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### Number and algebra

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### Working mathematically

### For younger learners

### Advanced mathematics

# Reasoning, Justifying, Convincing and Proof - Advanced

### Binomial Coefficients

### To Prove or Not to Prove

### Iffy Logic

### Magic Squares II

### Unit Interval

### Proofs with Pictures

### Network Trees

### Proof: A Brief Historical Survey

### Dalmatians

### There's a Limit

### Picturing Pythagorean Triples

### Iff

### Pent

### Quad in Quad

### Kite in a SquareLive

### Sixational

### Some Circuits in Graph or Network Theory

### Impossible Sums

### Proof Sorter - Quadratic Equation

### A Knight's Journey

### Curve Fitter

### Difference of Odd Squares

### Always Perfect

### Calculating with Cosines

### An Introduction to Proof by Contradiction

### Leonardo's Problem

### A Long Time at the Till

### Big and Small Numbers in Physics - Group Task

### Three Ways

### Integration Matcher

### Quadratic Harmony

### Pair Squares

### A Computer Program to Find Magic Squares

### More Dicey Decisions

### Napoleon's Hat

### Sixty-seven Squared

### Big, Bigger, Biggest

### Continued Fractions II

### Prime Sequences

### Code to Zero

### Diverging

### An Alphanumeric

### Summats Clear

### Dangerous Driver?

### Fixing It

### Euclid's Algorithm II

### Flexi Quad Tan

### Direct Logic

### Tetra Inequalities

### Prime AP

### Middle Man

### Magic W Wrap Up

### Without Calculus

### Notty Logic

### Proof Sorter - Geometric Sequence

### Polynomial Relations

### Power Quady

### Pythagorean Golden Means

### Staircase

### Proof Sorter - the Square Root of 2 Is Irrational

### Trig Rules OK

### Stonehenge

### Where Do We Get Our Feet Wet?

### Look Before You Leap

### Transitivity

### Proof Sorter - Sum of an Arithmetic Sequence

### Fibonacci Factors

### Stats Statements

### Basic Rhythms

### What's a Group?

### The Clue Is in the Question

### Shape and Territory

### Eyes Down

### Modulus Arithmetic and a Solution to Dirisibly Yours

### Why Stop at Three by One

### Golden Eggs

### Can it Be?

### Polite Numbers

### Fibonacci Fashion

### On the Importance of Pedantry

### Square Pair Circles

### Impossible Triangles?

### Binomial

### Pythagorean Fibs

### Exhaustion

### And So on - and on -and On

### Be Reasonable

### Sums of Squares and Sums of Cubes

### Generally Geometric

### Telescoping Functions

### Binary Squares

### Sperner's Lemma

### Discrete Trends

### OK! Now Prove It

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Age 14 to 18

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.

Age 14 to 18

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

Age 14 to 18

Challenge Level

Can you rearrange the cards to make a series of correct mathematical statements?

Age 14 to 18

An article which gives an account of some properties of magic squares.

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?

Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Age 14 to 18

Challenge Level

Explore some of the different types of network, and prove a result about network trees.

Age 14 to 18

If you think that mathematical proof is really clearcut and universal then you should read this article.

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.

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?

Age 14 to 18

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.

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?

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.

Age 14 to 18

Challenge Level

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

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?

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.

Age 14 to 18

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

Age 14 to 18

Challenge Level

Which numbers cannot be written as the sum of two or more consecutive numbers?

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.

Age 14 to 18

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

Age 14 to 18

ShortChallenge Level

This problem challenges you to find cubic equations which satisfy different conditions.

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?

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.

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?

Age 14 to 18

An introduction to proof by contradiction, a powerful method of mathematical proof.

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?

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?

Age 16 to 18

Challenge Level

Work in groups to try to create the best approximations to these physical quantities.

Age 16 to 18

Challenge Level

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

Age 16 to 18

Challenge Level

Can you match the charts of these functions to the charts of their integrals?

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.

Age 16 to 18

Challenge Level

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.

Age 16 to 18

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.

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?

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?

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?

Age 16 to 18

Challenge Level

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

Age 16 to 18

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

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?

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.

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.

Age 16 to 18

Freddie Manners, of Packwood Haugh School in Shropshire solved an alphanumeric without using the extra information supplied and this article explains his reasoning.

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.

Age 16 to 18

Challenge Level

Was it possible that this dangerous driving penalty was issued in error?

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?

Age 16 to 18

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.

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.

Age 16 to 18

Challenge Level

Can you work through these direct proofs, using our interactive proof sorters?

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?

Age 16 to 18

Challenge Level

What can you say about the common difference of an AP where every term is prime?

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?

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.

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.

Age 16 to 18

Challenge Level

Have a go at being mathematically negative, by negating these statements.

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?

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.

Age 16 to 18

Challenge Level

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

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.

Age 16 to 18

Challenge Level

Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?

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.

Age 16 to 18

Challenge Level

Change the squares in this diagram and spot the property that stays the same for the triangles. Explain...

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.

Age 16 to 18

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

Age 16 to 18

Challenge Level

Relate these algebraic expressions to geometrical diagrams.

Age 16 to 18

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.

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

Age 16 to 18

Challenge Level

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

Age 16 to 18

Challenge Level

Are these statistical statements sometimes, always or never true? Or it is impossible to say?

Age 16 to 18

Challenge Level

Explore a number pattern which has the same symmetries in different bases.

Age 16 to 18

Challenge Level

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.

Age 16 to 18

Challenge Level

Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?

Age 16 to 18

Challenge Level

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Age 16 to 18

Challenge Level

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?

Age 16 to 18

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.

Age 16 to 18

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.

Age 16 to 18

Challenge Level

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

Age 16 to 18

Challenge Level

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

Age 16 to 18

Challenge Level

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?

Age 16 to 18

Challenge Level

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?

Age 16 to 18

A introduction to how patterns can be deceiving, and what is and is not a proof.

Age 16 to 18

Challenge Level

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.

Age 16 to 18

Challenge Level

Which of these triangular jigsaws are impossible to finish?

Age 16 to 18

Challenge Level

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

Age 16 to 18

Challenge Level

What have Fibonacci numbers got to do with Pythagorean triples?

Age 16 to 18

Challenge Level

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

Age 16 to 18

Challenge Level

Can you find the value of this function involving algebraic fractions for x=2000?

Age 16 to 18

Challenge Level

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

Age 16 to 18

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.

Age 16 to 18

Challenge Level

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

Age 16 to 18

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.

Age 16 to 18

Challenge Level

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

Age 16 to 18

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.

Age 16 to 18

Challenge Level

Find the maximum value of n to the power 1/n and prove that it is a maximum.

Age 16 to 18

Challenge Level

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?