This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

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?

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

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

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.

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?

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

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.

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

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

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?

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

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

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

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

This challenge is about finding the difference between numbers which have the same tens digit.

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?

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?

This is a game for two players. Can you find out how to be the first to get to 12 o'clock?

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

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

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?

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

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

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.

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

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?

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

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

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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.

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?

Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

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.

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

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

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

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?

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.

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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

It starts quite simple but great opportunities for number discoveries and patterns!