Are these statements 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?
Are these statements relating to odd and even numbers 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.
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?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Ben’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?
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.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
An investigation that gives you the opportunity to make and justify predictions.
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 encourages you to explore dividing a three-digit number by a single-digit number.
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.
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?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Can you find all the ways to get 15 at the top of this triangle of numbers?
Got It game for an adult and child. How can you play so that you know you will always win?
Can you explain how this card trick works?
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?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.
This task follows on from Build it Up and takes the ideas into three dimensions!
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?
In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 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?
This challenge asks you to imagine a snake coiling on itself.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
This activity involves rounding four-digit numbers to the nearest thousand.
Delight your friends with this cunning trick! Can you explain how it works?
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?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
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.
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.
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?
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 explain the strategy for winning this game with any target?
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.
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.
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?
Find out what a "fault-free" rectangle is and try to make some of your own.
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 the sum of all three-digit numbers each of whose digits is odd.
How many centimetres of rope will I need to make another mat just like the one I have here?
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.
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?