Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
This activity involves rounding four-digit numbers to the nearest thousand.
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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 find the values at the vertices when you know the values on
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?
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.
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Delight your friends with this cunning trick! Can you explain how
Can you explain how this card trick works?
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?
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
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 are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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?
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?
This challenge asks you to imagine a snake coiling on itself.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
How many centimetres of rope will I need to make another mat just
like the one I have here?
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?
Polygons drawn on square dotty paper have dots on their perimeter
(p) and often internal (i) ones as well. Find a relationship
between p, i and the area of the polygons.
Can you find sets of sloping lines that enclose a square?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
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.
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?
It starts quite simple but great opportunities for number discoveries and patterns!
What happens when you round these numbers to the nearest whole number?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
Consider all two digit numbers (10, 11, . . . ,99). In writing down
all these numbers, which digits occur least often, and which occur
most often ? What about three digit numbers, four digit numbers. . . .
What happens when you round these three-digit numbers to the nearest 100?
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.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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
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?
Charlie has moved between countries and the average income of both
has increased. How can this be so?
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
Find out what a "fault-free" rectangle is and try to make some of
It would be nice to have a strategy for disentangling any tangled
A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?
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 you have a large supply of 3kg and 8kg weights. How many of
each weight would you need for the average (mean) of the weights to
be 6kg? What other averages could you have?
Can you tangle yourself up and reach any fraction?