What happens when you add three numbers together? Will your answer be odd or even? How do you know?

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

Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

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

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

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

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

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

In this maze of hexagons, you start in the centre at 0. The next hexagon must be a multiple of 2 and the next a multiple of 5. What are the possible paths you could take?

Find out about Magic Squares in this article written for students. Why are they magic?!

Throughout these challenges, the touching faces of any adjacent dice must have the same number. Can you find a way of making the total on the top come to each number from 11 to 18 inclusive?

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?

What happens when you try and fit the triomino pieces into these two grids?

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

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.

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.

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

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.

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

Use the information to describe these marbles. What colours must be on marbles that sparkle when rolling but are dark inside?

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?

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.

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

Can you find the chosen number from the grid using the clues?

Here are some rods that are different colours. How could I make a yellow rod using white and red rods?

This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?

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

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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

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.

How will you go about finding all the jigsaw pieces that have one peg and one hole?

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

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

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

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

What is the best way to shunt these carriages so that each train can continue its journey?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

How many trains can you make which are the same length as Matt's and Katie's, using rods that are identical?

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

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?

Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?