Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
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
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
Think of a number, square it and subtract your starting number. Is
the number you’re left with odd or even? How do the images
help to explain this?
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?
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?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Find out what a "fault-free" rectangle is and try to make some of
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Delight your friends with this cunning trick! Can you explain how
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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.
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?
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
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?
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?
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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.
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?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
This article for teachers describes several games, found on the
site, all of which have a related structure that can be used to
develop the skills of strategic planning.
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
Got It game for an adult and child. How can you play so that you know you will always win?
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
Find the sum of all three-digit numbers each of whose digits is
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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?
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.
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?
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?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
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?
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
This activity involves rounding four-digit numbers to the nearest thousand.