Introduction
It's common if we heard parent
saying that "My children hate math!".
What do our children like the most? Children are more interested
in art and literature. Sometimes they complain about having to
take math and science. The
usual argument may goes something like this:
"This is so stupid.
I hate this
stuff and
I"m never going to use it ever again. I'm going to be
writer, and never solve another physics problem in my
life! Why do I
have to learn this stuff?"

Learning English with fun! 

Huuhhh!
How to deal with issues like this? You are also the same
(while you are children!) Ha..ha..haa..
OK. One thing you can do is to keep explain to them about the importance
of math in life.
Yaahh, try to explain that
the logic
learned through mathematics and the problem solving
skills obtained through science are invaluable tools for
the rest of life. 
Have Fun
With Lipo Interactive CD.
Learning English for Primary School Children. 


Math Around Us!
If they're younger, you
can
show them how math and physics,
geometry, chemistry are ALL AROUND US all the time
(patterns, symmetry, shapes, numbers, codes, going to
the grocery store checkout, counting up the scores in a
bowling game, calculating the average scores at the
Olympics for figure skating or whatever, even measuring
when baking or buying something by the pound etc) and
if they learn how its applied in
everyday life they might appreciate it more because they
know they can actually use it.



Many jobs need
mathematics. A photographer will talk about angles,
perspective, depth and background, sizing/resizing of
pictures, distance between subject and photographer
(feet, height all that stuff), that's just an example.


Take into account that, if a kid
says "I want to be a photographer" well there is math in being a
photographer  talk about angles, perspective, depth and
background, sizing/resizing of pictures, distance between
subject and photographer (feet, height all that stuff), that's
just an example. Tell them that the number of people in
the world who make a living as photographers is extremely small
compared to the number of people who make a living using math
and science.
Attracting Children to Learn Math
How to attract children to learn
math? How about using
mathematical games?
Is there any mathematical games for children? Is there any
mathematical games that can help teachers to develop children's
understanding on number concept?
Ads: Do
you know about 'number sense' mathematical games for children?
Click the picture to
get more information!
Math and Mathematical Games
Mathematical games develop mathematical communication
as
children
explain and justify their moves to one another. Communication is
an essential part of mathematics and mathematics education
because it is a “way of sharing ideas and clarifying
understanding".
In addition, games can motivate students and engage them in thinking about,
and applying, concepts and skills. Games give pupils an
opportunity to communicate their ideas and justify their
thinking.
In using
games, the teacher plays an important role in encouraging pupils
to explain their thinking and in keeping them focused on
mathematical ideas. Asking them to explain and justify their
moves during a trial round of the game played as a whole class
demonstrates the type of thinking and communicating. That is
important for students to use later when they play the game in
pairs.
Games
contribute to the development of knowledge by having a positive
affect on the atmosphere in the class which in turn produces a
better mental attitude towards math in the children. Educational
games provide a unique opportunity for integrating the
cognitive, affective and social aspects of learning.
Using
Games Successfully
The
success of the mathematical games as learning tools depends on the teacher's
talent in asking probing, open questions and ultimately how well
the teacher establishes a classroom climate that encourages
experimentation. Ultimately the focus must be on cognitive
processes rather than on the correctness of final outcomes. The
process by which 'wrong' answers are reached should be valued as
much as processes producing 'right' answers.
Ernest
(1986) claims that the success of mathematics teaching depends
to a large extent on the active involvement of the learner and
playing games demands involvement. Games cannot be played
passively: players have to be actively involved. For this reason
psychologists including Piaget, Bruner and Dienes suggest games
have a very important part to play in learning, particularly in
the learning of mathematics.

Myrummy is a game for math!

Selecting
a Game to Use

When
considering what games to use it is vital that the context which
they are to be used is considered. The thinking behind each game
should be analyzed and matched to the learning objectives that
are to be met.



Looking at
some of the questions which children should ask themselves when
starting to play a game, and putting them under a mathematical
heading gives a good idea to the higher order skills involved. 
Form of question 
Mathematical
heading 
How do I play this? 
Interpretation 
What is the best way of playing? 
Optimisation 
How can I make sure of winning? 
Analysis 
What happens if...? 
Variation 
What are the chances of...? 
Probability 
Form of statement 
Mathematical idea 
This game is the same as... 
Isomorphism 
You can win by... 
A
particular case 
This works with all these games. 
Generalisation 
Look, I can show you it does... 
Proving 
I
record the game like this... 
Symbolisation and Notation 
Clearly
the strategy to be used is the decision of the teacher, and is
dependent upon various factors like the ability of the children,
their motivation and sociability, the ethos of the school, and
the degree of control that the teacher has.


Methods of
Playing Games
When
mathematical games
are used, it is important that they are played properly for
three reasons:

#1:
First of all there is the intrinsic mathematics which
is always present.

#2: Second, there is the high level of interest
and motivation which playing games generates.

#3: Third, and perhaps
most important, is the higher order understanding of the problem
to be solved, which can only be gained by playing through
different games.
From this there are lines of attack which can
be used when analyzing games and trying to find a winning
strategy. The teacher should demonstrate these and develop the
skill in the pupils.
For instance try making it simpler in some
way, usually by making it smaller. If the full game is played on
a grid of five by five cells, start the pupils playing and
finding solutions on a three by three grid. If a solution cannot
be found for the simpler version, it is very unlikely they will
for the more complex case.

Articles:

Different Types of Games

It is
important to remember that not all children like playing
games, especially if they have weaker social skills.
Others
may not like playing games of any type because they do not like
the competition. However, these pupils seem to be a minority.

Children with weak number
skills will not enjoy activities where this puts them at a
disadvantage so using mathematical games which are
noncompetitive or involve an element of chance are best.
Games in
which chance plays a part can be
helpful by giving weaker players a more level playing field.
Such games are also a good introduction to the topic of
probability.
Example of
different types of games.

#1 Pontoon
Pontoon is
an excellent game for practicing addition and subtraction of
numbers up to 21. Because it contains an element of chance,
pupils can play happily with the rest of the class without being
at a disadvantage because they are weaker at maths. The gambling
aspect can be left out of the game, but playing with matchsticks
or other small objects adds more fun and extra counting
practice.




#2 Dice and Counters
Any game
which involves throwing dice and moving counters helps build
confidence with numbers. These games can be made more difficult
by using two dice and working out the move by adding the two
numbers or finding the difference between them. The numbers can
also be multiplied together but this often means that the games
are completed too quickly unless, like Monopoly, it involves
travelling around a board many times.


#3 ScoreKeeping
Score
keeping comes into many games. Mostly this just means adding
numbers together but some games are more complicated. Scrabble
and darts both involve multiplying by 2 and 3. The darts game
301 provides excellent practice at subtraction and this can be
developed to work with negative numbers.


#4 Pairs
Games
involving pairing cards can be very flexible. For instance the
pairs of cards can form the two halves of an equation, marked
with two equivalent fractions or a percentage and its decimal
equivalent.


#5 Computer Games
When
choosing computers and electronic games, they should contain the
following features:

Several levels of difficulty so that differentiation is
allowed.

The
ability to practice one skill at a time.

More
than one attempt allowed before the correct answer is given.
This allows time for rethinking and means that accidentally
pressing the wrong key isn't a disaster.

A
response to a wrong answer which is less interesting than
the response to a correct one. Many games fail on this
point.


Think Tank is Student Activity Cards for Computation
and Number Sense!

Personal
Experience
When
choosing any games or mathematical games it is important to remember that
they stop
being fun if they are used all the time. From this Researcher's
own experience, having used games whilst teaching a higher set
year seven class and a bottom set year eight class, it can be
seen that games can play an important part in teaching math.
On three
occasions with year eight, whilst working on a unit about
variables which involved expanding brackets, I used a
multiplication bingo game. The idea initially was to increase
pupil's quick recall of basic multiplication facts. The first
time proved to be a very difficult lesson. The pupils were very
keen to play but did not abide by the rules calling out answers
to the multiplications drawn from the bucket. Although this
defeated the object of the game I decided that they should
continue as it was good practice for the next time.
One week
later when we played the game again, they took it much
more seriously and did not call out answers. There was a new
atmosphere of competition in the class and I was very happy with
the response to the game. In the next lesson I was able to try a
few variations on the theme to test the pupils' knowledge
further. To start I discussed with the class what was actually
involved in playing the game. Once we agreed that we were
marking off numbers that were the product of two others I then
asked the question 'Which numbers would you not want on your
bingo card?' I was very pleased that the correct response came
from a pupil and in this way I was able to introduce the idea of
'prime numbers' which was new to a lot of the class. On
preprepared blank bingo cards I instructed the pupils to write
ten numbers of their own choice between zero and fifty. By
keeping the cards after the game I was able to check through
their choice of numbers to see if they understood the concept of
prime numbers. A further dimension to playing the game was added
when I decided that instead of pulling multiplication cards out
of a bucket, I would point to a number on a pupil's card and
they would have to shout out the requisite multiplication pair.
This not only gave the game a new feel but more importantly gave
the pupils practice at the reverse operation of factorizing.
In this
case, playing the games especially mathematical games was beneficial to the
children. In the
bottom set class there were a few children who were very
disillusioned with maths. However, even if their number skills
had not improved, it became very clear when on numerous
occasions it was asked if the game could be played again that
their confidence and attitude towards maths lessons had.
When using
games and mathematical games with the year seven class it became immediately evident
that they were more receptive to playing/working in this
manner. This could be because of the difference in top and
bottom sets or because the children were used to playing games in
primary school and had not lost the skill yet.
Conclusion
From the
research that this Researcher has made and the  all be it
limited  experience at using games and mathematical in teaching,
it seems that
they are very beneficial to the children learning under the right
conditions. The mathematical games for example, must be appropriate for their use in class
and the teacher must be clear about the objectives of the game.
Certain games are
best used for out of class activities such as math clubs. Games
such as myrummy have a clear link to
a mathematical topic and are a very useful classroom activity.
Most
importantly the children need to be practiced at using games and
solving problems analytically so that they do not waste valuable
time. This method of learning is common in primary school but is
not carried on through secondary. If time is dedicated to these
types of activities in the schemes of work throughout Key Stages
3 and 4, the pupils' problem solving skills become finely tuned,
the teacher is freed from the feeling that they are 'losing
time' by playing games and methods and strategies for future
course work can be developed at an early age.
Top of the
page
Appendix
Mathematical games
Mathematical games include many topics which are a part of
recreational mathematics, but can also cover topics such as the mathematics of games, and playing games with mathematics
As far as twoplayer games are considered, what distinguishes a mathematical game from ordinary games is the emphasis on mathematical
analysis of the game, rather than actually
playing it.
Mathematical Games was the title of a longrunning column on the subject by
Martin Gardner in
Scientific American. He inspired several new generations of mathematicians and scientists through his interest in mathematical recreations.
Mathematical Games was succeeded by
Metamagical Themas, a similarly distinguished but shorterrunning column by
Douglas Hofstadter, and afterwards by
Mathematical Recreations, a column by
Ian Stewart.
Additional Information:
Card Game
A card game is any game using playing cards, either traditional or gamespecific.
There are also some card games that require multiple standard decks. In this scenario, a "deck" refers to a set of 52 cards or a single deck, while a "pack" or "shoe" (Blackjack) refers to the collection of "decks" as a whole.
The Deck or Pack
A card game is played with a
deck (common in the US), or pack (common in the UK), of cards intended for that game. The deck consists of a fixed number of pieces of printed cardboard known as
cards. The cards in a deck are identical in size and shape. Each card has two sides, the
face and the back. The backs of the cards in a deck are indistinguishable. The faces of the cards in a deck may all be unique, or may include duplicates, depending on the game. In either case, any card is readily identifiable by its face. The set of cards that make up the deck are known to all of the players using that deck.
Although many games have special decks of cards, the 52 card pack is known as the standard deck, and is used in a wide variety of games. It consists of 52 cards, each card having a suit (one of spades, hearts, diamonds and clubs) and a rank (a number between 2 and 10, or one of jack, queen, king and ace). For any combination of one suit and one rank, there is exactly one card in the standard deck having that suit and rank. In addition to games that use the standard deck, there are also games that use some modification of the standard deck, for example all cards of rank lower than some rank (e.g., a pinochle deck), or adding a special card, joker, to the standard deck. Many European regions have their own variants of the standard deck having different names and imagery for suits, or having a different set of ranks in the cards.
There are also some card games that require multiple standard decks. In this scenario, a "deck" refers to a set of 52 cards or a single deck, while a "pack" or "shoe" (Blackjack) refers to the collection of "decks" as a whole.
The Ideal
Dealing is done either clockwise or counterclockwise. If this is omitted from the rules, then it should be assumed to be:
 clockwise for games from North America, North and West Europe and Russia;
 counterclockwise for South and East Europe and Asia, also for Swiss games.
A player is chosen to deal. That person takes all of the cards in the pack, stacks them together so that they are all the same way up and the same way round, and shuffles them. There are various techniques of shuffling, all intended to put the cards into a random order. During the shuffle, the dealer holds the cards so that he or she and the other players cannot see any of their faces.
Shuffling should continue until the chance of a card remaining next to the one that was originally next to is small. In practice, many dealers do not shuffle for long enough to achieve this.
After the shuffle, the dealer offers the deck to another player to
cut the deck. If the deal is clockwise, this is the player to the dealer's right; if counterclockwise, it is the player to the dealer's left. The invitation to cut is made by placing the pack, face downward, on the table near the player who is to cut: who then lifts the upper portion of the pack clear of the lower portion and places it alongside. The formerly lower portion is then replaced on top of the formerly upper portion.
The dealer then deals the cards. This is done by dealer holding the pack, facedown, in one hand, and removing cards from the top of it with her other hand to distribute to the players, placing them facedown on the table in front of the players to whom they are dealt. The rules of the game will specify the details of the deal. It normally starts with the players next to the dealer in the direction of play (left in a clockwise game; right in an anticlockwise one), and continues in the same direction around the table. The cards may be dealt one at a time, or in groups. Unless the rules specify otherwise, assume that the cards are dealt one at a time. Unless the rules specify otherwise, assume that all the cards are dealt out; but in many card games, some remain undealt, and are left face down in the middle of the table, forming the talon, skat, or stock. The player who received the first card from the deal may be known as eldest hand, or as forehand. A card may be taken from either the mother deck or the discarded deck upon a players turn. A card taken from the mother deck must be either replaced in the player's hand for another card, or placed on the discard deck. Cards from the discard deck may be handled without breach of the noreturn rule and may be placed back. This maneuver does not count towards a turn for a player. Play as usual resumes as usual after the player has committed to a regular move.
The set of cards dealt to a player is known as his or her
hand.
Throughout the shuffle, cut, and deal, the dealer should arrange that the players are unable to see the faces of any of the cards. The players should not try to see any of the faces. Should a card accidentally become exposed (visible to all), then normally any player can demand a redeal  that is, all the cards are gathered up, and the shuffle, cut and deal are repeated. Should a player accidentally see a card (other than one dealt to herself) she should admit this.
It is dishonest to try to see cards as they are dealt, or to take advantage of having seen a card accidentally.
When the deal is complete, all players pick up their cards and hold them in such a way that the faces can be seen by the holder of the cards but not the other players. It is helpful to fan one's cards out so that (if they have corner indices) all their values can be seen at once. In most games it is also useful to sort one's hand, rearranging the cards in a way appropriate to the game. For example in a trick taking game it is easier to have all one's cards of the same suit together, whereas in a
rummy game one might sort them by rank or by potential combinations.
The Rules
A new card game starts in a small way, either as someone's invention, or as a modification of an existing game. Those playing it may agree to change the rules as they wish. The rules that they agree on become the "house rules" under which they play the game. A set of house rules may be accepted as valid by a group of players wherever they play. It may also be accepted as governing all play within a particular house, café, or club.
When a game becomes sufficiently popular, so that people often play it with strangers, there is a need for a generally accepted set of rules. This is often met by a particular set of house rules becoming generally recognised. For example, when
whist became popular in 18thcentury England, players in the Portland Club agreed on a set of house rules for use on its premises. Players in some other clubs then agreed to follow the "Portland Club" rules, rather than go to the trouble of codifying and printing their own sets of rules. The Portland Club rules eventually became generally accepted throughout England.
Whist is a classic card game which was played widely in the 18th and 19th centuries. Although the rules are extremely simple there is enormous scope for scientific play and since the only information known at the start of play is the player's thirteen cards the game is difficult to play well.
There is nothing "official" about this process. If you decide to play whist seriously, it would be sensible to learn the Portland Club rules, so that you can play with other people who already know these rules. But if you only play whist with your family, you are likely to ignore these rules, and just use what rules you choose. And if you play whist seriously with a group of friends, you are still perfectly free to devise your own set of rules, should you want to.
It is sometimes said that the "official" or "correct" sets of rules governing a card game are those "in Hoyle". Edmond Hoyle was an 18thcentury Englishman who published a number of books about card games. His books were popular, especially his treatise on how to become a good whist player. After (and even before) his death, many publishers have taken advantage of his popularity by placing his name on their books of rules. The presence of his name on a rule book has no significance at all. The rules given in the book may be no more than the opinion of the author.
If there is a sense in which a card game can have an "official" set of rules, it is when that card game has an "official" governing body. For example, the rules of tournament bridge are governed by the World Bridge Federation, and by local bodies in various countries such as the American Contract Bridge League in the USA, and the English Bridge Union in England. The rules of
skat are governed by The International Skat Players Association and in Germany by the Deutsche Skatverband which publishes the
Skatordnung. The rules of French tarot are governed by the Fédération Française de Tarot. But there is no compulsion to follow the rules put out by these organisations. If you and your friends decide to play a game by a set of rules unknown to the game's official body, you are doing nothing illegal.
Skat is the most popular card game in Germany and Silesia. It is also played in American regions with large German populations, such as Wisconsin and Texas. It is a three or fourplayer game of tricks using a 32card deck. The deck of 32 cards consists of the cards 7, 8, 9, 10, jack, queen, king and ace in the suits diamonds, hearts, spades and clubs. There are no
jokers. The Joker is a special card found in most modern decks of playing cards.
Many widelyplayed card games have no official regulating body. An example is Canasta.
Canasta is a matching card game in which the object is to create melds of cards of the same rank and then
go out by playing or discarding all the cards in your hand.
Rules Infractions
An infraction is any action which is against the rules of the game, such as playing a card when it is not one's turn to play and the accidental exposure of a card.
In many official sets of rules for card games, the rules specifying the penalties for various infractions occupy more pages than the rules specifying how to play correctly. This is tedious, but necessary for games that are played seriously. Players who intend to play a card game at a high level generally ensure before beginning that all agree on the penalties to be used. When playing privately, this will normally be a question of agreeing house rules. In a tournament there will probably be a tournament director who will enforce the rules when required and arbitrate in cases of doubt.
If a player breaks the rules of a game deliberately, this is cheating. Most card players would refuse to play cards with a known cheat. The rest of this section is therefore about accidental infractions, caused by ignorance, clumsiness, inattention, etc.
As the same game is played repeatedly among a group of players, precedents build up about how a particular infraction of the rules should be handled. For example, "Sheila just led a card when it wasn't her turn. Last week when Jo did that, we agreed ... etc.". Sets of such precedents tend to become established among groups of players, and to be regarded as part of the house rules. Sets of house rules become formalised, as described in the previous section. Therefore, for some games, there is a "proper" way of handling infractions of the rules. But for many games, without governing bodies, there is no standard way of handling infractions.
In many circumstances, there is no need for special rules dealing with what happens after an infraction. As a general principle, the person who broke a rule should not benefit by it, and the other players should not lose by it. An exception to this may be made in games with fixed partnerships, in which it may be felt that the partner(s) of the person who broke a rule should also not benefit. The penalty for an accidental infraction should be as mild as reasonable, consistent with there being no possible benefit to the person responsible.
Source:
http://en.wikipedia.org/wiki/Card_game
Mathematical
Puzzles
Mathematical
puzzles vary from the simple to deep problems which are still
unsolved. The whole history of mathematics is interwoven with
mathematical games which have led to the study of many areas of
mathematics. Number games, geometrical puzzles, network problems and
combinatorial problems are among the best known types of puzzles.
The Rhind papyrus
shows that early Egyptian mathematics was largely based on puzzle
type problems. For example the papyrus, written in around 1850 BC,
contains a rather familiar type of puzzle.
Seven houses
contain seven cats. Each cat kills seven mice. Each mouse had
eaten seven ears of grain. Each ear of grain would have produced
seven hekats of wheat. What is the total of all of these?
Similar problems
appear in
Fibonacci's
Liber Abaci written in 1202 and the familiar
St Ives Riddle of the 18^{th} Century based on the same idea.
Greek mathematics
produced many classic puzzles. Perhaps the most famous are from
Archimedes in his book
The Sandreckoner where he gives
the Cattle Problem.
If thou art
diligent and wise, O Stranger, compute the number of cattle of
the Sun...
In some
interpretations of the problem the number of cattle turns out to be
a number with 206545 digits!
Archimedes also invented a division of a square into 14 pieces
leading to a game similar to Tangrams involving making figures from
the 14 pieces. Tangrams are of Chinese origin and require little
mathematical skill. It is interesting however to see how many convex
figures you can make from the 7 tangram pieces. Note again the
number 7 which seems to have been associated with magical
properties. They were to have a new popularity when
Dodgson, writing as Lewis Carroll, introduced Alice type
characters.
Fibonacci, already mentioned above, is famed for his invention
of the sequence 1, 1, 2, 3, 5, 8, 13, ... where each number is the
sum of the previous two. In fact a wealth of mathematics has arisen
from this sequence and today a Journal is devoted to topics related
to the sequence. Here is the famous Rabbit Problem.
A certain
man put a pair of rabbits in a place surrounded on all sides by
a wall. How many pairs of rabbits can be produced from that pair
in a year if it is supposed that every month each pair begins a
new pair which from the second month on becomes productive?
Fibonacci writes out the first 13 terms of the sequence but does
not give the recurrence relation which generates it.
One of the earliest
mentions of Chess in puzzles is by the Arabic mathematician Ibn
Kallikan who, in 1256, poses the problem of the grains of wheat, 1
on the first square of the chess board, 2 on the second, 4 on the
third, 8 on the fourth etc. One of the earliest problem involving
chess pieces is due to Guarini di Forli who in 1512 asked how two
white and two black knights could be interchanged if they are placed
at the corners of a 3 3
board (using normal knight's moves).
Magic squares
involve using all the numbers 1, 2, 3, ..., n^{2} to
fill the squares of an n
x n board so that each
row, each column and both main diagonals sum to the same number.
They are claimed to go back as far as 2200 BC when the Chinese
called them loshu. In the early 16^{th} Century
Cornelius Agrippa constructed squares for n = 3, 4, 5, 6, 7,
8, 9 which he associated with the seven planets then known
(including the Sun and the Moon).
Dürer's famous engraving of
Melancholia made in 1514
includes a picture of a magic square.
The number of magic
squares of a given order is still an unsolved problem. There are 880
squares of size 4 and 275305224 squares of size 5, but the number of
larger squares is still unknown.
Durer's square shown above is symmetrical and other conditions
were also studied such as the condition that all the diagonals
(traced as if the square was on a torus) added to the same number as
the row and column sum.
Euler studied this type of square known as a pandiagonal square.
No pandiagonal square of order 2(2n + 1) can exist but they
do for all other orders. For n = 4 there are 880 magic
squares of which 48 are pandiagonal.
Veblen in 1908 used matrix methods to study magic squares.
Other early
inventors of games included
Recorde and
Cardan.
Cardan invented a game consisting of a number of rings on a bar.
It appears in the 1550 edition of his book De Subtililate .
The rings were arranged so that only the ring A at one end
could be taken on and off without problems. To take any other off
the ring next to it towards A had to be on the bar and all others
towards A had to be off the bar. To take all the rings off
requires (2^{n+1}  1)/3 moves if n is odd and (2^{n+1}
 2)/3 moves if n is even. This problem is similar to the
Towers of Hanoi described below. In fact
Lucas (the inventor of the Towers of Hanoi) gives a pretty
solution to
Cardan's Ring Puzzle using binary arithmetic.
Tartaglia, who with
Cardan jointly discovered the algebraic solution of the cubic,
was another famous inventor of mathematical recreations. He invented
many arithmetical problems, and contributed to problems with
weighing masses with the smallest number of weights and Ferry Boat
type problems which now have solutions using graph theory.
Bachet was famed as a poet, translator and early mathematician
of the French Academy. He is best known for his translation of 1621
of
Diophantus's
Arithmetica. This is the book which
Fermat was reading when he inscribed the margin with his famous
Last Theorem.
Bachet, however, is also famed as a collector of mathematical
puzzles which he published in 1612 Problèmes plaisans et
délectables qui font par les nombres . It contains many of the
problems referred to above, river crossing problems, weighing
problems, number tricks, magic squares etc. Here is an example of
one of
Bachet's weighing problems
What is the
least number of weights that can be used on a scale pan to weigh
any integral number of pounds from 1 to 40
inclusive, if the weights can be placed in either of the scale
pans?
Euler is perhaps the mathematician whose puzzles have led to the
most deep mathematical disciplines. In addition to magic square
problems and number problems he considered the Knight's Tour
of the chess board, the Thirty Six Officers problem and the
Seven Bridges of Königsberg.
Euler was not the first to examine the Knight's Tour problem.
De Moivre and
Montmort had looked at it and solved the problem in the early
years of the 18^{th} Century after the question had been
posed by
Taylor.
Ozanam and
Montucla quote the solutions of both
De Moivre and
Montmort.
Euler, in 1759 following a suggestion of
L Bertrand of Geneva, was the first to make a serious
mathematical analysis of it, introducing concepts which were to
become important in graph theory.
Lagrange also contributed to the understanding of the Knight's
Tour problem, as did
Vandermonde.
The Seven
Bridges of Königsberg heralds the beginning of graph theory and
topology.
The Thirty Six
Officers Problem, posed by
Euler in 1779, asks if it is possible to arrange 6 regiments
consisting of 6 officers each of different ranks in a 6
x 6 square so that no rank or
regiment will be repeated in any row or column. The problem is
insoluble but it has led to important work in combinatorics.
Another famous
chess board problem is the Eight queens problem. This problem
asks in how many ways 8 queens can be placed on a chess board so
that no two attack each other. The generalised problem, in how many
ways can n queens be placed on an n
x n board so that no
two attack each other, was posed by Franz Nauck in 1850. In 1874
Günther and
Glaisher described methods for solving this problem based on
determinants. There is a unique solution (up to symmetry) to the 6
x 6 problem and the puzzle,
in the form of a wooden board with 36 holes into which pins were
placed, was sold on the streets of London for one penny.
In 1857
Hamilton described his
Icosian game at a meeting of the
British Association in Dublin. It was sold to J. Jacques and Sons,
makers of high quality chess sets, for
25 and patented in London
in 1859. The game is related to
Euler's Knight's Tour problem since, in today's terminology, it
asks for a Hamiltonian circuit in a certain graph. The game was a
failure and sold very few copies.
Another famous
problem was
Kirkman's
School Girl Problem. The problem, posed in
1850, asks how 15 school girls can walk in 5 rows of 3 each for 7
days so that no girl walks with any other girl in the same triplet
more than once. In fact, provided n is divisible by 3, we can ask
the more general question about n school girls walking for (n
 1)/2 days so that no girl walks with any other girl in the same
triplet more than once. Solutions for n = 9, 15, 27 were
given in 1850 and much work was done on the problem thereafter. It
is important in the modern theory of combinatorics.
Around this time
two professional inventors of mathematical puzzles,
Sam Loyd and
Henry Ernest Dudeney, were entertaining the world with a large
number of mathematical games and recreations.
Loyd's most famous game was the 15 puzzle.
Loyd was also famous for his chess puzzles. He invented a number
of puzzles, some of which are very hard, which he published in the
first number of the American Chess Journal.
Edouard Lucas invented the Towers of Hanoi in 1883.
The game of
pentominoes is of more recent invention. The problem of tiling an 8
x 8 square with a square hole
in the centre was solved in 1935. This problem was shown by computer
to have exactly 65 solutions in 1958. In 1953 more general
polyominoes were introduced. It is still an unsolved problem how
many distinct polyominoes of each order there are. There are 12
pentominoes, 35 hexominoes and 108 heptominoes (including one rather
dubious one with a hole in the middle!). Puzzles with polyominoes
were invented by Solomon W. Golomb, a mathematician and electrical
engineer at Southern California University.
There is a
3dimensional version of pentominoes where cubes are used as the
basic elements instead of squares. A 3 x 4
x 5 rectangular prism can be
made from the 3dimensional pentominoes. Closely related to these is
Piet Hein's Soma Cubes. This consists of 7 pieces, 6 pieces
consisting of 4 small cubes and one of 3 small cubes. The aim of
this game is to assemble a 3 x
3 x 3 cube. This can in fact
be done in 230 essentially different ways!
A slightly older
game (1921) but still a cube game is due to
P A MacMahon and called 30 Coloured Cubes Puzzle. There are 30
cubes which have all possible permutations of precisely 6 colours
for their faces. (Can you prove there are exactly 30 such cubes?)
Choose a cube at random and then choose 8 other cubes to make a 2 x 2
x 2 cube with the same
arrangement of colours for it's faces as the first chosen cube. Each
face of the 2 x 2
x 2 cube has to be a single
colour and the interior faces have to match in colour.
Raymond Smullyan, a mathematical logician, composed a number of
chess problems of a very different type from those usually composed.
They are know known as problems of retrograde analysis and their
object is to deduce the past history of a game rather than the
future of a game which is the conventional problem. Problems of
retrograde analysis are problems in mathematical logic.
One of the most
important of the modern professional puzzle inventors and collectors
is Martin Gardner who wrote an extremely good column in Scientific American for about 30 years, stopping about four
years ago. He published some of
Smullyan's retrograde analysis chees problems in 1973. He also
reported on a computing game in 1973. Of course the advent of
personal computers has made both the writing and playing of
mathematical games for computers an important new direction. The
game Gardner reported on was 'Spirolaterals' devised by Frank Olds
with only 3 or 4 lines of code.
The most famous of
recent puzzles in the of Rubik's cube invented by the Hungarian Ernö
Rubik. It's fame is incredible. Invented in 1974, patented in 1975
it was put on the market in Hungary in 1977. However it did not
really begin as a craze until 1981. By 1982 10 million cubes had
been sold in Hungary, more than the population of the country. It is
estimated that 100 million were sold worldwide. It is really a
group theory puzzle, although not many people realise this.
The cube consists
of 3 x 3
x 3 smaller cubes which, in
the initial configuration, are coloured so that the 6 faces of the
large cube are coloured in 6 distinct colours. The 9 cubes forming
one face can be rotated through 45 degress.
There are 43,252,003,274,489,856,000 different arrangements of the
small cubes, only one of these arrangements being the initial
position. Solving the cube shows the importance of conjugates and
commutators in a group.
References (13 books/articles)