Can you work out how to win this game of Nim? Does it matter if you
go first or second?
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?
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
Find out what a "fault-free" rectangle is and try to make some of
Can you explain how this card trick works?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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?
In each of the pictures the invitation is for you to: Count what
you see. Identify how you think the pattern would continue.
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.
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
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 asks you to imagine a snake coiling on itself.
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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.
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
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.
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
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?
Can you find the values at the vertices when you know the values on
A collection of games on the NIM theme
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Start with two numbers. This is the start of a sequence. The next
number is the average of the last two numbers. Continue the
sequence. What will happen if you carry on for ever?
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?
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?
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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.
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?
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
Imagine starting with one yellow cube and covering it all over with
a single layer of red cubes, and then covering that cube with a
layer of blue cubes. How many red and blue cubes would you need?
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.
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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
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.
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.