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?

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.

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

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?"

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?

Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E

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

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

How many solutions can you find to this sum? Each of the different letters stands for a different number.

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

I added together some of my neighbours house numbers. Can you explain the patterns I noticed?

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

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

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?

Special clue numbers related to the difference between numbers in two adjacent cells and values of the stars in the "constellation" make this a doubly interesting problem.

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.

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

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.

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?

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?

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

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

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?

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

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

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

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

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original 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?

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?

A challenging activity focusing on finding all possible ways of stacking rods.

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.

Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.

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

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

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

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?

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

Take three whole numbers. The differences between them give you three new numbers. Find the differences between the new numbers and keep repeating this. What happens?

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.

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

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

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

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

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

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

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

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.

A few extra challenges set by some young NRICH members.

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.