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?
This challenge asks you to imagine a snake coiling on itself.
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?
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?
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.
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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.
Take a look at the multiplication square. The first eleven triangle
numbers have been identified. Can you see a pattern? Does the
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?
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?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
This activity involves rounding four-digit numbers to the nearest thousand.
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 a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
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 find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Can you tangle yourself up and reach any fraction?
It would be nice to have a strategy for disentangling any tangled
Are these statements always true, sometimes true or never true?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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.
Can you find all the ways to get 15 at the top of this triangle of numbers?
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?
This task follows on from Build it Up and takes the ideas into three dimensions!
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
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 challenge focuses on finding the sum and difference of pairs of two-digit numbers.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.