Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
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?
Delight your friends with this cunning trick! Can you explain how
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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.
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
This challenge asks you to imagine a snake coiling on itself.
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Can you explain how this card trick works?
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?
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
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.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Find out what a "fault-free" rectangle is and try to make some of
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
An investigation that gives you the opportunity to make and justify
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
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?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
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 centimetres of rope will I need to make another mat just
like the one I have here?
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 ...?
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?
What happens when you round these three-digit numbers to the nearest 100?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Find the sum of all three-digit numbers each of whose digits is
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 and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?
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?
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.
What happens when you round these numbers to the nearest whole number?
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?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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?
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
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 see how to build a harmonic triangle? Can you work out the next two rows?