# Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical reasoning & proof ### Napoleon's Hat

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

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR? ### Proof Sorter - Geometric Series

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

Can you correctly order the steps in the proof of the formula for the sum of a geometric series? ### Direct Logic

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

Can you work through these direct proofs, using our interactive proof sorters? ### 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. ##### Age 14 to 16 Challenge Level:

Four jewellers share their stock. Can you work out the relative values of their gems? ### Thousand Words

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

Here the diagram says it all. Can you find the diagram? ### 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. . . . ### Towering Trapeziums

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

Can you find the areas of the trapezia in this sequence? ### Areas and Ratios

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

Do you have enough information to work out the area of the shaded quadrilateral? ### Pythagorean Triples II

##### Age 11 to 16

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

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

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle? ### Middle Man

##### Age 16 to 18 Challenge 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? ### 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? ### Iffy Logic

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

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

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

Have a go at being mathematically negative, by negating these statements. ### 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! ### Proof of Pick's Theorem

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

Follow the hints and prove Pick's Theorem. ### 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? ### Can it Be

##### Age 16 to 18 Challenge Level: ### 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. ### Proof Sorter - Sum of an AP

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

Use this interactivity to sort out the steps of the proof of the formula for the sum of an arithmetic series. The 'thermometer' will tell you how you are doing ### 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. ### The Triangle Game

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

Can you discover whether this is a fair game? ### 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? ### 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. ### 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. ### Matter of Scale

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

Prove Pythagoras' Theorem using enlargements and scale factors. ### 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. . . . ### Picture Story

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

Can you see how this picture illustrates the formula for the sum of the first six cube 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. ### 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. . . . ### 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? ### 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. ### Mind Your Ps and Qs

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

Sort these mathematical propositions into a series of 8 correct statements. ### Plus or Minus

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

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$. ### 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? ### 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? ### 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. ### 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. ### 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. ### 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. ### 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/...)). ### 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. ### 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. ### Pareq Exists

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

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines. ### Magic Squares II

##### Age 14 to 18

An article which gives an account of some properties of magic squares. ### 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. ### Little and Large

##### Age 16 to 18 Challenge 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?