Are these statements relating to odd and even numbers always true, sometimes true or never true?

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

Here are two kinds of spirals for you to explore. What do you notice?

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

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.

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?

Florence, Ethan and Alma have each added together two 'next-door' numbers. What is the same about their answers?

Are these statements always true, sometimes true or never true?

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

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?

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

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

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

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.

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?

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?

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

Are these statements relating to calculation and properties of shapes always true, sometimes true or never true?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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?

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

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?

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

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?

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

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

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

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

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

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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

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

This task follows on from Build it Up and takes the ideas into three dimensions!

Can you find all the ways to get 15 at the top of this triangle of numbers?

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?

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

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.

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

A game for 2 players with similaritlies to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

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?

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?

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

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.

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.

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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