Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Delight your friends with this cunning trick! Can you explain how
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.
Can you explain how this card trick works?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Can you work out how to win this game of Nim? Does it matter if you go first or second?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
This challenge asks you to imagine a snake coiling on itself.
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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?
It starts quite simple but great opportunities for number discoveries and patterns!
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?
Find out what a "fault-free" rectangle is and try to make some of
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
Here are two kinds of spirals for you to explore. What do you notice?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Charlie has moved between countries and the average income of both
has increased. How can this be so?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
It would be nice to have a strategy for disentangling any tangled
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.
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
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.
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.
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.
Got It game for an adult and child. How can you play so that you know you will always win?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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 find the values at the vertices when you know the values on
Can you tangle yourself up and reach any fraction?
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?
An investigation that gives you the opportunity to make and justify
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
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?
What happens when you round these three-digit numbers to the nearest 100?