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

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

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?

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.

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.

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

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

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

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

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?

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

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

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?

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?

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?

How many centimetres of rope will I need to make another mat just like the one I have here?

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

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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?

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

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

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.

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?

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

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?

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.

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?

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

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

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

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?

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

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

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?

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

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

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

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

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.

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

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

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

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?

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?

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