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

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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.

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Rolling Coins

Age 14 to 16 Challenge Level:

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

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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?

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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. . . .

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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.

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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?

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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?

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Pythagorean Triples II

Age 11 to 16

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

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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!

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The Triangle Game

Age 11 to 16 Challenge Level:

Can you discover whether this is a fair game?

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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.

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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?

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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.

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Towering Trapeziums

Age 14 to 16 Challenge Level:

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

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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?

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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?

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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. . . .

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Cyclic Quadrilaterals

Age 11 to 16 Challenge Level:

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

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Matter of Scale

Age 14 to 16 Challenge Level:

Prove Pythagoras' Theorem using enlargements and scale factors.

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Leonardo's Problem

Age 14 to 18 Challenge Level:

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

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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.

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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?

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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. . . .

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Gift of Gems

Age 14 to 16 Challenge Level:

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

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Stonehenge

Age 16 to 18 Challenge Level:

Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.

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Triangle Incircle Iteration

Age 14 to 16 Challenge Level:

Keep constructing triangles in the incircle of the previous triangle. What happens?

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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.

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Proximity

Age 14 to 16 Challenge Level:

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

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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.

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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.

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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.

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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.

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Proofs with Pictures

Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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Shape and Territory

Age 16 to 18 Challenge Level:

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

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Quadratic Harmony

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.

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Problem Solving, Using and Applying and Functional Mathematics

Age 5 to 18 Challenge 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.

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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?

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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?

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Dalmatians

Age 14 to 18 Challenge 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.

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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. . . .

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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?

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AMGM

Age 14 to 16 Challenge Level:

Can you use the diagram to prove the AM-GM inequality?

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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.

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Pythagoras Proofs

Age 14 to 16 Challenge Level:

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

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An Introduction to Number Theory

Age 16 to 18

An introduction to some beautiful results of Number Theory

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Air Nets

Age 7 to 18 Challenge Level:

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

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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?

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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?

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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.

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Folding Fractions

Age 14 to 16 Challenge Level:

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