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#### Resources tagged with Mathematical reasoning & proof similar to Function Pyramids:

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### There are 185 results

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

### The Great Weights Puzzle

##### Age 14 to 16 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?

### Knight Defeated

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

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .

### Symmetric Tangles

##### Age 14 to 16

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

### Plus or Minus

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

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

### 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.

### Cube Net

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

How many tours visit each vertex of a cube once and only once? How many return to the starting point?

### Rational Roots

##### Age 16 to 18 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.

### Natural Sum

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

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. . . .

### Problem Solving, Using and Applying and Functional Mathematics

##### Age 5 to 18 Challenge Level:

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

### Pythagorean Triples II

##### Age 11 to 16

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

### Picture Story

##### Age 11 to 16 Challenge Level:

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

### 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

### Proof of Pick's Theorem

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

Follow the hints and prove Pick's Theorem.

### More Number Sandwiches

##### Age 11 to 16 Challenge Level:

When is it impossible to make number sandwiches?

### Stonehenge

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

Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.

### 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!

### Whole Number Dynamics I

##### Age 14 to 18

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

### Ordered Sums

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

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

### Triangular Intersection

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

What is the largest number of intersection points that a triangle and a quadrilateral can have?

### Proximity

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

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

### Multiplication Square

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

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

### Russian Cubes

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

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

### Some Circuits in Graph or Network Theory

##### Age 14 to 18

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

### Janine's Conjecture

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

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

### The Root Cause

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

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

### Sprouts Explained

##### Age 7 to 18

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. . . .

### Postage

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

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. . . .

### Number Rules - OK

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

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

### 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.

##### 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.

### 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.

### Square Pair Circles

##### Age 16 to 18 Challenge 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.

### Polite Numbers

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

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?

### 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.

### Rolling Coins

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

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. . . .

### Doodles

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

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

### N000ughty Thoughts

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

How many noughts are at the end of these giant numbers?

### Golden Eggs

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

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

### 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?

### Perfectly Square

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

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

### Three Frogs

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

Three frogs started jumping randomly over any adjacent frog. Is it possible for them to finish up in the same order they started?

### Air Nets

##### Age 7 to 18 Challenge Level:

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

### Iffy Logic

##### Age 14 to 18 Challenge Level:

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

### No Right Angle Here

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

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

### How Many Solutions?

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

Find all the solutions to the this equation.

### Dodgy Proofs

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

These proofs are wrong. Can you see why?

### Advent Calendar 2011 - Secondary

##### Age 11 to 18 Challenge Level:

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

### Classifying Solids Using Angle Deficiency

##### Age 11 to 16 Challenge Level:

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

### Magic Squares II

##### Age 14 to 18

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