Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

To avoid losing think of another very well known game where the patterns of play are similar.

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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.

Show that all pentagonal numbers are one third of a triangular number.

Can you describe this route to infinity? Where will the arrows take you next?

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?

Can you find a rule which connects consecutive triangular numbers?

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm.

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. . . .

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

Can you make a tetrahedron whose faces all have the same perimeter?

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.

This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

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?

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?

Square It game for an adult and child. Can you come up with a way of always winning this game?

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo!

Can you find a rule which relates triangular numbers to square numbers?

Every day at noon a boat leaves Le Havre for New York while another boat leaves New York for Le Havre. The ocean crossing takes seven days. How many boats will each boat cross during their journey?

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

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?

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

I found these clocks in the Arts Centre at the University of Warwick intriguing - do they really need four clocks and what times would be ambiguous with only two or three of them?

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

When dice land edge-up, we usually roll again. But what if we didn't...?

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

See if you can anticipate successive 'generations' of the two animals shown here.

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

What can you see? What do you notice? What questions can you ask?