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#### Resources tagged with Mathematical reasoning & proof similar to Kissing:

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

### Golden Eggs

##### Stage: 5 Challenge Level:

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

### Thousand Words

##### Stage: 5 Challenge Level:

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

### Proof Sorter - Quadratic Equation

##### 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.

### 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.

### 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!

### Round and Round

##### Stage: 4 Challenge Level:

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

### Fitting In

##### Stage: 4 Challenge Level:

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

### 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.

### 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/...)).

### Pent

##### Stage: 4 and 5 Challenge Level:

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

### Pythagorean Triples II

##### Stage: 3 and 4

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

### Plus or Minus

##### Stage: 5 Challenge Level:

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

### Rolling Coins

##### Stage: 4 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. . . .

### 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?

### More Sums of Squares

##### Stage: 5

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

### Rotating Triangle

##### Stage: 3 and 4 Challenge Level:

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

### 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.

##### 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.

### 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?

##### Stage: 5 Challenge Level:

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

### Diophantine N-tuples

##### Stage: 4 Challenge Level:

Take any whole number q. Calculate q^2 - 1. Factorize q^2-1 to give two factors a and b (not necessarily q+1 and q-1). Put c = a + b + 2q . Then you will find that ab+1 , bc+1 and ca+1 are all. . . .

### 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.

### 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.

### 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.

### 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.

### Dodgy Proofs

##### Stage: 5 Challenge Level:

These proofs are wrong. Can you see why?

### Circle Box

##### Stage: 4 Challenge Level:

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

### 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?

### Polite Numbers

##### Stage: 5 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?

### There's a Limit

##### Stage: 4 and 5 Challenge Level:

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

### 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.

### Long Short

##### Stage: 4 Challenge Level:

A quadrilateral inscribed in a unit circle has sides of lengths s1, s2, s3 and s4 where s1 ≤ s2 ≤ s3 ≤ s4. Find a quadrilateral of this type for which s1= sqrt2 and show s1 cannot. . . .

### Kite in a Square

##### Stage: 4 Challenge Level:

Can you make sense of the three methods to work out the area of the kite in the square?

### 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.

### 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?

### Square Pair Circles

##### Stage: 5 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.

### 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.

### 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.

### Calculating with Cosines

##### Stage: 4 and 5 Challenge Level:

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

### Rhombus in Rectangle

##### Stage: 4 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.

### 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

### Salinon

##### Stage: 4 Challenge Level:

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

### 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.

### Pythagorean Golden Means

##### Stage: 5 Challenge Level:

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.

### 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. . . .

### Notty Logic

##### Stage: 5 Challenge Level:

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

### Pythagoras Proofs

##### Stage: 4 Challenge Level:

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

### Zig Zag

##### Stage: 4 Challenge Level:

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

### An Introduction to Number Theory

##### Stage: 5

An introduction to some beautiful results of Number Theory

### Magic Squares II

##### Stage: 4 and 5

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