Are these statements always true, sometimes true or never true?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
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?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
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?
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 challenge encourages you to explore dividing a three-digit number by a single-digit number.
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.
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 investigation that gives you the opportunity to make and justify
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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?
This challenge asks you to imagine a snake coiling on itself.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Delight your friends with this cunning trick! Can you explain how
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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?
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
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?
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
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?
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.
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.
Can you explain how this card trick works?
This activity involves rounding four-digit numbers to the nearest thousand.
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
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
This task follows on from Build it Up and takes the ideas into three dimensions!
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?
Can you explain the strategy for winning this game with any target?
Find the sum of all three-digit numbers each of whose digits is
Can you find all the ways to get 15 at the top of this triangle of numbers?
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?
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.
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?
Got It game for an adult and child. How can you play so that you know you will always win?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
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?
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?
Find out what a "fault-free" rectangle is and try to make some of
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?