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?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. 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.

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

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.

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?

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

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

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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

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

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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?

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.

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.

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

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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?

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

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

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

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

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.

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?

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

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. . . .

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.

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

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

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?

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

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?

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

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

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?

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.

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

Can you find the values at the vertices when you know the values on the edges?

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?

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

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?