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?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Find out what a "fault-free" rectangle is and try to make some of
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?
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
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.
Can you explain how this card trick works?
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?
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?
Can you find the values at the vertices when you know the values on
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
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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.
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
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?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
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?
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.
This challenge asks you to imagine a snake coiling on itself.
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.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Delight your friends with this cunning trick! Can you explain how
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?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
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.
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Find the sum of all three-digit numbers each of whose digits is
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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.
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?
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.
Start with any number of counters in any number of piles. 2 players
take it in turns to remove any number of counters from a single
pile. The winner is the player to take the last counter.
Can you tangle yourself up and reach any fraction?
A collection of games on the NIM theme
It would be nice to have a strategy for disentangling any tangled
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
It starts quite simple but great opportunities for number discoveries and patterns!
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.