A counter is placed in the bottom right hand corner of a grid. You
toss a coin and move the star according to the following rules: ...
What is the probability that you end up in the top left-hand. . . .
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
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.
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.
Take a look at the multiplication square. The first eleven triangle
numbers have been identified. Can you see a pattern? Does the
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
It would be nice to have a strategy for disentangling any tangled
Can you tangle yourself up and reach any fraction?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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 game for 2 players
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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.
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.
Can you find sets of sloping lines that enclose a square?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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. . . .
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.
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?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
To avoid losing think of another very well known game where the
patterns of play are similar.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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. . . .
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
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?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Charlie has moved between countries and the average income of both
has increased. How can this be so?
It starts quite simple but great opportunities for number discoveries and patterns!
Can you find the values at the vertices when you know the values on
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Can you describe this route to infinity? Where will the arrows take you next?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Find some examples of pairs of numbers such that their sum is a
factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and
16 is a factor of 48.
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. . . .
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
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?
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. . . .
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?
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 an efficient method to work out how many handshakes
there would be if hundreds of people met?
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?