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Resources tagged with Mathematical reasoning & proof similar to Pythagorean Fibs:

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

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Golden Eggs

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

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

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Proof Sorter - Quadratic Equation

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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Plus or Minus

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

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

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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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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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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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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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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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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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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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Square Mean

Stage: 4 Challenge Level: Challenge Level:1

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

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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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Target Six

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

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

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Proof Sorter - Geometric Series

Stage: 5 Challenge Level: Challenge Level:1

This is an interactivity in which you have to sort into the correct order the steps in the proof of the formula for the sum of a geometric series.

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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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Direct Logic

Stage: 5 Challenge Level: Challenge Level:1

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

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Mind Your Ps and Qs

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

Sort these mathematical propositions into a series of 8 correct statements.

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Contrary Logic

Stage: 5 Challenge Level: Challenge Level:1

Can you invert the logic to prove these statements?

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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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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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The Golden Ratio, Fibonacci Numbers and Continued Fractions.

Stage: 4

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

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Thousand Words

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

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

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Integral Inequality

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

An inequality involving integrals of squares of functions.

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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/...)).

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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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Square Pair Circles

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

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.

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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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Power Quady

Stage: 5 Challenge Level: Challenge Level:1

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

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Rational Roots

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

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.

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How Many Solutions?

Stage: 5 Challenge Level: Challenge Level:1

Find all the solutions to the this equation.

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Road Maker 2

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

Can you work out where the blue-and-red brick roads end?

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Proof Sorter - Sum of an AP

Stage: 5 Challenge Level: Challenge Level:1

Use this interactivity to sort out the steps of the proof of the formula for the sum of an arithmetic series. The 'thermometer' will tell you how you are doing

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Tetra Inequalities

Stage: 5 Challenge Level: Challenge Level:1

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

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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.

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Diophantine N-tuples

Stage: 4 Challenge Level: Challenge Level:1

Can you explain why a sequence of operations always gives you perfect squares?

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More Sums of Squares

Stage: 5

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

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Transitivity

Stage: 5

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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Composite Notions

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

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

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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.

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Euler's Squares

Stage: 4 Challenge Level: Challenge Level:1

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

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A Computer Program to Find Magic Squares

Stage: 5

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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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.

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Proofs with Pictures

Stage: 5

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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Pythagorean Golden Means

Stage: 5 Challenge Level: Challenge Level:1

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.

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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.

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Sums of Squares and Sums of Cubes

Stage: 5

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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Mediant

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

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

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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.

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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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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.