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#### Resources tagged with Mathematical reasoning & proof similar to Pythagoras’ Comma:

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Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical reasoning & proof ### Matter of Scale

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

Prove Pythagoras' Theorem using enlargements and scale factors. ### 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. ### 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? ### Elevenses

##### Age 11 to 14 Challenge Level:

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results? ### Folding Fractions

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

What fractions can you divide the diagonal of a square into by simple folding? ##### Age 11 to 14 Challenge Level:

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true. ### Unit Fractions

##### Age 11 to 14 Challenge Level:

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation. ### 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. ### 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. . . . ### Mod 3

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

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3. ### Gift of Gems

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

Four jewellers share their stock. Can you work out the relative values of their gems? ### 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. ### Why 24?

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

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results. ### Take Three from Five

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

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him? ### 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. ### What Numbers Can We Make Now?

##### Age 11 to 14 Challenge Level:

Imagine we have four bags containing numbers from a sequence. What numbers can we make now? ### Euler's Squares

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

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square... ### 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? ### 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? ### What Numbers Can We Make?

##### Age 11 to 14 Challenge Level:

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make? ### Proofs with Pictures

##### Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities. ### 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 ### 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? ### 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. ### The Triangle Game

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

Can you discover whether this is a fair game? ### 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... ### Königsberg

##### Age 11 to 14 Challenge 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? ### 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. ### Pattern of Islands

##### Age 11 to 14 Challenge Level:

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island... ### 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. ##### Age 14 to 16 Challenge Level:

Kyle and his teacher disagree about his test score - who is right? ### Magic Squares II

##### Age 14 to 18

An article which gives an account of some properties of magic squares. ### 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. ### A Knight's Journey

##### Age 14 to 18

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition. ### A Biggy

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

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power. ### Sixational

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

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . . ### More Number Sandwiches

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

When is it impossible to make number sandwiches? ### 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. ### 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? ### 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? ### Always Perfect

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

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square. ### Ratty

##### Age 11 to 14 Challenge Level:

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation? ### Children at Large

##### Age 11 to 14 Challenge Level:

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children? ### 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. . . . ### Pythagorean Triples II

##### Age 11 to 16

This is the second article on right-angled triangles whose edge lengths are whole numbers. ### 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! ### 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. . . . ### Calculating with Cosines

##### Age 14 to 18 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? ### 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.