# Resources tagged with: Mathematical reasoning & proof

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### There are 171 results

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

### Pent

##### Age 14 to 18 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.

### Matter of Scale

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

Prove Pythagoras' Theorem using enlargements and scale factors.

### 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.

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

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

### Folding Fractions

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

What fractions can you divide the diagonal of a square into by simple folding?

### Folding Squares

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

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?

### 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.

### Pythagoras Proofs

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

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

### Parallel Universe

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

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

### Same Length

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

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

### Kite in a Square

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

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

### Towering Trapeziums

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

Can you find the areas of the trapezia in this sequence?

### Fitting In

##### Age 14 to 16 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. . . .

### 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.

### 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?

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

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

### Proof Sorter - Quadratic Equation

##### Age 14 to 18 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.

### Target Six

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

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

### Salinon

##### Age 14 to 16 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?

### The Golden Ratio, Fibonacci Numbers and Continued Fractions.

##### Age 14 to 16

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.

### Plus or Minus

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

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

### Square Mean

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

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

### Areas and Ratios

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

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

### 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/...)).

### Encircling

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

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

### Pythagorean Golden Means

##### Age 16 to 18 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.

### 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.

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

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

### The Pillar of Chios

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

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

### 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.

### Similarly So

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

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

### 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.

### Pair Squares

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

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.

### Big, Bigger, Biggest

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

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

### 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.

### 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.

### 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.

### Proofs with Pictures

##### Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

### Common Divisor

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

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

### Long Short

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

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

### Our Ages

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

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

### There's a Limit

##### Age 14 to 18 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?

### Thousand Words

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

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

##### 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.

### 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?

### Composite Notions

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

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

### 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?

### Round and Round

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

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

### 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.