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 NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

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.

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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?

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?

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?

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

in how many ways can you place the numbers 1, 2, 3 … 9 in the nine regions of the Olympic Emblem (5 overlapping circles) so that the amount in each ring is the same?

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

It is possible to identify a particular card out of a pack of 15 with the use of some mathematical reasoning. What is this reasoning and can it be applied to other numbers of cards?

60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?

Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

Given the products of diagonally opposite cells - can you complete this Sudoku?

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?

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

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

This Sudoku requires you to do some working backwards before working forwards.

Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?

This Sudoku puzzle can be solved with the help of small clue-numbers on the border lines between pairs of neighbouring squares of the grid.

Have a go at this game which has been inspired by the Big Internet Math-Off 2019. Can you gain more columns of lily pads than your opponent?

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

By selecting digits for an addition grid, what targets can you make?

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

You need to find the values of the stars before you can apply normal Sudoku rules.

A Sudoku that uses transformations as supporting clues.

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.

This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?

If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

Use the clues about the shaded areas to help solve this sudoku

Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".

This is a variation of sudoku which contains a set of special clue-numbers. Each set of 4 small digits stands for the numbers in the four cells of the grid adjacent to this set.

Each of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4.

A monkey with peaches, keeps a fraction of them each day, gives the rest away, and then eats one. How long can his peaches last?

Two sudokus in one. Challenge yourself to make the necessary connections.

Find out about Magic Squares in this article written for students. Why are they magic?!

The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . .

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

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

Two sudokus in one. Challenge yourself to make the necessary connections.