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Resources tagged with Mathematical reasoning & proof similar to Proofs with Pictures:

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Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical reasoning & proof

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Proofs with Pictures

Stage: 5

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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Middle Man

Stage: 5 Challenge Level: Challenge Level:1

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?

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Yih or Luk Tsut K'i or Three Men's Morris

Stage: 3, 4 and 5 Challenge Level: Challenge Level:1

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. . . .

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Picture Story

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

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Natural Sum

Stage: 4 Challenge Level: Challenge Level:1

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

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Sprouts Explained

Stage: 2, 3, 4 and 5

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. . . .

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The Triangle Game

Stage: 3 and 4 Challenge Level: Challenge Level:1

Can you discover whether this is a fair game?

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Sperner's Lemma

Stage: 5

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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Dodgy Proofs

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

These proofs are wrong. Can you see why?

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AMGM

Stage: 4 Challenge Level: Challenge Level:1

Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?

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Three Frogs

Stage: 4 Challenge Level: Challenge Level:1

Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .

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Rotating Triangle

Stage: 3 and 4 Challenge Level: Challenge Level:1

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

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Particularly General

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

By proving these particular identities, prove the existence of general cases.

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Binomial

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Whole Number Dynamics II

Stage: 4 and 5

This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.

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Whole Number Dynamics III

Stage: 4 and 5

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

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Iffy Logic

Stage: 4 Short Challenge Level: Challenge Level:1

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

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To Prove or Not to Prove

Stage: 4 and 5

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

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Diverging

Stage: 5 Challenge Level: Challenge Level:1

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.

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What Numbers Can We Make Now?

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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Whole Number Dynamics IV

Stage: 4 and 5

Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?

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A Knight's Journey

Stage: 4 and 5

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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The Clue Is in the Question

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .

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Whole Number Dynamics I

Stage: 4 and 5

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

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Geometric Parabola

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

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Breaking the Equation ' Empirical Argument = Proof '

Stage: 2, 3, 4 and 5

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.

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Whole Number Dynamics V

Stage: 4 and 5

The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.

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Calculating with Cosines

Stage: 5 Challenge Level: Challenge Level:1

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?

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Picturing Pythagorean Triples

Stage: 4 and 5

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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Try to Win

Stage: 5

Solve this famous unsolved problem and win a prize. Take a positive integer N. If even, divide by 2; if odd, multiply by 3 and add 1. Iterate. Prove that the sequence always goes to 4,2,1,4,2,1...

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Notty Logic

Stage: 5 Challenge Level: Challenge Level:1

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

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Telescoping Functions

Stage: 5

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?

Stage: 5

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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L-triominoes

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

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Magic Squares II

Stage: 4 and 5

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

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Magic W Wrap Up

Stage: 5 Challenge Level: Challenge Level:1

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.

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Postage

Stage: 4 Challenge Level: Challenge Level:1

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

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Quadratic Harmony

Stage: 5 Challenge Level: Challenge Level:1

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.

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Mechanical Integration

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

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Polite Numbers

Stage: 5 Challenge Level: Challenge Level:1

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

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Always Perfect

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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Rolling Coins

Stage: 4 Challenge Level: Challenge Level:1

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

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Modular Fractions

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

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Pythagorean Triples II

Stage: 3 and 4

This is the second article on right-angled triangles whose edge lengths are whole numbers.

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Can it Be

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Proof of Pick's Theorem

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Follow the hints and prove Pick's Theorem.

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Pythagorean Triples I

Stage: 3 and 4

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

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Angle Trisection

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

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Pair Squares

Stage: 5 Challenge Level: Challenge Level:1

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.

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Modulus Arithmetic and a Solution to Differences

Stage: 5

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.