The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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
Can you explain how this card trick works?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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.
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?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Here are two kinds of spirals for you to explore. What do you notice?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
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?
We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4
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?
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?
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.
This challenge asks you to imagine a snake coiling on itself.
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.
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
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 are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Can you find the values at the vertices when you know the values on
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
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.
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
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?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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 tangle yourself up and reach any fraction?
It would be nice to have a strategy for disentangling any tangled
Charlie has moved between countries and the average income of both
has increased. How can this be so?
Can all unit fractions be written as the sum of two unit fractions?
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
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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 Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?