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?

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

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?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

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

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?

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

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 will you go about finding all the jigsaw pieces that have one peg and one hole?

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

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?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

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?

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

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

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

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

The clues for this Sudoku are the product of the numbers in adjacent squares.

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?

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

The Zargoes use almost the same alphabet as English. What does this birthday message say?

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

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

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.

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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?

These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?