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. . . .

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....?

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?

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?

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?

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.

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

Solve the equations to identify the clue numbers in this Sudoku problem.

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?

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?

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?

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?

A few extra challenges set by some young NRICH members.

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 pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?

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

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.

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

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

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

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

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.

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

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?

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.

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

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?

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

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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?

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

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

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?

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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

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

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.

A Sudoku that uses transformations as supporting clues.