If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
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?
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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
To avoid losing think of another very well known game where the
patterns of play are similar.
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.
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
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.
In how many ways can you arrange three dice side by side on a
surface so that the sum of the numbers on each of the four faces
(top, bottom, front and back) is equal?
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. . . .
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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 collection of games on the NIM theme
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
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?
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 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 the values at the vertices when you know the values on
the edges of these multiplication arithmagons?
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 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?
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. . . .
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
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?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable.
Decide which of these diagrams are traversable.
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.
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
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?
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. . . .
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?
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. . . .
Can you tangle yourself up and reach any fraction?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
Can all unit fractions be written as the sum of two unit fractions?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
It starts quite simple but great opportunities for number discoveries and patterns!
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?