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?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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.
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Delight your friends with this cunning trick! Can you explain how
Here are two kinds of spirals for you to explore. What do you notice?
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?
Can you explain how this card trick works?
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 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?
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.
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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 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?
This activity involves rounding four-digit numbers to the nearest thousand.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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.
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.
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?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
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?
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.
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
This task follows on from Build it Up and takes the ideas into three dimensions!
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?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
Can you find all the ways to get 15 at the top of this triangle of numbers?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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. . . .
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?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
The number of plants in Mr McGregor's magic potting shed increases
overnight. He'd like to put the same number of plants in each of
his gardens, planting one garden each day. How can he do it?
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?
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Can you tangle yourself up and reach any fraction?