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.

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and record your findings in truth tables.

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

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. . . .

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

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. . . .

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

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

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.

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.

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?

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

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

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?

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.

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

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

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

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?

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.

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?

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

By proving these particular identities, prove the existence of general cases.

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

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.

Kyle and his teacher disagree about his test score - who is right?

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...

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?

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.

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.

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.

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

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.

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. . . .

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

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

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

When is it impossible to make number sandwiches?

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.

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

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.

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

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

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.