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?
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?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?
Stop the Clock game for an adult and child. How can you make sure you always win this game?
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?
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?
This is a game for two players. Can you find out how to be the first to get to 12 o'clock?
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
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?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
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?
Here are two kinds of spirals for you to explore. What do you notice?
This activity involves rounding four-digit numbers to the nearest thousand.
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
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?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
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 challenge asks you to imagine a snake coiling on itself.
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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.
Find out what a "fault-free" rectangle is and try to make some of your own.
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?
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.
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.
Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.
Find the sum of all three-digit numbers each of whose digits is odd.
Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.
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?
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?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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.
This challenge is about finding the difference between numbers which have the same tens digit.
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?
How many centimetres of rope will I need to make another mat just like the one I have here?
What happens when you round these three-digit numbers to the nearest 100?