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?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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 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?
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?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
Can you find the values at the vertices when you know the values on
Find out what a "fault-free" rectangle is and try to make some of
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?
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
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 see why 2 by 2 could be 5? Can you predict what 2 by 10
Try adding together the dates of all the days in one week. Now
multiply the first date by 7 and add 21. Can you explain what
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
This challenge asks you to imagine a snake coiling on itself.
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Can you explain how this card trick works?
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.
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.
It would be nice to have a strategy for disentangling any tangled
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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.
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.
Can you tangle yourself up and reach any fraction?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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. . . .
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
It starts quite simple but great opportunities for number discoveries and patterns!
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
This activity involves rounding four-digit numbers to the nearest thousand.
In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.
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.