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

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

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

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?

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

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?

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

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

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

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

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.

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?

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

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.

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

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

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.

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.

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?

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

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.

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

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

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?

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?

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?

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?

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

A few extra challenges set by some young NRICH members.

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

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

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?

Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?

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?

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

A Sudoku that uses transformations as supporting clues.

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.

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

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

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

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

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

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?

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.

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.