article
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.
article
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.
problem
Air nets
Age
7 to 18
Challenge level
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
article
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.
article
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.
game
Yih or Luk tsut k'i or Three Men's Morris
Age
11 to 18
Challenge level
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.
problem
The bridges of Konigsberg
Age
11 to 18
Challenge level
Investigate how networks can be used to solve a problem for the 18th Century inhabitants of Konigsberg.
problem
What does it all add up to?
Age
11 to 18
Challenge level
If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?
article
An introduction to proof by contradiction
An introduction to proof by contradiction, a powerful method of mathematical proof.
article
Proof: a brief historical survey
If you think that mathematical proof is really clearcut and
universal then you should read this article.
article
Binomial coefficients
An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.
article
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.
article
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.
article
To prove or not to prove
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
article
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.
problem
Road maker
Age
14 to 18
Challenge level
Which of these roads will satisfy a Munchkin builder?
problem
Iffy logic
Age
14 to 18
Challenge level
Can you rearrange the cards to make a series of correct mathematical statements?
problem
Common divisor
Age
14 to 18
Challenge level
Can you find out what numbers divide these expressions? Can you prove that they are always divisors?
problem
Network trees
Age
14 to 18
Challenge level
Explore some of the different types of network, and prove a result about network trees.
problem
Summing geometric progressions
Age
14 to 18
Challenge level
Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?
problem
Unit interval
Age
14 to 18
Challenge level
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
problem
There's a limit
Age
14 to 18
Challenge level
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?
problem
Mega quadratic equations
Age
14 to 18
Challenge level
What do you get when you raise a quadratic to the power of a quadratic?
problem
IFFY triangles
Age
14 to 18
Challenge level
Can you prove these triangle theorems both ways?
problem
Dalmatians
Age
14 to 18
Challenge level
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.
interactivity
Proof sorter - quadratic equation
Age
14 to 18
Challenge level
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.
problem
Always perfect
Age
14 to 18
Challenge level
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
problem
Curve fitter
Age
14 to 18
Challenge level
This problem challenges you to find cubic equations which satisfy different conditions.
problem
Back fitter
Age
14 to 18
Challenge level
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?
problem
Calculating with cosines
Age
14 to 18
Challenge level
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?
problem
Always two
Age
14 to 18
Challenge level
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.
problem
Quad in quad
Age
14 to 18
Challenge level
Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
problem
Kite in a square
Age
14 to 18
Challenge level
Can you make sense of the three methods to work out what fraction of the total area is shaded?
problem
Sixational
Age
14 to 18
Challenge level
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.
problem
Impossible sums
Age
14 to 18
Challenge level
Which numbers cannot be written as the sum of two or more consecutive numbers?
problem
Iff
Age
14 to 18
Challenge level
Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
problem
Difference of odd squares
Age
14 to 18
Challenge level
$40$ can be written as $7^2 - 3^2.$ Which other numbers can be written as the difference of squares of odd numbers?
problem
Pent
Age
14 to 18
Challenge level
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.
problem
The converse of Pythagoras
Age
14 to 18
Challenge level
Can you prove that triangles are right-angled when $a^2+b^2=c^2$?
problem
Leonardo's problem
Age
14 to 18
Challenge level
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?
problem
A long time at the till
Age
14 to 18
Challenge level
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?
article
An introduction to number theory
An introduction to some beautiful results in Number Theory.
article
On the importance of pedantry
A introduction to how patterns can be deceiving, and what is and is not a proof.
article
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.
article
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.
article
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.
article
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.
article
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.
article
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.
article
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.
article
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/...)).
article
Fractional calculus III
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
article
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.
article
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.
article
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.
article
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.
article
Powerful properties
Yatir from Israel wrote this article on numbers that can be written as $ 2^n-n $ where n is a positive integer.
article
The kth sum of n numbers
Yatir from Israel describes his method for summing a series of
triangle numbers.
article
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.
problem
Big and small numbers in physics - group task
Age
16 to 18
Challenge level
Work in groups to try to create the best approximations to these
physical quantities.
problem
Sixty-seven squared
Age
16 to 18
Challenge level
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?
interactivity
Proof sorter - geometric sequence
Age
16 to 18
Challenge level
Can you correctly order the steps in the proof of the formula for the sum of the first n terms in a geometric sequence?
interactivity
Proof sorter - the square root of 2 is irrational
Age
16 to 18
Challenge level
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.
interactivity
Proof sorter - sum of an arithmetic sequence
Age
16 to 18
Challenge level
Put the steps of this proof in order to find the formula for the sum of an arithmetic sequence
problem
Stats statements
Age
16 to 18
Challenge level
Are these statistical statements sometimes, always or never true?
Or it is impossible to say?
problem
Fixing it
Age
16 to 18
Challenge level
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?
problem
Integration matcher
Age
16 to 18
Challenge level
Can you match the charts of these functions to the charts of their integrals?
problem
Fibonacci factors
Age
16 to 18
Challenge level
For which values of n is the Fibonacci number fn even? Which
Fibonnaci numbers are divisible by 3?
problem
Prime sequences
Age
16 to 18
Challenge level
This group tasks allows you to search for arithmetic progressions
in the prime numbers. How many of the challenges will you discover
for yourself?
problem
Contrary logic
Age
16 to 18
Challenge level
Can you invert the logic to prove these statements?
problem
Dangerous driver?
Age
16 to 18
Challenge level
Was it possible that this dangerous driving penalty was issued in
error?
problem
Direct logic
Age
16 to 18
Challenge level
Can you work through these direct proofs, using our interactive proof sorters?
problem
Flexi quad tan
Age
16 to 18
Challenge level
As a quadrilateral Q is deformed (keeping the edge lengths constnt)
the diagonals and the angle X between them change. Prove that the
area of Q is proportional to tanX.
problem
NOTty logic
Age
16 to 18
Challenge level
Have a go at being mathematically negative, by negating these statements.
problem
Napoleon's hat
Age
16 to 18
Challenge level
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?
problem
Summats clear
Age
16 to 18
Challenge level
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.
problem
Model solutions
Age
16 to 18
Challenge level
How do these modelling assumption affect the solutions?
problem
Diverging
Age
16 to 18
Challenge level
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.
problem
Polynomial relations
Age
16 to 18
Challenge level
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.
problem
Tetra inequalities
Age
16 to 18
Challenge level
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?
problem
Stonehenge
Age
16 to 18
Challenge level
Explain why, when moving heavy objects on rollers, the object moves
twice as fast as the rollers. Try a similar experiment yourself.
problem
How many solutions?
Age
16 to 18
Challenge level
Find all the solutions to the this equation.
problem
More dicey decisions
Age
16 to 18
Challenge level
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
problem
Staircase
Age
16 to 18
Challenge level
Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?
problem
Code to zero
Age
16 to 18
Challenge level
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.
problem
Quadratic harmony
Age
16 to 18
Challenge level
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.
problem
Without calculus
Age
16 to 18
Challenge level
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.
problem
Polynomial interpolation
Age
16 to 18
Challenge level
Can you fit polynomials through these points?
problem
Pythagorean golden means
Age
16 to 18
Challenge level
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.
problem
Middle man
Age
16 to 18
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?
problem
Three ways
Age
16 to 18
Challenge level
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
problem
Look before you leap
Age
16 to 18
Challenge level
Relate these algebraic expressions to geometrical diagrams.
problem
Big, bigger, biggest
Age
16 to 18
Challenge level
Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?
problem
Parabella
Age
16 to 18
Challenge level
This is a beautiful result involving a parabola and parallels.
problem
Trig rules OK
Age
16 to 18
Challenge level
Change the squares in this diagram and spot the property that stays the same for the triangles. Explain...
problem
Prime AP
Age
16 to 18
Challenge level
What can you say about the common difference of an AP where every term is prime?
problem
Magic W wrap up
Age
16 to 18
Challenge level
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.