How many centimetres of rope will I need to make another mat just
like the one I have here?
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Find the sum of all three-digit numbers each of whose digits is
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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?
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.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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 explain how this card trick works?
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
Delight your friends with this cunning trick! Can you explain how
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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 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 put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Here are two kinds of spirals for you to explore. What do you notice?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
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.
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
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
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
What would be the smallest number of moves needed to move a Knight
from a chess set from one corner to the opposite corner of a 99 by
99 square board?
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.
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?
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.
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 find all the ways to get 15 at the top of this triangle of numbers?
This task follows on from Build it Up and takes the ideas into three dimensions!
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
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?
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?
What happens when you round these numbers to the nearest whole number?
What happens when you round these three-digit numbers to the nearest 100?
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?
Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the
numbers on each circle add up to the same amount. Can you find the
rule for giving another set of six numbers?
This challenge asks you to imagine a snake coiling on itself.
This activity involves rounding four-digit numbers to the nearest thousand.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.