A monkey with peaches, keeps a fraction of them each day, gives the rest away, and then eats one. How long can his peaches last?

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

Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?

The items in the shopping basket add and multiply to give the same amount. What could their prices be?

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

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

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

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

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?

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?

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

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?

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

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

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

A Sudoku that uses transformations as supporting clues.

My dice has inky marks on each face. Can you find the route it has taken? What does each face look like?

What could the half time scores have been in these Olympic hockey matches?

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.

This article for primary teachers suggests ways in which to help children become better at working systematically.

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 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.

Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

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?

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

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

A few extra challenges set by some young NRICH members.

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

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

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

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?

This Sudoku puzzle can be solved with the help of small clue-numbers on the border lines between pairs of neighbouring squares of the grid.

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?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

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

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

This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.

This Sudoku, based on differences. Using the one clue number can you find the solution?