A collection of games on the NIM theme
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.
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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.
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
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
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten.
Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
Can you describe this route to infinity? Where will the arrows take you next?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
List any 3 numbers. It is always possible to find a subset of
adjacent numbers that add up to a multiple of 3. Can you explain
why and prove it?
Think of a number, add one, double it, take away 3, add the number
you first thought of, add 7, divide by 3 and take away the number
you first thought of. You should now be left with 2. How do I. . . .
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?
Choose any 3 digits and make a 6 digit number by repeating the 3
digits in the same order (e.g. 594594). Explain why whatever digits
you choose the number will always be divisible by 7, 11 and 13.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
Can you find sets of sloping lines that enclose a square?
Consider all two digit numbers (10, 11, . . . ,99). In writing down
all these numbers, which digits occur least often, and which occur
most often ? What about three digit numbers, four digit numbers. . . .
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.
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. . . .
Charlie has moved between countries and the average income of both
has increased. How can this be so?
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?
It starts quite simple but great opportunities for number discoveries and patterns!
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?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Take a look at the multiplication square. The first eleven triangle
numbers have been identified. Can you see a pattern? Does the
Can you tangle yourself up and reach any fraction?
It would be nice to have a strategy for disentangling any tangled
Can you find the values at the vertices when you know the values on
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.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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.
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?
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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?
Delight your friends with this cunning trick! Can you explain how
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 why 2 by 2 could be 5? Can you predict what 2 by 10
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?