Reasoning, Justifying, Convincing and Proof - Advanced
Binomial Coefficients
An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.
To Prove or Not to Prove
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
Magic Squares II
An article which gives an account of some properties of magic squares.
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?
Proof: A Brief Historical Survey
If you think that mathematical proof is really clearcut and universal then you should read this article.
Proofs with Pictures
Some diagrammatic 'proofs' of algebraic identities and inequalities.
Network Trees
Explore some of the different types of network, and prove a result about network trees.
Iffy Logic
Can you rearrange the cards to make a series of correct mathematical statements?
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?
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.
Iff
Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
Curve Fitter
This problem challenges you to find cubic equations which satisfy different conditions.
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?
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.
Quad in Quad
Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
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.
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.
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.
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.
Kite in a Square
Can you make sense of the three methods to work out what fraction of the total area is shaded?
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?
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?
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?
An Introduction to Proof by Contradiction
An introduction to proof by contradiction, a powerful method of mathematical proof.
Three Ways
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
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.
Direct Logic
Can you work through these direct proofs, using our interactive proof sorters?
Pair Squares
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.
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.
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?
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 $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?
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/...)).
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.
Notty Logic
Have a go at being mathematically negative, by negating these statements.
Diverging
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.
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.
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.
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?
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.
Flexi Quad Tan
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.
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?
Prime AP
What can you say about the common difference of an AP where every term is prime?
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?
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?
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.
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.
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?
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.
Stats Statements
Are these statistical statements sometimes, always or never true? Or it is impossible to say?
Power Quady
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
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.
Integration Matcher
Can you match the charts of these functions to the charts of their integrals?
Staircase
Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?
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.
Stonehenge
Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.
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.
Look Before You Leap
Relate these algebraic expressions to geometrical diagrams.
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.
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?
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
Fibonacci Factors
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
Basic Rhythms
Explore a number pattern which has the same symmetries in different bases.
What's a Group?
Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.
Shape and Territory
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
Eyes Down
The symbol [ ] means 'the integer part of'. Can the numbers [2x]; 2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three different values?
Impossible Triangles?
Which of these triangular jigsaws are impossible to finish?
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.
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.
Golden Eggs
Find a connection between the shape of a special ellipse and an infinite string of nested square roots.
Can it Be?
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
Polite Numbers
A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?
Fibonacci Fashion
What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?
Particularly General
By proving these particular identities, prove the existence of general cases.
Square Pair Circles
Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.
The Clue Is in the Question
Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?
Binomial
By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
Mind Your Ps and Qs
Sort these mathematical propositions into a series of 8 correct statements.
Pythagorean Fibs
What have Fibonacci numbers got to do with Pythagorean triples?
Exhaustion
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
And So on - and on -and On
Can you find the value of this function involving algebraic fractions for x=2000?
Be Reasonable
Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.
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.
Generally Geometric
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
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.
Binary Squares
If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?
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.
Discrete Trends
Find the maximum value of n to the power 1/n and prove that it is a maximum.
On the Importance of Pedantry
A introduction to how patterns can be deceiving, and what is and is not a proof.
OK! Now Prove It
Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?
Little and Large
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?
