Mathematical Games

Mathematical Games
A Rationale for their Use in the
Teaching of Mathematics in School
What would be the advantages?
•Motivation. Games generate enthusiasm, excitement, total
involvement and enjoyment and, over a period of time, should
enhance pupils’ attitudes towards the subject.
•Variety. Games add variety to the overall mathematics
curriculum, by bringing another varied approach into the teaching
of the subject.
•Discussion. Games encourage discussion.
•Co-operation. Even competitive games can encourage cooperation.
•Active involvement. Games encourage the active involvement
of children, making them more receptive to learning and
increasing their motivation. Active involvement not only
enhances learning, but according to some psychologists is
essential for learning to take place at all. For this reason
psychologists including Piaget, Bruner and Dienes suggest that
games have a very important part to play in learning,
particularly in the learning of mathematics. Of these three,
Zoltan P. Dienes goes furthest by suggesting that all
mathematics teaching should begin with games. Although
Dienes may be overstating his case, he is a man well worth
listening to.
Dienes has not only carried out an extensive programme of
classroom research, he has also developed some of the best
apparatus available for teaching mathematics, including the
multi-base arithmetic blocks, the algebraical experience
materials, logic blocks and the number balance.
I have claimed that, if we can teach mathematics through
games, then there are many desirable by-products. But can
mathematics be taught effectively using games?
Leaving aside general aims such as those above, the major
purpose of teaching mathematics is the attainment of
Let us focus on three type of objective.
1. The Reinforcement and Practice of Skills
Much of mathematics teaching revolves around giving children
practice in newly acquired skills, or in reinforcing and further
developing skills. Games provide a way of taking the drudgery
out of the practice of skills, and indeed of making the practice
more effective.
(See Steeplechase on the handout.)
2. The Acquisition and Development of Concepts
See Fair/Unfair Games on the handout.
See Edith Biggs’ research project as discussed in the handout.
See Steeplechase on the handout.
The sample studies discussed in the handout show how games
can play a vital part in aiding children to first acquire and then
to further develop mathematical concepts.
3. The Development of Problem Solving Strategies
HMI have specified the following problem solving strategies
as distinct objectives of mathematics teaching:
•Trial and error methods
•Simplifying difficult tasks
•Looking for pattern
•Making and testing hypotheses
•Proving and disproving
Mathematical games can foster the development of most, if not
all, of these strategies and higher level skills.
Analysing Games
Level 1: Local reasoning
Each time we make a move we have to ask ourselves what the
immediate consequences of that move are likely to be: “If I go
there, then he/she will …”. This kind of reasoning is local in the
sense that we apply it to one little bit of the whole game at a time.
Such reasoning is important, but it ignores long-term effects. A
move may be locally safe, yet guarantee defeat in the long run!
Level 2: The search for global rules
Global rules or strategies are those which influence one’s playing
of the game as a whole.
Level 3: Being absolutely sure
Here we need some kind of mathematical proof that one’s
strategy really does control play in the way one thinks it does.
This is a game for two players.
Players take turns to choose any whole number from 1 to 10.
They keep a running total of all the chosen numbers.
The first player to make this total reach exactly 100 wins.
Sample Game:
Player 1’s choice
Player 2’s choice
Running Total
So Player 1 wins!
Play the game a few times with your neighbour.
(Play & modify)
Can you find a winning strategy?
Try to modify the game in some way, e.g.
- suppose the first to 100 loses and overshooting is not
- suppose you can only choose a number between 5 and
The Spiral Game
This is a game for two players. Place
a counter on the dot marked “Start”.
Now take it in turns to move the
counter between 1 and 6 dots inwards
along the spiral. The first player to
reach the dot marked “Finish” wins.
Try to find a winning strategy.
Change in some way the rule for
moving, and investigate winning
First One Home
This game is for two players.
You will need to draw a large
grid like the one shown.
Place a counter on any square
of your grid.
Now take it in turns to slide the
counter any number of squares
due West, South, or SouthWest.
The first player to reach the
square marked “End” is the
Pin Them Down!
A game for 2 players.
Each player places his/her counters
as shown.
The players take it in turns to slide
one of their counters up or down
the board any number of spaces.
No jumping is allowed.
The aim is to prevent your
opponent from being able to move
by trapping his/her counters.
    
    
Domino Square
This is a game for 2 players.
You will need a supply of 8
dominoes or 8 paper rectangles.
Each player, in turn, places a domino
on the square grid, so that it covers
two horizontally or vertically
adjacent squares.
After a domino has been placed, it
cannot be moved.
The last player to be able to place a
domino on the grid wins.
This is a game for 2 players.
Arrange a pile of counters
arbitrarily into 2 heaps.
Each player in turn can remove as
many counters as (s)he likes from
one of the heaps. (S)he, can if
(s)he wishes, remove all the
counters in a heap, but (s)he must
take at least one.
The winner is the player who takes the last counter.
Try to find a winning strategy.
Now change the game in some way and analyse your own version.
Laser Wars
Two tanks are armed with laser
beams that annihilate anything which
lies to the North, South, East or West
of them. They move alternately. At
each move a tank can move any
number of squares North, South, East
or West but it cannot move across or
into the path of the opponent's laser
A player loses when he is unable to
move on his turn.
Play the game on the board shown, using two objects to represent “tanks”.
Try to find a winning strategy which works wherever the tanks are placed to
start with. Try to change the game in some way.
This is like an old 14th century game for 2 players, in which a ball is
thrown at a number of wooden pins standing side by side. The size of
the ball is such that it can knock down either a single pin or two pins
standing next to teach other. Players alternately roll a ball and the
person who knocks over the last pin (or pair of pins) wins.
Try to find a winning strategy.
(Assume that you can always hit the pin or pins that you aim for, and
that no one is ever allowed to miss).
Now try changing the rules
(Alternative Presentation)
Draw out a connected line of cells, such as:
Decide who is to be O and who is to be X. Players take
On each turn, a player must put his/her mark in either 1
square or 2 adjacent squares.
No square may be used twice.
The player who makes the last mark, or marks, is the
Towers of Hanoi
A puzzle for one person.
In a temple at Benares there were three rods and one rod held 64 discs of
gold, all of different diameters, placed so that the largest lay at the bottom
and the others, in decreasing order of size, rested upon it. The priests were
set the task of moving the discs, one at a time, so that eventually the discs
would rest in the same order on the other rod. At no one time could a disc
be placed upon a smaller one.
About how long do you think the task would take them, assuming that they
were to work without stopping and that the time taken to move a disc from
one rod to another was five seconds on average?
What is the least number of moves necessary to move two, three, four, ...,
sixty-four, ..., n, ... discs from one rod to another? Can you prove the
This is a game for two players.
All that is needed is a plain
piece of paper and a pencil.
To start, mark a number of dots on the paper; it is best to begin
with three dots, but try any number from 2 to 8.
Each player takes it in turn to draw a line which joins one dot to
any other dot, or to itself, and then places a new dot anywhere on
this line. These restrictions must be observed:
(a) The line must not cross itself or any other line, nor pass
through any other dot;
(b) No dot may have more than three lines coming from it.
The winner is the last person able to play.
Is there a rule which determines the number of moves which can
be made in any game?
Draw a 3x3 grid as used for Noughts and
Decide who is to be O and who is to be
Players take turns putting their own
marks in, only one mark at a time.
The player who first gets three of his
marks in a straight line, vertically,
horizontally or diagonally, loses the
Draw out a grid of 3 x 3 squares.
Decide who is to be O and who is to
be X.
Players take turns to put their marks
in as many squares as they like
provided that the squares used are
all in the same straight line
(vertically or horizontally). They
do not have to be next to each other.
No square may be used twice.
The player who makes the last
mark, or marks, is the winner.
Draw out a connected “chain” of
cells. The actual number is not
important. Decide who is to be O
and who is to be X.
Players take turns to put their
mark in 1, 2 or 3 adjacent cells.
Each cell may have only 1 mark.
The player who marks in the last
cell, or cells, is the winner.
Cat & Mouse
A game for two players.
The board is made up of 27
‘holes’ connected by ‘passages’.
Each player has a counter or
marker of their own. One player
is the mouse, the other player is
the cat.
At the beginning the cat goes on C
and the mouse goes on M.
Players then take turns moving their own
markers. Cat goes first.
Moves are made from one hole to the next along the passages.
The cat captures the mouse if it can move into the same hole as the
mouse. The mouse tries to avoid being caught!
A good mouse is never caught!
A game for two players.
Place 4 counters or markers
on the spots shown circled.
Players take turns moving these markers.
In their turn, a player must move one marker
by sliding it along the line (towards the centre)
a distance of 1, 2 or 3 spots.
Markers may not jump or overtake, and no spot may have more
than 1 marker on it.
A player moving a marker on to the centre spot takes off that
The winner is the player who takes off the last marker.
Square Dance
A game for two players.
Players take opposite corners
and place 2 of their own counters
or markers on the dots which are
circled. One marker on each dot.
Players take turns moving one of
their own markers at each turn.
A marker may be moved 1, 2 or 3
dots forwards or backwards around the square.
Two markers cannot be put on the same dot, and they cannot jump
or pass each other.
A player who is unable to make a move loses the game.
A game for two players.
Place a counter or marker on every
square except the one with the star.
Players take turns.
At each turn a player must move
one marker by jumping over one
other marker into an empty square.
This move may be up, down, or
across, but not diagonally.
The marker that has been jumped
over is removed.
The last player who is able to make
a jump wins the game.
Take One!
Take Two!
A game for two players.
Place a counter or marker in
each of the 25 circles.
Players take turns removing
exactly one pair of markers
at each turn.
The pair removed must be in
two circles that touch.
The player who removes the
last pair wins the game.
One counter is needed. The first
player starts by placing the counter
on one of the numbers and saying
that number. Then, starting with the
second player, each player in turn
moves the counter by sliding it along
a straight line to another number and
saying the total so far. When a total
of (say) 23 is reached, that player
wins. If the total exceeds 23, the
player loses.
Star Pick
Place a counter on each spot.
Players take turns picking up
these counters. In his turn a
player may pick up 1 or 2
counters. A player may only
pick up 2 counters provided
that they are connected by a
single straight line. The
winner is the player who
picks up the last counter(s).
Poly Pick
Place a counter on each spot. One
player is White, the other is Black.
Players take turns picking up
the counters. In his turn
a player may pick up
1 or 2 counters.
A player may only
pick up 2 counters
provided that they are
connected by a single
straight line of the player’s
own colour. The winner is the player
who picks up the last counter(s).
‘Modern’ Seega
This is a game played by young Egyptians today.
Two players each have three pieces,
which are set up at either end of a
3 by 3 board.
Playing alternately, you can move
a piece one or two squares in any
direction (including diagonally)
but must not pass over another piece.
The winner is the first to get three pieces in a straight line
(diagonals included) other than along the original starting line.
This is an ancient Chinese game.
Alternately, players remove counters with the
option of removing:
(a) any number of counters from one pile.
or (b) the same number of counters from each
The winner is the player who removes
the last counter(s).
Can you form a winning strategy for
the first player?
Alquerque (Africa)
Two players each have 12 pieces,
starting in the positions shown.
Pieces can move along a line to an
empty point. Pieces can be captured
by being jumped over onto an empty
point. More than one capture can be
made in one move, and the direction
of movement can also be changed.
If a player misses a chance to capture an opponent’s piece, then the
offending piece can be removed from the board. The winner is the
first person to capture all of the opponent’s pieces.
Dara (Nigeria)
The board consists of 5 rows of 6
holes. Each player has 12 pieces,
which are placed, in turn, into the
holes. Once all of the pieces have
been placed moves are made. A piece
can be moved into an adjacent empty
hole (not diagonally). When a line of
3 is formed the player removes one
of the opponent’s pieces from the
board. The game ends when a player
is unable to make a line of 3 pieces.
This game is played by the
Dakarkari people using
stones, pieces of pottery or
shaped sticks.
Exchange Kono (Korea)
Each player has 8 pieces,
with the starting position
as shown. The players
take it in turns to move a
counter one space
diagonally onto a black
spot. The aim is to be the
first to occupy the
opponent’s starting
positions. There are no
jumps or captures.
Fox and Geese (Iceland)
This game was played by the
Vikings. There are 13 geese and
1 fox. The geese start in the
positions shown; the fox starts
on any empty spot. Geese move
first, along a line. The fox kills
a goose by jumping over it to a
vacant point. The geese win if
they surround the fox. The fox
wins if there are so many geese
killed that it cannot be
Go Bang (Japan)
In Japan the most popular
game is Go, with
professional players earning
a lot of money. Go Bang is
a simpler version of Go,
arriving in England in 1885.
Counters are placed
alternately on the
intersections of a 10x10
square board. The aim is to
form 5 counters in a row in
any direction.
Kungser (Tibet)
A battle game between 2 Princes and 24
Lamas. The Princes and Lamas are placed
as shown. The first player (the Prince) can
move a Prince one space or capture a Lama
by jumping over it to an empty space
beyond. The second player (the Lama) plays
by placing a Lama on the board until all 24
have been used. Then the second player
continues by moving Lamas on the board.
The Prince wins if only 8 Lamas remain.
The Lama wins if the Princes are trapped.
Multiple captures are allowed.
Mu Torere (New Zealand)
Played by the Ngati Porou people, this is the only native Maori
board game known. The board would
have been marked on the ground with
twigs or stones used as counters.
Two players have 4 pieces (perepere)
each placed on adjacent points of the star
(kewai). The aim is to block the opponent
from moving. The centre space is called the
putahi. Moves can be made (a) from one kewai to
an adjacent empty kewai, (b) from the putahi to a kewai, (c) from a
kewai to the putahi as long as either one or both of the adjacent points
is occupied by an enemy piece. Only one piece can occupy the same
place at the same time. Jumping is not allowed.
Shap Luk Kon Tseung Kwan
One player is the general and the other controls the 16
soldiers. They can all move one step along any line in
any direction. The general can enter the triangle at the
top but the soldiers
cannot. The general and the soldiers
can capture. The general can capture
two soldiers by moving to an empty
point between them. Both soldiers are
removed from the board. If the soldiers
can position themselves so that they are
directly beside the general on the same
line the general is captured and loses.
If the general is trapped inside the
triangle he is captured and loses.
The Fifteen Game
Draw a strip of cells and write in the numbers 1 to 9, as shown.
Players take turns claiming a number – perhaps by putting their
initial(s) in that cell – but they must allow the number to be seen
clearly. The winner is the first player to claim three numbers which
add up to 15. The player may actually possess more than three
numbers, but only three of them can be counted.
This may look pretty uninteresting, but do not be deceived!
(Make 15 analysis)
Difficult to analyse? How many ways are there
of making 15 with three numbers chosen from
the above selection? This should reveal that
some numbers are ‘better’ than others. Try
setting out the nine numbers as a magic square
and have players select their numbers from that
by crossing them out with their own distinctive
signs (like maybe an O and an X). What game
are you really playing? Is it now easier to
analyse? Does it even need analysing? This is a
very practical example of an isomorphism.
Draw a grid of any convenient size and shape.
Players take turns putting their own distinctive mark (say an O and
an X) in any cell. The only restriction is that no two cells which are
side by side, touching along a common edge, may have the SAME
type of mark in them. The winner is the last player who is able to
make an allowed mark.
End to End
Draw a strip of any convenient number of cells.
Place a counter in one end cell.
Players take turns advancing the counter towards the other end. In
one turn a player may advance the counter 1, 2 or 3 cells. The
winner is the player who actually moves the counter into the end
cell. As a variation, it could be that the player who is forced to
move into the end cell is the loser.
Odd wins
Draw a strip of 13 cells.
Players start at opposite ends. In turns players put their own marks
in 1, 2 or 3 cells. Players must fill cells as they work from their
own ends; no blanks may be left. When all the cells have been
occupied, then the winner is the player who has made an ODD
number of marks.
Variation 1
Allow the marks to be placed anywhere, with the single
restriction that, if 2 or 3 cells are filled in during one turn then
they must be adjacent cells.
(This is much more difficult to analyse.)
Variation 2
Play it on a rectangular grid (provided it has an ODD number of
cells). Again, multiple entries are only allowed
in a set of cells connected by edges. In addition, an entry of 3
marks should be allowed only in a straight line of cells.
Analysis now becomes possible only by computer.
Capture the Numbers
Write out the numbers 1 to 12. These are the numbers which are to
be captured. Use 2 dice. Players take turns throwing the two dice
and adding the two top numbers together to make a target value. A
player may then capture either
(a) one number which is equal in value to the target value; or
(b) two numbers which add up to the target value.
A number is captured by drawing a ring around it and identifying
which player captured it. Sometimes no captures are possible. The
game stops when it is clear that no more captures can be made.
The winner is the player who has captured the most numbers.
Played like Noughts and Crosses on a 3 by 3
grid but, before each player has a turn he or
she rolls a die to determine the mark which is
to be made. If the die shows an odd number
then the player has to put an O in the grid. If
the die shows an even number then the player
has to put an X in the grid. The winner is the
first player to make a line of either Os or Xs.
Played on a 4 by 4 grid.
Players take turns putting either a O or an X in any cell.
(Note that neither player has their own particular mark.)
The winner is the first player to make a line of either 3 Os or 3
Xs in any direction (vertically, horizontally or diagonally).
Winners or Losers
Draw a strip of any convenient number of cells.
Decide which player is to go first. The other player decides
whether they shall play for winners or losers.
They both start from the SAME end, filling in 1, 2 or 3 cells
during their turn, but the filling in must be continuous; NO
blanks may be left.
Eventually all the cells must be filled in. The player who fills in
the LAST cell(s) wins if they are playing winners
loses if they are playing losers.
The Game of Euclid
This is based on the Euclidean algorithm for
finding the highest common factor of two
positive integers a, b. Each move changes the
current pair of numbers a, b by subtracting some
multiple of the smaller (say b) from the larger
(say a) to get a new pair of numbers a-kb, b.
Negative numbers are forbidden. The first player
to produce a pair involving zero is deemed to be
the winner. The art lies in choosing the right
multiple of the smaller number to subtract each
27, 8
A 11, 8
B 3, 8
A 3, 5
B 3, 2
A 1, 2
B 1, 0
B wins this
Could A
have done
Nine cards
Two players have a pack of 9 cards. They take turns to remove 1,
2 or 3 cards from the pack; but a player must never remove the
same number of cards as the previous player. The winner is the
one who either takes the last card or leaves the other player with
no valid move.
What about different sized packs?
Nine Men’s Morris
Two players (black and white) each have
9 counters to place at any of the 24 points
on the board. In part one, players play
alternately placing men on the board. In
part two, a turn is taken by moving a piece
to an adjacent vacant point. The object is
to form a “mill” or row of three – each
time this is done, a player may remove
one of the opponent’s men. The winner
either blocks all his opponents men or
reduces their number to two.
Three Men’s Morris
This simpler version of Nine Men’s Morris is played on a board
with 9 points, with players having 4 counters each. Here the aim
is simply to get three men in a line.
In the cathedrals of Norwich, Canterbury,
Gloucester, Salisbury and Westminster
Abbey there are boards cut into the
cloister seats doubtless to relieve the
tedium of long services!
Ko-no (China & India)
One player has counters at A,
B; the other at X,Y. A move
must be to the vacant space –
a player loses when he is
blocked. Teachers may be
surprised at the problem
solving strategies which this
simple game develops in
young children.
Mathematical Recreations
For example:
A wolf, a goat and a cabbage must be moved across a river in a
boat holding only one besides the ferryman. How must he carry
them across so that the goat shall not eat the cabbage, nor the
wolf the goat? From: Problems for the Quickening of the Mind
by Alcuin of York (c. 775).
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