15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

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

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.

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.

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

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

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

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.

How many solutions can you find to this sum? Each of the different letters stands for a different number.

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

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

The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?

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?

An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.

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?

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

Solve the equations to identify the clue numbers in this Sudoku problem.

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

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.

Find the values of the nine letters in the sum: FOOT + BALL = GAME

Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.

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

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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?

Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

How many different triangles can you make on a circular pegboard that has nine pegs?

What is the best way to shunt these carriages so that each train can continue its journey?

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

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

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

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?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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.

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?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?