# Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Thinking Mathematically > Mathematical reasoning & proof ### Picture Story

##### Age 14 to 16Challenge Level

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

##### Age 14 to 16Challenge Level

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

##### Age 11 to 14Challenge Level

Which hexagons tessellate? ### Matter of Scale

##### Age 14 to 16Challenge Level

Prove Pythagoras' Theorem using enlargements and scale factors. ### Proof Sorter - Quadratic Equation

##### Age 14 to 18Challenge 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 11 to 16Challenge Level

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem? ##### Age 11 to 14Challenge Level

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . . ### Disappearing Square

##### Age 11 to 14Challenge Level

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . . ### L-triominoes

##### Age 14 to 16Challenge Level

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

##### Age 14 to 16Challenge 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. . . . ### Problem Solving, Using and Applying and Functional Mathematics

##### Age 5 to 18Challenge 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. ### Magic Squares II

##### Age 14 to 18

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

##### Age 14 to 16Challenge Level

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

##### Age 11 to 16Challenge Level

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

##### Age 14 to 16Challenge 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. ##### Age 14 to 16Challenge Level

Kyle and his teacher disagree about his test score - who is right? ### More Number Pyramids

##### Age 11 to 14Challenge Level

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge... ### The Triangle Game

##### Age 11 to 16Challenge Level

Can you discover whether this is a fair game? ### Tower of Hanoi

##### Age 11 to 14Challenge Level

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice. ### Concrete Wheel

##### Age 11 to 14Challenge Level

A huge wheel is rolling past your window. What do you see? ### The Bridges of Konigsberg

##### Age 11 to 18Challenge Level

Investigate how networks can be used to solve a problem for the 18th Century inhabitants of Konigsberg. ### Convex Polygons

##### Age 11 to 14Challenge Level

Show that among the interior angles of a convex polygon there cannot be more than three acute angles. ### Circle Box

##### Age 14 to 16Challenge 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? ### The Pillar of Chios

##### Age 14 to 16Challenge 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. ### Towering Trapeziums

##### Age 14 to 16Challenge Level

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

##### Age 14 to 18Challenge Level

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence. ### 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. ### Go Forth and Generalise

##### Age 11 to 14

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important. ### A Chordingly

##### Age 11 to 14Challenge Level

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle. ##### Age 7 to 14Challenge Level

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed? ### Clocked

##### Age 11 to 14Challenge Level

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours? ### Proof: A Brief Historical Survey

##### Age 14 to 18

If you think that mathematical proof is really clearcut and universal then you should read this article. ### 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. . . . ### Round and Round

##### Age 14 to 16Challenge Level

Prove that the shaded area of the semicircle is equal to the area of the inner circle. ### Coins on a Plate

##### Age 11 to 14Challenge Level

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle. ### Rhombus in Rectangle

##### Age 14 to 16Challenge 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. ### DOTS Division

##### Age 14 to 16Challenge Level

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}. ### Long Short

##### Age 14 to 16Challenge Level

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

##### Age 11 to 14Challenge Level

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps? ### Proofs with Pictures

##### Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities. ### Angle Trisection

##### Age 14 to 16Challenge Level

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

##### Age 11 to 16

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

##### Age 14 to 16Challenge 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. . . . ### Tourism

##### Age 11 to 14Challenge Level

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable. ### 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. ### Iffy Logic

##### Age 14 to 18Challenge Level

Can you rearrange the cards to make a series of correct mathematical statements? ### Kite in a Square

##### Age 14 to 16Challenge Level

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