What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

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.

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?

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.

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?

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

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.

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

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

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

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

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?

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?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?

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

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

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

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?

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

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.

A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"

Countries from across the world competed in a sports tournament. Can you devise an efficient strategy to work out the order in which they finished?

The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?

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?

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

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

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 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.

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?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

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

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.

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.

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.

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

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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?

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

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

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

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

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

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

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