MATHEMATICAL GAMES FOR CHILDREN

 

"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."

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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?"

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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.

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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? 

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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.

 

 

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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 non-competitive 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 Score-Keeping

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.

 

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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 pre-prepared 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.

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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 two-player 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 long-running 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 shorter-running column by Douglas Hofstadter, and afterwards by Mathematical Recreations, a column by Ian Stewart.

 

Teaching Math through Games

Students learn mathematics through the experiences that teachers provide. Teachers must know and understand deeply the mathematics they are teaching and understand and be committed to their students as learners of mathematics and as human beings. Mathematical games can foster mathematical communication as students explain and justify their moves to one another. In addition, games can motivate students and engage them in thinking about and applying concepts and skills.

There is no one "right way" to teach. Nevertheless, much is known about effective mathematics teaching. Selecting and using suitable curricular materials, using appropriate instructional tools and techniques to support learning, and pursuing continuous self-improvement are actions good teachers take every day.

Teaching Mathematical concepts with Card Games

The act of playing card games goes back at least 14 centuries. The longevity of card games shows that people have such fascinating with the various types of card games.  With mathematical games using cards, we give students the opportunity to learn how to play the games with other peers and to realize that not only that they have fun, but they also had the opportunity to increase their sense of numbers (please refer to our article about number sense).

There are three different types of card games to emphasize specific learning process:

  • Memory Card
  • Rummy
  • Hearts

If a teacher has never played cards then it would be quite difficult. The memory game, however, could be used by anyone without any prior knowledge. The other two games (Gin Rummy and Hearts) require the teacher’s background and familiarity of the games to be strong. MyRummy is a true card games designed to occupy children when they are looking for something to do. MyRummy is fun and easy to learn either by teachers (to explain the rule of game) or children (to play the game).

Record keeping by the students will be a very important component within the classroom and that is where the teacher’s focus should be (not on the rules of play). The student’s need for competency in mathematics before teaching the games is minimal.

Students and children in general, have a knack at being able to pick up on how a game is played. With MyRummy card games, students will know how to count without a calculator, with a low range of numbers (1 to 20). My Rummy is designed from simple to complex learning principle. It has different level of sophistication and arranged in hierarchical order to suit the needs of different group of learners - fast, moderate or average and late or slow learner.

 


 

Additional Information: Card Game

A card game is any game using playing cards, either traditional or game-specific.

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 counter-clockwise, 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, face-down, in one hand, and removing cards from the top of it with her other hand to distribute to the players, placing them face-down 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 no-return 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 18th-century 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 18th-century 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 four-player game of tricks using a 32-card 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 widely-played 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 18th 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 cross 3 board (using normal knight's moves).

Magic squares involve using all the numbers 1, 2, 3, ..., n2 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 lo-shu. In the early 16th 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 (2n+1 - 1)/3 moves if n is odd and (2n+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 18th 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 pounds25 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 3-dimensional 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 3-dimensional 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 world-wide. 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)

 

 

 

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