Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?

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.

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . .

A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

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?

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?

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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

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

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

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

A game for 2 people. Take turns joining two dots, until your opponent is unable to move.

A game for two players based on a game from the Somali people of Africa. The first player to pick all the other's 'pumpkins' is the winner.

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

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you. . . .

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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. . . .

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.

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

A huge wheel is rolling past your window. What do you see?

Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . .

An ancient game for two from Egypt. You'll need twelve distinctive 'stones' each to play. You could chalk out the board on the ground - do ask permission first.

In how many different ways can I colour the five edges of a pentagon red, blue and green so that no two adjacent edges are the same colour?

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?

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

A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?

How many different symmetrical shapes can you make by shading triangles or squares?

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

This is a simple version of an ancient game played all over the world. It is also called Mancala. What tactics will increase your chances of winning?

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

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

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

How many moves does it take to swap over some red and blue frogs? Do you have a method?

How many ways can you write the word EUROMATHS by starting at the top left hand corner and taking the next letter by stepping one step down or one step to the right in a 5x5 array?

This game for two, was played in ancient Egypt as far back as 1400 BC. The game was taken by the Moors to Spain, where it is mentioned in 13th century manuscripts, and the Spanish name Alquerque. . . .

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.

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.

This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.

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.