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

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

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?

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

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.

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

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?

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

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

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?

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

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

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

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?

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.

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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

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?

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

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

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

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?

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

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?

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

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

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

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?

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

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

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?

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 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.

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?

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

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.

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

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.

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?

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?

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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.

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?

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?