# Resources tagged with: Mathematical reasoning & proof

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### There are 173 results

Broad Topics > Thinking Mathematically > Mathematical reasoning & proof

### Thousand Words

##### Age 16 to 18Challenge Level

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

### To Prove or Not to Prove

##### Age 14 to 18

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

### Golden Eggs

##### Age 16 to 18Challenge Level

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

### Contrary Logic

##### Age 16 to 18Challenge Level

Can you invert the logic to prove these statements?

### Iffy Logic

##### Age 14 to 18Challenge Level

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

### Plus or Minus

##### Age 16 to 18Challenge Level

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

### Dodgy Proofs

##### Age 16 to 18Challenge Level

These proofs are wrong. Can you see why?

### Breaking the Equation ' Empirical Argument = Proof '

##### Age 7 to 18

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.

### Without Calculus

##### Age 16 to 18Challenge 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.

### Sperner's Lemma

##### 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.

### Rational Roots

##### Age 16 to 18Challenge 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.

### Target Six

##### Age 16 to 18Challenge Level

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

### The Clue Is in the Question

##### Age 16 to 18Challenge Level

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

### Proof: A Brief Historical Survey

##### Age 14 to 18

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

### Proofs with Pictures

##### Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

### Impossible Sandwiches

##### Age 11 to 18

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.

### Square Pair Circles

##### Age 16 to 18Challenge 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.

### Continued Fractions II

##### 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/...)).

### Proof Sorter - Quadratic Equation

##### Age 14 to 18Challenge 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.

### Integral Inequality

##### Age 16 to 18Challenge Level

An inequality involving integrals of squares of functions.

### An Introduction to Number Theory

##### Age 16 to 18

An introduction to some beautiful results in Number Theory.

### Dalmatians

##### Age 14 to 18Challenge 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.

### Magic Squares II

##### Age 14 to 18

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

##### Age 16 to 18Challenge Level

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

### How Many Solutions?

##### Age 16 to 18Challenge Level

Find all the solutions to the this equation.

##### Age 16 to 18 ShortChallenge Level

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

### Where Do We Get Our Feet Wet?

##### 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.

### Tetra Inequalities

##### Age 16 to 18Challenge Level

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.

### Proof of Pick's Theorem

##### Age 16 to 18Challenge Level

Follow the hints and prove Pick's Theorem.

### Big, Bigger, Biggest

##### Age 16 to 18Challenge Level

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

### Generally Geometric

##### Age 16 to 18Challenge Level

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

### Polite Numbers

##### Age 16 to 18Challenge 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?

### Unit Interval

##### Age 14 to 18Challenge Level

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

### Little and Large

##### Age 16 to 18Challenge Level

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

### Square Mean

##### Age 14 to 16Challenge Level

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

### Exhaustion

##### Age 16 to 18Challenge Level

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

### Sums of Squares and Sums of Cubes

##### 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.

### Telescoping Functions

##### 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.

### Polynomial Relations

##### Age 16 to 18Challenge 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.

### Euclid's Algorithm II

##### 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.

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

##### Age 11 to 18Challenge 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. . . .

### Picturing Pythagorean Triples

##### 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.

### Binomial

##### Age 16 to 18Challenge Level

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

### Impossible Triangles?

##### Age 16 to 18Challenge Level

Which of these triangular jigsaws are impossible to finish?

### Fractional Calculus III

##### Age 16 to 18

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

##### Age 16 to 18Challenge 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.

### Perfectly Square

##### Age 14 to 16Challenge Level

The sums of the squares of three related numbers is also a perfect square - can you explain why?

### Middle Man

##### Age 16 to 18Challenge 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?