Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

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

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

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

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

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

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

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

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?

Ben’s class were making cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

An investigation that gives you the opportunity to make and justify predictions.

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

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

Can you work out how to win this game of Nim? Does it matter if you go first or second?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

Can you find the values at the vertices when you know the values on the edges?

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

This activity involves rounding four-digit numbers to the nearest thousand.

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

What happens when you round these three-digit numbers to the nearest 100?

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?

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.

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.

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

How many moves does it take to swap over some red and blue frogs? Do you have a method?