
Curve Fitter
This problem challenges you to find cubic equations which satisfy different conditions.
Proof Sorter - Quadratic Equation
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.
Difference of Odd Squares
$40$ can be written as $7^2 - 3^2.$ Which other numbers can be written as the difference of squares of odd numbers?
Back Fitter
10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?
Iff
Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
Pent
The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
The Converse of Pythagoras
Can you prove that triangles are right-angled when $a^2+b^2=c^2$?
Negative Powers
What does this number mean? Which order of 1, 2, 3 and 4 makes the highest value? Which makes the lowest?
Always Perfect
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Calculating With Cosines
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?
Leonardo's Problem
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?
A Long Time at the Till
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 Computer Program to Find Magic Squares
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.
An Alphanumeric
Freddie Manners, of Packwood Haugh School in Shropshire solved an alphanumeric without using the extra information supplied and this article explains his reasoning.
Powerful Properties
Yatir from Israel wrote this article on numbers that can be written as $ 2^n-n $ where n is a positive integer.
The Kth Sum of N Numbers
Yatir from Israel describes his method for summing a series of triangle numbers.
Euclid's Algorithm II
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.
An Introduction to Number Theory
An introduction to some beautiful results in Number Theory.
On the Importance of Pedantry
A introduction to how patterns can be deceiving, and what is and is not a proof.
Telescoping Functions
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.
Where Do We Get Our Feet Wet?
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Why Stop at Three by One
Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.
Modulus Arithmetic and a Solution to Differences
Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.
Sums of Squares and Sums of Cubes
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.
Transitivity
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.
Modulus Arithmetic and a Solution to Dirisibly Yours
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.
Continued Fractions II
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
Fractional Calculus III
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
Sperner's Lemma
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.
Euler's Formula and Topology
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 relationship between Euler's formula and angle deficiency of polyhedra.
Three Ways
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
Polynomial Interpolation
Can you fit polynomials through these points?
Tetra Inequalities
Can you prove that in every tetrahedron there is a vertex where the three edges meeting at that vertex have lengths which could be the sides of a triangle?
Big, Bigger, Biggest
Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?
Contrary Logic
Can you invert the logic to prove these statements?
Direct Logic
Can you work through these direct proofs, using our interactive proof sorters?
Staircase
Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?
Prime AP
What can you say about the common difference of an AP where every term is prime?
NOTty Logic
Have a go at being mathematically negative, by negating these statements.
Quadratic Harmony
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.
Proving the Laws of Logarithms
Here you have an opportunity to explore the proofs of the laws of logarithms.
Integration Matcher
Can you match the charts of these functions to the charts of their integrals?
Prime Sequences
This group tasks allows you to search for arithmetic progressions in the prime numbers. How many of the challenges will you discover for yourself?
Dangerous Driver?
Was it possible that this dangerous driving penalty was issued in error?
Middle Man
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?
Look Before You Leap
Relate these algebraic expressions to geometrical diagrams.
Proof Sorter - Geometric Sequence
Can you correctly order the steps in the proof of the formula for the sum of the first n terms in a geometric sequence?
Adding Odd Numbers (part 2)
Can you use Proof by Induction to establish what will happen when you add more and more odd numbers?
Parabella
This is a beautiful result involving a parabola and parallels.
Proof Sorter - The Square Root of 2 Is Irrational
Try this interactivity to familiarise yourself with the proof that the square root of 2 is irrational. Sort the steps of the proof into the correct order.
Model Solutions
How do these modelling assumption affect the solutions?
Proof Sorter - Sum of an Arithmetic Sequence
Put the steps of this proof in order to find the formula for the sum of an arithmetic sequence
Two Cubic Equations
Which statement correctly describes the real roots of the equation?
Trig Rules OK
Change the squares in this diagram and spot the property that stays the same for the triangles. Explain...
Fixing It
A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?
Fred's Maths Problems
If the statement is not true, what can we say will be true?
Magic W Wrap Up
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.
Tetra Slice
Can you prove that a quadrilateral drawn inside a tetrahedron is a parallelogram?
Sixty-Seven Squared
Evaluate these powers of 67. What do you notice? Can you convince someone what the answer would be to (a million sixes followed by a 7) squared?
T for Tan
Can you find a way to prove the trig identities using a diagram?
Napoleon's Hat
Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?
Trig Identity
In this short challenge, can you use angle properties in a circle to figure out some trig identities?