Problem solving is at the heart of the NRICH site. All the problems
give learners opportunities to learn, develop or use mathematical
concepts and skills. Read here for more information.
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.
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?
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.
A game for 2 players with similaritlies to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
A collection of games on the NIM theme
It starts quite simple but great opportunities for number discoveries and patterns!
This challenge asks you to imagine a snake coiling on itself.
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
A package contains a set of resources designed to develop
pupils’ mathematical thinking. This package places a
particular emphasis on “generalising” and is designed
to meet the. . . .
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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 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 work out how to win this game of Nim? Does it matter if you go first or second?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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.
If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?
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?
Two children made up a game as they walked along the garden paths.
Can you find out their scores? Can you find some paths of your own?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
This problem challenges you to find out how many odd numbers there
are between pairs of numbers. Can you find a pair of numbers that
has four odds between them?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
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?
Got It game for an adult and child. How can you play so that you know you will always win?
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?
Stop the Clock game for an adult and child. How can you make sure you always win this game?
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
This challenge is about finding the difference between numbers which have the same tens digit.
This activity focuses on rounding to the nearest 10.
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?
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?
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?
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
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?
Find the sum of all three-digit numbers each of whose digits is
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.