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.
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.
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.
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?
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.
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.
A Powerful Matrix
What happens when you find the powers of this matrix?
Common Divisor
Can you find out what numbers divide these expressions? Can you prove that they are always divisors?
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?
Which Spinners?
Can you work out which spinners were used to generate the frequency charts?
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.
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?
Iffy Logic
Can you rearrange the cards to make a series of correct mathematical statements?
IFFY Triangles
Can you prove these triangle theorems both ways?
Summing Geometric Progressions
Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?
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?
Road Maker
Which of these roads will satisfy a Munchkin builder?
Network Trees
Explore some of the different types of network, and prove a result about network trees.
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?
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.
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?
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?
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.
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?
The Converse of Pythagoras
Can you prove that triangles are right-angled when $a^2+b^2=c^2$?
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?
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.
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.
Tetra Slice
Can you prove that a quadrilateral drawn inside a tetrahedron is a parallelogram?
Staircase
Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?
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.
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?
Summats Clear
Find the sum, f(n), of the first n terms of the sequence: 0, 1, 1, 2, 2, 3, 3........p, p, p +1, p + 1,..... Prove that f(a + b) - f(a - b) = ab.
Trig Identity
In this short challenge, can you use angle properties in a circle to figure out some trig identities?
Unit Interval
Can you prove our inequality holds for all values of x and y between 0 and 1?
Polynomial Relations
Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.
Gradient Match
What can you deduce about the gradients of curves linking (0,0), (8,8) and (4,6)?
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?
Stonehenge
Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.
Turning to Calculus
Get started with calculus by exploring the connections between the sign of a curve and the sign of its gradient.
How Many Solutions?
Find all the solutions to the this equation.
Look Before You Leap
Relate these algebraic expressions to geometrical diagrams.
Patterns of Inflection
Find the relationship between the locations of points of inflection, maxima and minima of functions.
Parabella
This is a beautiful result involving a parabola and parallels.
Code to Zero
Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.
Seriesly
Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!
Archimedes Numerical Roots
How did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
Trig Rules OK
Change the squares in this diagram and spot the property that stays the same for the triangles. Explain...
Without Calculus
Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.
More Dicey Decisions
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
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.
Pythagorean Golden Means
Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.
Stats Statements
Are these statistical statements sometimes, always or never true? Or it is impossible to say?
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?
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?
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?
T for Tan
Can you find a way to prove the trig identities using a diagram?
Direct Logic
Can you work through these direct proofs, using our interactive proof sorters?
Prime AP
What can you say about the common difference of an AP where every term is prime?