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?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Are these statements relating to odd and even numbers always true, sometimes true or never true?
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.
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?
Here are two kinds of spirals for you to explore. What do you notice?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
This challenge asks you to imagine a snake coiling on itself.
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?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
This activity involves rounding four-digit numbers to the nearest thousand.
Can you explain how this card trick works?
Delight your friends with this cunning trick! Can you explain how
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Are these statements always true, sometimes true or never true?
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.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Choose any 3 digits and make a 6 digit number by repeating the 3
digits in the same order (e.g. 594594). Explain why whatever digits
you choose the number will always be divisible by 7, 11 and 13.
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.
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
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?
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?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
How many centimetres of rope will I need to make another mat just
like the one I have here?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
List any 3 numbers. It is always possible to find a subset of
adjacent numbers that add up to a multiple of 3. Can you explain
why and prove it?
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?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Find some examples of pairs of numbers such that their sum is a
factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and
16 is a factor of 48.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
An investigation that gives you the opportunity to make and justify
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?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
Find the sum of all three-digit numbers each of whose digits is
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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?
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
This task follows on from Build it Up and takes the ideas into three dimensions!
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.