Can you work out how to win this game of Nim? Does it matter if you go first or second?
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.
Can you continue this pattern of triangles and begin to predict how
many sticks are used for each new "layer"?
Delight your friends with this cunning trick! Can you explain how
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?
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?
Can you explain how this card trick works?
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
This challenge asks you to imagine a snake coiling on itself.
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
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?
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.
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
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.
Ben’s class were making cutting up number tracks. First they
cut them into twos and added up the numbers on each piece. What
patterns could they see?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
It starts quite simple but great opportunities for number discoveries and patterns!
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 would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
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?
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.
How many centimetres of rope will I need to make another mat just
like the one I have here?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
An investigation that gives you the opportunity to make and justify
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?
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?
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?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
Can you see how to build a harmonic triangle? Can you work out the
next two rows?
Find out what a "fault-free" rectangle is and try to make some of
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind