Other tags that relate to Pareq Exists
Circle properties and circle theorems. Factors and multiples. Mathematical reasoning & proof. Triangles. Similarity and congruence. Pythagoras' theorem. Generalising. Visualising. Constructions. Creating and manipulating expressions and formulae.There are 160 results
Broad Topics > Thinking Mathematically > Mathematical reasoning & proof
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.
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?
Pythagorean Triples II
Age 11 to 16
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Chameleons
Age 11 to 14
Challenge Level
Whenever two chameleons of different colours meet they change colour to the third colour. Describe the shortest sequence of meetings in which all the chameleons change to green if you start with 12. . . .
Folding Fractions
Age 14 to 16
Challenge Level
What fractions can you divide the diagonal of a square into by simple folding?
Lens Angle
Age 14 to 16
Challenge Level
Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.
Find the Fake
Age 14 to 16
Challenge Level
There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?
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?
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!
Converse
Age 14 to 16
Challenge Level
Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?
Cosines Rule
Age 14 to 16
Challenge Level
Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.
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?
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?
Zig Zag
Age 14 to 16
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?
A Chordingly
Age 11 to 14
Challenge Level
Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.
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.
Gift of Gems
Age 14 to 16
Challenge Level
Four jewellers share their stock. Can you work out the relative values of their gems?
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.
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. . . .
Towering Trapeziums
Age 14 to 16
Challenge Level
Can you find the areas of the trapezia in this sequence?
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?
Con Tricks
Age 11 to 14
Here are some examples of 'cons', and see if you can figure out where the trick is.
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.
Cross-country Race
Age 14 to 16
Challenge Level
Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?
Is it Magic or Is it Maths?
Age 11 to 14
Challenge Level
Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . .
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?
Thirty Nine, Seventy Five
Age 11 to 14
Challenge Level
We have exactly 100 coins. There are five different values of coins. We have decided to buy a piece of computer software for 39.75. We have the correct money, not a penny more, not a penny less! Can. . . .
Always the Same
Age 11 to 14
Challenge Level
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?
Coins on a Plate
Age 11 to 14
Challenge 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.
Volume of a Pyramid and a Cone
Age 11 to 14
These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.
Matter of Scale
Age 14 to 16
Challenge Level
Prove Pythagoras' Theorem using enlargements and scale factors.
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.
Disappearing Square
Age 11 to 14
Challenge 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. . . .
Circle Box
Age 14 to 16
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?
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.
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?
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.
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.
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?
Diophantine N-tuples
Age 14 to 16
Challenge Level
Can you explain why a sequence of operations always gives you perfect squares?
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.
Russian Cubes
Age 14 to 16
Challenge Level
I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?
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?
More Mathematical Mysteries
Age 11 to 14
Challenge Level
Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .
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?
Logic
Age 7 to 14
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
Cyclic Quad Jigsaw
Age 14 to 16
Challenge Level
A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?
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.
Tourism
Age 11 to 14
Challenge 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.
NRICH is part of the family of activities in the Millennium Mathematics Project.