Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Delight your friends with this cunning trick! Can you explain how
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
What happens when you round these numbers to the nearest whole number?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
What happens when you round these three-digit numbers to the nearest 100?
How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?
Can you explain how this card trick works?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
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.
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?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?
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.
Find out what a "fault-free" rectangle is and try to make some of
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
An investigation that gives you the opportunity to make and justify
Can you find all the ways to get 15 at the top of this triangle of numbers?
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
What happens if you join every second point on this circle? How
about every third point? Try with different steps and see if you
can predict what will happen.
Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
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?
Find the sum of all three-digit numbers each of whose digits is
We can arrange dots in a similar way to the 5 on a dice and they
usually sit quite well into a rectangular shape. How many
altogether in this 3 by 5? What happens for other sizes?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the
numbers on each circle add up to the same amount. Can you find the
rule for giving another set of six numbers?
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 explain the strategy for winning this game with any target?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
How can you arrange these 10 matches in four piles so that when you
move one match from three of the piles into the fourth, you end up
with the same arrangement?
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?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
This task follows on from Build it Up and takes the ideas into three dimensions!