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?

Find out what a "fault-free" rectangle is and try to make some of your own.

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

This challenge extends the Plants investigation so now four or more children are involved.

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?

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

Can you find all the different ways of lining up these Cuisenaire rods?

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

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?

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?

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.

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.

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.

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.

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

The Zargoes use almost the same alphabet as English. What does this birthday message say?

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

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?

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

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

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

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

A few extra challenges set by some young NRICH members.

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

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

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

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.

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

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?

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

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

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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.

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 all the different triangles on these peg boards, and find their angles?

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

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

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

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?

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?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

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.

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

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?

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

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