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

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

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

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

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

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

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

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?

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?

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?

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?

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

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?

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.

Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

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?

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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?

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

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

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?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

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

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.

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.

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.

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

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

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

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

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

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?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

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

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?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?

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

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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 practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.