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. . . .
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?
Explore the effect of reflecting in two parallel mirror lines.
Explore the effect of reflecting in two intersecting mirror lines.
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?
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. . . .
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
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?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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.
Charlie has moved between countries and the average income of both
has increased. How can this be so?
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 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 game for 2 players
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.
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 collection of games on the NIM theme
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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 are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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.
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. . . .
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?
With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
Draw a square. A second square of the same size slides around the
first always maintaining contact and keeping the same orientation.
How far does the dot travel?
Got It game for an adult and child. How can you play so that you know you will always win?
Explore the effect of combining enlargements.
Can you find the values at the vertices when you know the values on
To avoid losing think of another very well known game where the
patterns of play are similar.
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. . . .
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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. . . .
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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. . . .
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?
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
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?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Can you describe this route to infinity? Where will the arrows take you next?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?