Reasoning, Justifying, Convincing and Proof - Advanced

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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.

$Breaking the Equation ' \Empirical Argument = Proof '$

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.

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.

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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.

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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.

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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?

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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.

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An introduction to proof by contradiction

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

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Proof: a brief historical survey

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

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Binomial coefficients

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

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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.

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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.

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Magic squares II

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

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Proofs with pictures

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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To prove or not to prove

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

problem

Which spinners?

Age

14 to 18

Challenge level

Can you work out which spinners were used to generate the frequency charts?

problem

Iffy logic

Age

14 to 18

Challenge level

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

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

Network trees

Age

14 to 18

Challenge level

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

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Perception versus reality

Age

14 to 18

Challenge level

Infographics are a powerful way of communicating statistical information. Can you come up with your own?

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

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

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.

problem

A powerful matrix

Age

14 to 18

Challenge level

What happens when you find the powers of this matrix?

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?

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IFFY triangles

Age

14 to 18

Challenge level

Can you prove these triangle theorems both ways?

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

Road maker

Age

14 to 18

Challenge level

Which of these roads will satisfy a Munchkin builder?

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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?

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

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

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

Impossible sums

Age

14 to 18

Challenge level

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

problem

Curve fitter

Age

14 to 18

Challenge level

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

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

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

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

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

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

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

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?

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?

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An introduction to number theory

An introduction to some beautiful results in Number Theory.

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On the importance of pedantry

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

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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.

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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.

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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.

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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.

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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.

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More sums of squares

Tom writes about expressing numbers as the sums of three squares.

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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.

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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.

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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/...)).

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Fractional calculus III

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

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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.

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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.

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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.

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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.

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Powerful properties

Yatir from Israel wrote this article on numbers that can be written as $ 2^n-n $ where n is a positive integer.

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The kth sum of n numbers

Yatir from Israel describes his method for summing a series of triangle numbers.

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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

Seriesly

Age

16 to 18

Challenge level

Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!

problem

Archimedes numerical roots

Age

16 to 18

Challenge level

How did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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

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

Pair squares

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.

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

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

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

Flexi quads

Age

16 to 18

Challenge level

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

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?

problem

Polynomial interpolation

Age

16 to 18

Challenge level

Can you fit polynomials through these points?

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?

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.

problem

Contrary logic

Age

16 to 18

Challenge level

Can you invert the logic to prove these statements?

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

Direct logic

Age

16 to 18

Challenge level

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

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

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

NOTty logic

Age

16 to 18

Challenge level

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

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

Proving the laws of logarithms

Age

16 to 18

Challenge level

Here you have an opportunity to explore the proofs of the laws of logarithms.

problem

Integration matcher

Age

16 to 18

Challenge level

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

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

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

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

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

Dangerous driver?

Age

16 to 18

Challenge level

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

problem

Solving by squaring

Age

16 to 18

Challenge level

Which of the statements is true?

problem

Look before you leap

Age

16 to 18

Challenge level

Relate these algebraic expressions to geometrical diagrams.

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.