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 loser is the player who takes the last counter.
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.
A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7.
Players take turns to remove any number of counters from a row. The
player left with the last counter looses.
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
A game for 2 players
To avoid losing think of another very well known game where the
patterns of play are similar.
Show that for any triangle it is always possible to construct 3
touching circles with centres at the vertices. Is it possible to
construct touching circles centred at the vertices of any polygon?
Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Can you describe this route to infinity? Where will the arrows take you next?
Find the vertices of a pentagon given the midpoints of its sides.
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
The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
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. . . .
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Jo made a cube from some smaller cubes, painted some of the faces
of the large cube, and then took it apart again. 45 small cubes had
no paint on them at all. How many small cubes did Jo use?
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. . . .
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Many numbers can be expressed as the difference of two perfect
squares. What do you notice about the numbers you CANNOT make?
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.
Can you find the values at the vertices when you know the values on
Can you find the values at the vertices when you know the values on
the edges of these multiplication arithmagons?
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?
Can you find sets of sloping lines that enclose a square?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Charlie has moved between countries and the average income of both
has increased. How can this be so?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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. . . .
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. . . .
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 would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
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 many moves does it take to swap over some red and blue frogs? Do you have a method?
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
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?
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
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. . . .