# Search by Topic

#### Resources tagged with Mathematical reasoning & proof similar to Farey Neighbours:

Filter by: Content type:
Age range:
Challenge level:

### There are 176 results

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

### Mod 3

##### Age 14 to 16 Challenge Level:

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

### Why 24?

##### Age 14 to 16 Challenge Level:

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

### Three Ways

##### Age 16 to 18 Challenge Level:

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

### 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.

### More Sums of Squares

##### Age 16 to 18

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

### Modulus Arithmetic and a Solution to Differences

##### Age 16 to 18

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

### Plus or Minus

##### Age 16 to 18 Challenge Level:

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

### Exhaustion

##### Age 16 to 18 Challenge Level:

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

### Without Calculus

##### Age 16 to 18 Challenge 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.

### Water Pistols

##### Age 16 to 18 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?

### Prime AP

##### Age 16 to 18 Challenge Level:

What can you say about the common difference of an AP where every term is prime?

### Particularly General

##### Age 16 to 18 Challenge Level:

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

### Modular Fractions

##### Age 16 to 18 Challenge Level:

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.

### How Many Solutions?

##### Age 16 to 18 Challenge Level:

Find all the solutions to the this equation.

### Modulus Arithmetic and a Solution to Dirisibly Yours

##### Age 16 to 18

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.

##### Age 14 to 16 Challenge Level:

Kyle and his teacher disagree about his test score - who is right?

### Rhombus in Rectangle

##### Age 14 to 16 Challenge Level:

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

### The Frieze Tree

##### Age 11 to 16

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

### Mouhefanggai

##### Age 14 to 16

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.

### A Computer Program to Find Magic Squares

##### Age 16 to 18

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.

### Transitivity

##### Age 16 to 18

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.

### 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.

### 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.

### Proofs with Pictures

##### Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

### 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/...)).

### 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.

### Angle Trisection

##### Age 14 to 16 Challenge Level:

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

### Take Three from Five

##### Age 14 to 16 Challenge Level:

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

### Pythagorean Triples I

##### Age 11 to 16

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!

##### Age 16 to 18 Challenge Level:

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

### Magic W Wrap Up

##### Age 16 to 18 Challenge Level:

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.

### Mechanical Integration

##### Age 16 to 18 Challenge Level:

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

##### 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.

### Proof of Pick's Theorem

##### Age 16 to 18 Challenge Level:

Follow the hints and prove Pick's Theorem.

### Pythagorean Triples II

##### Age 11 to 16

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

### 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.

### Magic Squares II

##### Age 14 to 18

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

### 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. . . .

### Matter of Scale

##### Age 14 to 16 Challenge Level:

Prove Pythagoras' Theorem using enlargements and scale factors.

### A Knight's Journey

##### Age 14 to 18

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

### 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.

### 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?

### Areas and Ratios

##### Age 16 to 18 Challenge Level:

Do you have enough information to work out the area of the shaded quadrilateral?

### L-triominoes

##### Age 14 to 16 Challenge Level:

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

### 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?

### The Clue Is in the Question

##### Age 16 to 18 Challenge Level:

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

### 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.

### Contrary Logic

##### Age 16 to 18 Challenge Level:

Can you invert the logic to prove these statements?

### Pythagoras Proofs

##### Age 14 to 16 Challenge Level:

Can you make sense of these three proofs of Pythagoras' Theorem?