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

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?

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

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

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

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

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.

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?

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?

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

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

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?

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

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

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?

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. . . .

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.

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?

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?

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

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.

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?

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?

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

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

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

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

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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.

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

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

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

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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

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

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?

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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

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

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

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

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

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

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

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

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?