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?

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?

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

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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

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

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

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

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

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.

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

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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

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.

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?

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

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

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.

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

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?

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

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

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

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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

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?

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?

Delight your friends with this cunning trick! Can you explain how it works?

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

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

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?

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?

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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?

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.

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

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

What happens when you round these three-digit numbers to the nearest 100?

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