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Resources tagged with Mathematical reasoning & proof similar to Plus or Minus:

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

Plus or Minus

Stage: 5 Challenge Level:

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

Golden Eggs

Stage: 5 Challenge Level:

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

Rational Roots

Stage: 5 Challenge Level:

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

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.

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.

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.

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.

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?

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.

Proof of Pick's Theorem

Stage: 5 Challenge Level:

Follow the hints and prove Pick's Theorem.

Stage: 5 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.

Stage: 4 and 5 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.

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.

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.

Binomial

Stage: 5 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

Square Mean

Stage: 4 Challenge Level:

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Mechanical Integration

Stage: 5 Challenge Level:

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

Target Six

Stage: 5 Challenge Level:

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

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.

Impossible Sandwiches

Stage: 3, 4 and 5

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.

Contrary Logic

Stage: 5 Challenge Level:

Can you invert the logic to prove these statements?

Direct Logic

Stage: 5 Challenge Level:

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

Iffy Logic

Stage: 4 and 5 Challenge Level:

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

Notty Logic

Stage: 5 Challenge Level:

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

The Clue Is in the Question

Stage: 5 Challenge Level:

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. . . .

Dodgy Proofs

Stage: 5 Challenge Level:

These proofs are wrong. Can you see why?

Stage: 3, 4 and 5 Challenge Level:

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

The Great Weights Puzzle

Stage: 4 Challenge Level:

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

Some Circuits in Graph or Network Theory

Stage: 4 and 5

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

Pareq Exists

Stage: 4 Challenge Level:

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

Integral Inequality

Stage: 5 Challenge Level:

An inequality involving integrals of squares of functions.

Recent Developments on S.P. Numbers

Stage: 5

Take a number, add its digits then multiply the digits together, then multiply these two results. If you get the same number it is an SP number.

Unit Interval

Stage: 4 and 5 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?

Water Pistols

Stage: 5 Challenge Level:

With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?

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. . . .

Thousand Words

Stage: 5 Challenge Level:

Here the diagram says it all. Can you find the diagram?

Magic Squares II

Stage: 4 and 5

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

Stage: 5 Challenge Level:

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

Tree Graphs

Stage: 4 Challenge Level:

A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree. . . .

Diverging

Stage: 5 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.

Find the Fake

Stage: 4 Challenge Level:

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?

How Many Solutions?

Stage: 5 Challenge Level:

Find all the solutions to the this equation.

An Introduction to Number Theory

Stage: 5

An introduction to some beautiful results of Number Theory

The Triangle Game

Stage: 3 and 4 Challenge Level:

Can you discover whether this is a fair game?

Continued Fractions II

Stage: 5

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

Modulus Arithmetic and a Solution to Dirisibly Yours

Stage: 5

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.

Proofs with Pictures

Stage: 5

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Euclid's Algorithm II

Stage: 5

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.

Mouhefanggai

Stage: 4

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.