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?

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

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?

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

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?

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

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

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

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

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?

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.

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

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?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

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?

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

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

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?

A few extra challenges set by some young NRICH members.

A pair of Sudokus with lots in common. In fact they are the same problem but rearranged. Can you find how they relate to solve them both?

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

A Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.

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

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

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.

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

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?

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?

Four friends must cross a bridge. How can they all cross it in just 17 minutes?

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

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

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

A man has 5 coins in his pocket. Given the clues, can you work out what the coins are?

Each clue in this Sudoku is the product of the two numbers in adjacent cells.

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

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?

Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

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

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

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.

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

Given the products of adjacent cells, can you complete this Sudoku?