The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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?
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.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Can you explain how this card trick works?
Delight your friends with this cunning trick! Can you explain how
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?
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?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
This challenge asks you to imagine a snake coiling on itself.
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 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?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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
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.
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
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?
Can you tangle yourself up and reach any fraction?
It would be nice to have a strategy for disentangling any tangled
Can you find the values at the vertices when you know the values on
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?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
Great Granddad is very proud of his telegram from the Queen
congratulating him on his hundredth birthday and he has friends who
are even older than he is... When was he born?
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?
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.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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.
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?
Explore the effect of combining enlargements.
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.
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?
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?
A package contains a set of resources designed to develop
pupils’ mathematical thinking. This package places a
particular emphasis on “generalising” and is designed
to meet the. . . .
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
This activity involves rounding four-digit numbers to the nearest thousand.
Can you describe this route to infinity? Where will the arrows take you next?