When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?

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

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?

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

Try out the lottery that is played in a far-away land. What is the chance of winning?

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

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

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

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?

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

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.

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

Each of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4.

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?

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

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.

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

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

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

Have a go at this game which has been inspired by the Big Internet Math-Off 2019. Can you gain more columns of lily pads than your opponent?

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?

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

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

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

A few extra challenges set by some young NRICH members.

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

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?

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.

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?

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.

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 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.

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?

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?

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?

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

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.

How long does it take to brush your teeth? Can you find the matching length of time?

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

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

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?

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

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

The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . .

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?

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?