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?

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

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

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

Got It game for an adult and child. How can you play so that you know you will always win?

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?

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.

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

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

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

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

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?

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?

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?

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?

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?

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.

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

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

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.

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?

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

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

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

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

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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.

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

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

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

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

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

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.

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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

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.

Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two 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?

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. . . .

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

What happens when you round these numbers to the nearest whole number?

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?