Ime Jogoseeducacao

Quem Somos?

Fri, 23 Dec 2011 00:07:09 -0200

Quem Somos?

O Instituto de Matemática e Estatística da Universidade Federal de Goiás existe desde 1966 (embora comoutronome antes de 1998), e desdeo início tem formado alunos que cursam faculdade para tornarem-se professores de matemática no ensino fundamental e médio.

Recentemente, na década de 1990, alguns professores tomaram interesse no uso de jogos – jogos que exigem lógica e raciocínio – enquanto auxílio pedagógica.

No ano 2000, o estudo de Jogos Matemáticos começou, e desde então tem se tornado tópico importante enquanto estudo em si e enquanto algo para uso na sala de aula. Em 2009 há cinco professores e funcionários da universidade que trabalham nesta área.

Eles são:

Professor	Email	Lattes
Bryon Richard Hall	bryon@mat.ufg.br
José Pedro Machado Ribeiro	pedro@mat.ufg.br
Maria Bethania Sardeiro dos Santos	bethania@mat.ufg.br
Silmara E.de Castro Carvalho	silmara@mat.ufg.br
Zaira da Cunha Melo Varizo	zaira@mat.ufg.br

Original

Apresentação

Fri, 23 Dec 2011 00:07:09 -0200

Apresentação

Jogos são um aspecto muito antigodo ser humano. Desde a pré-história tanto crianças como adultos têm se engajado freqüentemente em atividades que chamamos genericamente de jogos. Os jogos sempre tiveram um aspecto lúdico, de se engajar num combate imaginário sem conseqüências sérias como nos combates verdadeiros. Jogos semelhantes ao futebol são antigos e têm surgido em civilizações inteiramente distintas como na Grécia clássica, na China antiga e nos povos mais primitivos das Américas. Outros tipos de jogo teriam vínculo com drama e teatro. Finalmente, jogos com regras específicas que envolvem o raciocínio lógico também são antigos, como indicado pelos exemplos de xadrez e Go.

Jogos como estes têm vários aspectos que diferem-os de jogos esportivos ou artísticos. Sempre foi associado a jogos como xadrez a questão de capacidade de planejar. Pessoas fisicamente fracas, pessoas idosas ou, excepcionalmente crianças, podem ser excelentes jogadores de jogos que envolvem basicamente o raciocínio como instrumento fundamental. Este aspecto destes jogos nos leva a denotá-los jogos matemáticos.

Tratados sobre vários jogos matemáticos existem há séculos, porém sempre sobre jogos específicos e nunca sobre os princípios de jogos matemáticos. Desde 2007 existe um grupo de professores e alunos do Instituto de Matemática e Estatística da Universidade Federal de Goiás que estudo a teoria de jogos matemáticos e seu uso no ensino elementar e médio.

É comum não conhecer a diferença entre a teoria de jogos matemáticos e outra área da matemática conhecido simplesmente como a teoria dos jogos. Esta outra teoria dos jogos surgiu primeiro com o trabalho de matemáticos como John Nash, Morgenstern e von Neumann na primeira metade do século XX. Esta teoria aborda jogos com dois (ou mais) participantes que tentam determinar estratégias de jogo para maximizar seu desempenho. Nestes jogos “ganhar” significa marcar mais pontos de que os adversários, ou ganhar lucro monetário ou vantagem militar e coisas do tipo. Normalmente cada jogador não conhece o potencial ou situação exata dos adversários. Justamente por isso esta teoria serve para previsões econômicas e planejamento militar. Mas é uma situação totalmente diferente da de um jogo como xadrez, onde i) cada jogador tem pleno conhecimento da situação, ii) os jogadores jogam alternadamente, cada um respondendo às jogadas do adversário e, iii) cada jogador deseja simplesmente “fechar” o jogo, ou seja, não deixar alternativa alguma disponível ao adversário.

O Conceito de Jogo Matemático

Um grande número de jogos será discutido neste site. Em geral estes terão sempre os seguintes aspectos em comum:

1.sempre envolvem dois adversários (jogadores) 2.não existe empate 3.ganhar a partida significa ser o último a jogar

É verdade que no xadrez e vários outros jogos o empate é aceito, mas é um aspecto que preferimos evitar para a análise ficar mais bem definida. Também deve ser notado que existe outra opção a (3): o jogo pode ser definido de modo a ser vencedor aquele que não joga por último, denominada a versão misère.

Repare que no caso deste jogos

1)a sorte não entra no jogo; há transparência total; 2)os jogos envolvem somente percepção, raciocínio e lógica.

Jogos nos quais existem “exércitos” distintos, como em xadrez e damas, são chamados partesans. Os outros jogos, sem dois times distintos, são imparciais.

A Matemática de Jogos

Considere um jogo entre dois jogadores, A e B. Num jogo partesan dois “exércitos” se enfrentam em combates que se desenvolvem ao longo do jogo, em jogadas alternadas de A e B. Cada jogada é uma implementação de estratégia – com erro ou sem – e ao mesmo tempo uma resposta à jogada que seu adversário acabou de fazer. Num dado momento, comumente, o jogo chega a certo estágio no qual um dos jogadores – A, por exemplo – não tem como responder aos ataques de B. Se encontra numa situação de desvantagem, seja o que A fizer, se B não errar, B ganha. Obviamente também poderia ser o contrário – seja o que B fizer, A, se não errar, ganha a partida.

Chamamos um jogo com vantagem de A de positivo, e vantagem de B negativo. Os jogos de Hex e Domineering, expostos noutras páginas, são jogos partesans. O jogo de Hackenbush, a ser apresentado , é um jogo simples deste tipo. Em cada momento do jogo, o valor da configuração, ou seja, a situação do tabuleiro, pode ser positivo ou negativo.

É claro que no início de uma partida, não deve haver vantagem para ninguém. Mas em jogos matemáticos isso ocorre de uma maneira específica e bem-definida, matematicamente. Ou seja, formalmente não existe“sorte” num jogo matemático. Nadécada de trinta do século XX foi provado um teorema sobre jogos matemáticos por dois matemáticos europeus, independentemente, Sprague e Grundy. O teorema de Sprague-Grundy diz que em cada momento (configuração) de um jogo matemático, cada jogador terá uma estratégia otimal. Os dois jogadores seguindo estas estratégias, a situação é sempre do seguinte quadro

		Se A começa
		A ganha	A perde
Se B começa	B perde	positivo	zero
	B ganha	"fuzzy"	negativo

“Fuzzy”é uma palavra em inglês que significa borrada ou confusa. Neste site, usaremos a palavra fuzzy. Ela significa que quem joga primeiro tem vantagem – tanto para A como para B. Em jogos imparciais, sem times distintos (Nim e Brotas são dois exemplos), o valor de uma configuração é sempre ou zero ou fuzzy. Se não existem “times” distintos, nenhum jogador pode ter vantagem completa; o tabuleiro é de todos. Assim, em alguns jogos os valores sempre são fuzzy. Em termos matemáticos, os valores fuzzy não são números reais – chamam-se surreais. Jogos partesans são em certo sentido mais complicado de que isso. O valor de uma configuração (de tabuleiro) pode ser um número real positivo ou negativo, ou pode ser zero ou fuzzy.

Para ver detalhadamente um jogo imparcial, vá a - Nim , Kayles ou Brotas. Para ver um jogo partesan, vá aHackenbush ou Hackenbush Infantil.

Original

Hackenbush

Fri, 23 Dec 2011 00:07:08 -0200

Hackenbush

We discuss here a simple game called Hackenbush. In it are two colours: A is black (male) and B is red (female). Given a figure like the one below (Figure 1), A may erase any black coloured stroke and B any red one. If after a move other parts of the figure have become disconnected from the ground, they too will disappear. Hackenbush is partesan, as the two colours are distinct. A and B will alternate moves, the winner being who plays last.

Figure-1

Look at Figure 1 and we shall analyse the possible moves. Suppose A begins.If A eliminates the line 1 (black), B will answer eliminating line 1' (red), and that will cause the eradication of all lines (for disconnection from the ground) and therefore victory for B.

If A eliminates first the line 2 (and he must do something), B will respond eliminating her line 2' (red), leaving a situation with just one move for A and certain victory of B.

If B were to begin the game, she also has two alternatives. If she begins erasing line 1' (red), A will respond erasing his line 1 and winning. But if B begins with line 2', line 2 (black) will also disappear. A will follow with just one alternative (line 1) and B will answer and win erasing 1'.

We see that independent of who begins, B can always guartantee victory. Translating in the terms we used above, Figure 1 has a negative value.

To understand better the question of practical decisions to be made in playing a game, let's see in more detail how to calculate the value of a configuration in Hackenbush. A configuration which values zero describes a situation in which whoever begins will lose. As examples, think of i) an empty figure or ii) a figure with k lines of each colour, disconnected from each other. We shall say the a figure has positive value x if A has x more moves than B, and negative value x for the contrary: B has x more moves than A. So if A has n moves and B has m moves, the value of the configuration is n – m.

In the Figure 2a we have a configuration whose value we declare to be equal to ½. We justify this somewhat unusual fact by calling its value x and considering in Figure 2b a configuration which consists of “ 2x – 1” , that is two figures called x and another which we've already seen as – 1. We now show that 2x – 1 = 0, which implies that x is ½.

Figure-2a Figure-2b

Looking at Figure 2b, suppose that the player A (black) begins. He has only one real alternative. B responds eliminating the upper part of the remaining two stick figure, leaving one free black trace and one red. B (red) will obviously win the game.

If B begins, she eliminates one of the red upper parts and A responds eliminating the other double stick entirely. Once again there remains two independent traces, one black and one red, and A will win. If B had begun eliminating the independent red trace (on the right), A will respond in the same way and once again will win.

Thus, whoever begins the game at the right in Figure 2b, will lose. By definition that implies that its value is zero and therefore that x = ½.

There is an exact, general method of determining the value of a configuration. We comment that that does not imply that it is always easy to calculate! The method is general in that it may be applied to many different games. We illustrate the method for the game Hackenbush.

With advantage for A being called positive and for B negative, we see that on playing, A desires that the value of the configuration after his move is the greatest possible value, whereas B plays with the intention to make the value the smallest possible (and negative, if possible). So we examine the configuration and see which are A's best move and B's best move. We write the values between brackets with a bar between them. The value of the configuration is the “most simple” value contained between these two values – a concept to be explained in more detail later on. For example, in the case of Figure 2a each player has just one possible move. For A it means eliminating everything (value zero). For B, one black trace is left, of value one. Thus the value is calculated as { 0 | 1 } = ½, that is, ½ is the simplest value between zero and one.

Consider Figure 3. Once again, A and B have only one distinct move each. A will eliminate the whole figure (value zero) and B will leave as remaining exactly the figure that we proved of value ½ above. Thus the value of the configuration is { 0 | ½ } which has value ¼. The reader who desires to prove that more rigorously may design four repetitions of the figure and add one red trace alone beside them, and by calling

Figura 3
The figure in question of value x she will be examining the value of 4x – 1, which in a similar way to what we saw before may be shown to have value zero.

Repeating this calculation a number of times permits us to calculate the valuew of a more complex configuration.

Partesan games rarely have fuzzy configurations and in Hackenbush they do not exist. In impartial games they are very common, as in fact all impartial configurations always have value either zero or fuzzy.

To see an Impartial Game, go to Nim or Sprouts.

Nim

Fri, 23 Dec 2011 00:07:08 -0200

Nim

The game of Nim has origen in China, although not by this name. It was played in Europe as of the XVI century. Played with stones, coins, matches or any small object that exists in individual units. An arbitrary number of piles of these objects are formed, usually between three and six piles of from one to around twenty objects. A move consists of removing any number whatsoever (greater than zero) of objects from one of the piles. The player who wins is who takes the last object away. We see that Nim is impartial: there is no difference between being the first or second player, it mattering only who is the next to play. This will mean that the values of configurations will be zero or fuzzy, the fuzzy values being what is called surreal.

Suppose there are n piles of toothpicks, with p1 toothpicks in the first pile, p2 in the second pile, ..., pn in the n-th pile. At the end of the game nothing will remain and the value is zero. Observe that by our system of calculation this configuration has the value { | } or, if you prefer, { ? | ? }. So, in the informal sense we used above, we are saying the most simple number less than no specific number and greater than another is zero. This rather obtuse idea will be explained in more detail further on. For readers who have not seen these ideas before, consult Mathematical Games.

Now consider the configuration that consists of one pile of one toothpick. A and B have the same options, as always in impartial games, to retire this same toothpick. As the empty table, as seen above, has value zero, on doing the calculation of value we have { 0 | 0 }. What value is this? Evidently there is no real number that is simultaneously greater than and less than zero. So this value is not a real number; as in this case the first to play always wins, we see that in the table given above the value should not be neither positive nor negative, but fuzzy. This surreal number is designated *1, and is considered near zero but neither less than nor greater than zero. That is exactly the intention of the name: fuzzy.

A single pile of two toothpicks will permit two distinct moves for either A or B: the removal of one or two toothpicks. These moves result in configurations of values *1 and zero respectively. Thus the value of a pile of two is { 0, *1 | 0, *1} = *2. In general a pile of n toothpicks will have value { 0, *1, *2, ..., *(n-1) | 0, *1, ..., *(n-1) } = *n. Observe that the value is always the first fuzzy value that is not contained between brackets. That may sound trivial, but it is in fact a general rule called the rule of minimum excluded, or mex.

If instead we have two separate piles it isn't much harder. Two equal piles give certain victory to the second to play: whatever the first player does, she simply imitates the first player's move, thus maintaining after her move two equal piles. In terms of values, this implies that *k + *k = 0. Similarly, if we have two unequal piles the first to play guaratees his win by leaving after his move two equal piles.

Figure-1

Three piles, of one, two and three toothpicks respectively, implies in the loss of whoever begins the game, for whatever the first player plays, the second player can also create two equal piles after her move. Thus *1 + *2 + *3 = 0, and in consequence of this, *1 + *2 = *3, *1 + *3 = *2 and *2 + *3 = *1.

Charles L. Bouton published an article in 1901 which clarified once and for all the game (in fact, the name Nim was his invention). His method consisted of writing the number of toothpicks in each pile in binary notation: every whole number can be written using the numerals 0 and 1, expressing n as a sum of powers of two.

For example, 13 written in binary notation is 1101 = 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20. Now suppose we have two piles with n1 and n2 toothpicks. Writing n1 and n2 in binary notation, one above the other, we see that if they are equal there will be an even number of ones in each column. Thus after one player reduces one pile, altering the ones in a column, the second player may reciprocate restoring an even number on ones in each columns by returning to a situation of two equal piles.

That idea generalizes. Consider three piles as above, with one, two and three toothpicks in each pile. Writing in binary notation one above another we have

10
11
1

and we have an even number of ones in each column. As a given move by a player alters only one of the piles (numbers), whatever a player may do in this situation will leave an odd number of ones in at least one column. The second player restores the even number, and the game continues in this way until it ends – with victory for the second player. Bouton proved that if the number of ones in each column is even, victory is guaranteed for the second player (if she commits no errors!). Bouton considered sum of two ones or two zeros as zero, and sum of zero and one as equal to one. In this sense, specifically, he proved that

i) victory corresponds to sum zero;
ii) if a sum is zero, the next move will certainly yield a non-zero sum;
iii) if the sum is not zero, there always exists a move to make the sum after the move equal to zero.

To understand the facts ii) and iii), consider a situation of sum zero. For example, let there be four piles of 5, 2, 10 and 13 toothpicks. When written in binary notation we have 101, 10, 1010 and 1101. The sum is zero:

101
10
1010
1101
0000

Independently of what the next move is, one of the 0's or 1's will be altered, so that the number of 1's in at least one of the columns will no longer be an even number. Finally, consider the game with piles of 6, 3, 8 and 9 toothpicks. Writing the four numbers in binary notation we have

110
11
1000
1001
0100

Whoever plays first will certainly be able to change things so that there are an even number of ones in each column. Taking four toothpicks from the first pile will have that effect, leaving two in the pile with sum:

10
11
1000
1001
0000

The most important aspect of this idea is that it functions in many other games besides Nim. However, for pedagogical purposes we must ask how to use the analysis in activities of Nim from early on at school until the end of high school.

For that we have a pratical method, without an absolute guarantee, but frequently useful. Suppose for example the challenge of playing against an opponent a match with three piles of 4, 7 and 13 toothpicks. Instead of making a formal analysis via binary notation (but remembering the basic idea) we observe that only one pile has more than 8 toothpicks and therefore for certain the sum of the piles is non null. That means that the first to play can guarantee victory, but rather than doing the exact calculation of a winning move we simply reduce the pile of 13 to one of 5 or 6 (so that there are not two equal piles after our move). Let's say five and now considere the match from the second player's point of view.

With piles of 7, 5 and 4 matches the simplest logic says: all piles are smaller than 8. Therefore one should move so as to not produce two equal piles. In fact one of these moves – reducing the pile of four to one of two toothpicks, guarantees victory.

In 1910 the mathematician Moore proposed an alternative game to Nim, called Nimk, and published an article with respect to strategies of play. Nimk resembles Nim with the following difference: instead of taking any number of toothpicks from one pile, in Nimk we may remove any (same) number of toothpicks from one to k piles simultaneously. Consider Nim2 as an example. In this game one may remove any number of toothpicks from one or two piles simultaneously, always the same number of toothpicks from each pile.

Como é o Nimk no caso de haver somente duas pilhas?

Figure-2

In Nimk, as soon as there are only two piles, victory is guaranteed for the next player to move. The first strickly “non-trivial” game is one of three piles of one toothpick each – with certain victory for the second to play. In general the analysis is similar to that of Nim: express the number of toothpicks in each pile in binary notation and sum the columns, but sum them mod k+1 e not mod 2 (as in Nim). If the sum is zero there is victory guaranteed for the second player to move. If not, the first player may win by moving so as to make the sum after his move equal to zero (mod k+1).

For example, in Nim2 (with k+1 = 3), suppose we begin with four piles of 3, 6, 7 and 10 toothpicks respectively. Summing mod 3 we have

11
110
111
1010
1212

Thus the first to play can guarantee victory. We leave it to the reader to deduce which move will have this effect.

Kayles

Fri, 23 Dec 2011 00:07:08 -0200

Kayles

The game of Kayles formally uses one or more rows of bowling pins. Each player may knock down either one pin alone or an adjacent pair of pins (and must knock down at least one evry time). We denote a file of n pins by Kn, and the sum of two rows of n e m pins by Kn+Km.

As an example, a file of five pins permits five distinct moves: moves that transform K5 in K4, K3, K2+K2, K3+K1 or K2+K1, depending on the player knocking over one or two pins and their being in the middle or at the end of the initial row. Kayles was sold as game at the beginning of the twentieth century in England and is an impartial game. It resembles Nim except for the eventual divisão of a pile into two piles. As before, each player may interfere with only one pile at a time. The values are those of an impartial subtraction game, *n. In Kayles, with one, two, three or four pins in a row, the first to play can always win. With five pins in a row, the moves are those we listed above. If he plays K4 or K3, he loses. And the other moves?

Kayles is an example of a subtraction game of a different type. from those games S(n1, n2, ... nk). In Kayles we have, given m piles with n1, n2, ..., nm toothpicks in each pile, with the following rule:

i)a player may remove one or two toothpicks from any pile with that number or more of toothpicks, or
ii)remove 1 or 2 toothpicks from a pile, dividing the remaining toothpicks in two arbitrar piles

The analysis is done just as in Nim. Kayles has in fact the following mex sequence

n = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
G (n) = 0. 1 2 3 1 4 3 2 1 4 2 6 4 1 2 7 1 ...

Let's understand how this sequence was determined. A row of zero pins is worth zero. A row of one will also result in zero for each player, and therefore is worth *1. A row of two will result in either zero or *1, and is worth *2. A row of three will result in K2, K1 or K1 + K1 = 0, and therefore is worth *3. A row of four can give as alternatives K2 = *2, K3 = *3, K1 + K2 = *3 or K1 + K1 = 0. That means that K4 is worth *1. We leave as an exercize the confirmation of the values of five or more pins. We see that an appropriate name for the game would be (1,2)-Kayles: i.e. these rules can be generalized with complete freedom, resulting in a great number of new games.

An option we've never heard of for Kayles is to arrange the pins in connected rows, not in simple distinct rows. Below on the left we have a “double row” with three distinct moves allowed. One move would leave as the remaining figure the figure on the right.

What is the value of the figures above? Let's calculate the value of the figure on the right first: it may be transformed into any of the five figures below. Their values are illustrated below..

*1 + *1

*1 + *1 + *1

Simplifying, these values are 0, 0, *3, *3 and *1 respectively. Thefore, the figure above on the right has value *2. Notice that it's quite diferent from a simple mrow of two that has that same value.

Who are we

Fri, 23 Dec 2011 00:07:08 -0200

Who are we?

The Institute of Mathematics and Statistics of the Federal University of Goiás has existed since 1966 (though with another name before 1998), and since it began there have been students who study to become elementary and high-school teachers of mathematics.

Recently, in the 1990's, some professors here have become interested in the use of games – games which demand analysis and reason – as a pedagogical aid. In the year 2000, the study of Mathematical Games began and has since then become a topic for study in itself and as pedagogical technique for use in the classroom.

In 2009 there are five professors and staff of the university that work in this area. They are

Bryon Richard Hall
José Pedro ...
Maria Bethânia ...
Silmara ...
Zaíra ...

Introduction

Fri, 23 Dec 2011 00:07:08 -0200

Introdution

Games have existed in human society for a long time, at the very least since the neolithic era four to eight thousand years ago. The word “games” has a quite broad interpretation. All games have a ludic aspect and all are an imaginary combat between or among a group of adversaries. Athletic games like soccer have been played in distinct societies like ancient Greece and China, as well as in primitive Latin America. Other kinds of game are theatral games and musical games, also quite old. Finally, we have games with specific rules which involve rational thinking, like chess and Go.

Games like these are essentially different from sportive or artistic games. Even physically impaired people may be well prepared to play such games – they demand mental and not physical capacity. Games like these are denominated mathematical games.

Books about various different mathematical games have existed for centuries, but rarely about the principles behind mathematical games themselves. In 2007 a group of teachers and students at the Federal University of Goiás was created that began both the abstract study of mathematical games and the study of their use as a pedagogical tool at the elementary and high school level.

It's common to not understand the diference between the theory of mathematical games and the classical study game theory. This second area began in the first half of the twentieth century with the work of mathematicians like John Nash, Morgenstern and von Neumann. This theory considers games between two or more participants who search for strategies to maximize their profit or benefits in interactions with the others. To win these games implies gaining as many or more points than other players, generally in an economic or military sense. Normally, one player will not have complete information about his or her “adversaries”. Obviously this is very different from a game like chess or Go, where the information is complete and players play in sucession one of the other, and in which each player desires to simply “close” the game, denying his opponent any move at all.

The Concept of a Mathematical Game

A great number of games will be discussed in this site. In general all will have the following aspects in common:

1.they always involve two players
2.there is no tie possible
3.to win implies being the last to play

It's true that in chess and some other games, ties are possible, but it's an aspect that we prefer to ignore for now to have a more rigorous analysis of game play. Also, the aspect (3) is only one possibility, existing another: to win could mean to not be the last to play. Games like this are called games with the option misère, and will be considered in due time.
Notice that in these games

1.there is no role for luck in the game – there is total transperancy
2.the games involve only perception, reason and logic.

Besides the above there is one more possible distinction between games. The games that have distinct “armies”, like chess and checkers, are called partesan. Games that do not have distinct teams, like in Nim and Spouts, are impartial.

The Mathematics of Games

Consider a match between two players, A and B. In a “partesan” game two different armies will fight a battle throughout the game with alternating moves. Each move should be the implementation of a certain strategy – with or without error in planning – and simultaneously a response to the adversary's moves. In a certain moment the game will evolve into a situation in which one of the players, say A, has no possible response to player B. A would be in a situation of disadvantage: independent to what he or she does - if B commits no mistake, B will win. Obviously it could also be the opposite situation: whatever B does it is to no avail, A can certainly win.

We call a game with advantage to A positive and one with advantage to B negative. The games we will discuss called Hex and Domineering are partesan, as is Hackenbush (to be briefly discussed as well). In these moments of the game the value of the configuration will be positive or negative.

It should be clear that at the beginning of the match there is no intrinsec advantage on anyone's behalf. In mathematical games this occurs in a well-defined way, mathematically speaking. In other words, no “luck” is involved. In the 1930's a theorem was proven by two European mathematicians independently, Sprague and Grundy. The theorem of Sprague-Grundy says that in each moment along a match of a mathematical game each player will have an optimal strategy to follow. If both players follow their strategy the situation is always one of the following:

		If A plays first
		A wins	A loses
If B plays first	B loses	positive	zero
If B plays first	B wins	“ fuzzy ”	negative

By fuzzy we mean a value that, by what was said above, is neither positive nor negative nor zero – in short fuzzy corresponds to a different kind of value. It means that whoever plays first has an advantage. In impartial games, with no distinct armies in action, the value of a configuration will always be either zero or fuzzy, because there is no real difference between the players. It will always be a question of who plays next. These fuzzy values, mathematically, correspond to what are called surreal numbers.

Partesan games are more complicated. In them the value may be fuzzy or zero but may also be positive or negative.

To view a detailed fair game, go to - Nim , Kayles ou Sprouts. To see a game partesan go to-Hackenbush
or Hackenbush for Kids.

Hackenbush for Kids

Fri, 23 Dec 2011 00:07:08 -0200

Hackenbush for Kids

This game is presented in Berlycamp, Conway and Guy's book Winning Ways and is mathematically interesting as a game with positive or negative values (it's partesan) that may be infinitesimal, in contrast with the ordinary Hackenbush.

Draw a horizontal line as the “floor” e complete a figure using strokes of two colours, say blue and red. We'll say player A is blue (positive) and B is red (negative). So a figure that favours A will have a positive value, and one that favours B negative. If the figure favours whoever plays first it will have a fuzzy value, and if it favours the second to play it's value is zero.

As in Hackenbush, A and B alternate erasing one trace of their colour. A trace is considered connected to the floor if there is a sequence of strokes linking it to the floor. The rule of Hackenbush for Kids that distinguishes it from common Hackenbush is that no move may disconnect other parts of the figure from the floor: it's forbidden to create a disconnected body.

We'll call the value of a single isolated blue stroke +! and that of a red stroke -1. To attribute a value to a figure we do as done before in other sections. Consider the various options that each player may have and calculate the value of the result of that move. Between braces we put on the left the possible results of A and on the right those of B. The value of the figure will be the most simple “number” (real or surreal) contained between the highest left value and the smallest right value. The figure below, very simple indeed, will permit B to move leaving a figure of value 3 and A leaving one with value 2 – 1 = 1. So the configuration has value { 1 | 3 } = 2.

The figure below values zero. In fact, whovever begins the match will lose – confirming our definition of “value zero”.

{ -1 | 1} = 0

It won't always be that easy. The figure below allows only A de play, so the method says the value will be { -1 | }. What value is that? Zero – as whover begins will lose again!

{ -1 | }

For similar motives, the configuration below on the left is worth zero and the one to the right is worth – 1.

{ -3 | } = 0 { - 2 | 0, 1 } = - 1

More interesting are the following configurations.

{ -1, 0 | 1 } = ½ {-1, 0 | 0, 1} = *

On the right we have aconfiguration with a fuzzy value – something that does not exist in common Hackenbush. That means that the first player to play can win if she plays correctly. But its value is also such that the sum of it will any positive value whatsoever, for small as it may be, will be positive, and the sum with an arbitrarily small negative value is negative.

Subtraction Games

Fri, 23 Dec 2011 00:07:08 -0200

Subtraction Games

In general we may define a subtraction game in the following way: it is permitted in each move of a player to remove either n1, n2, n3, ..., nk toothpicks from any of the piles. Once again, the winner will be he who removes the last toothpick from the table. The value of a pile of n toothpicks is no longer *n, as seen in Nim, but is in fact calculated in the same way that that result was calculated: we determine the smallest number not included in a list of values obtained on a move. In other words, the mex rule continues valid in general subtraction games.

Consider a game in which either one, three or four toothpicks may be removed in each move. We designate this game by S(1,3,4). If there are no toothpicks at all, none may be removed and the value is mex { } = 0. From a pile of one toothpick, one may be removed, leaving none at all: mex {0} = *1, as “one” is the smallest number not contained in the set. From a pile of two, in this game, just one may be removed, so that mex {*1} = 0. From a pile of three toothpicks, one or three may be removed, leaving two or none: mex {0, 0} = *1. And in a pile of four, one, three or four may be removed, leaving three, one or none: mex {0, *1, *1} = 2. If we continue we find a sequence of values that will be written as a sequence called G(k). So for this game
G(k) = 0,1012320101232010123201012320...
which is in fact a periodic sequence, 0,1012320 of period seven.

The strategy to play is similar to that of Nim. The player wishes that the sum of the values of the piles be equal to zero after his move, for that implies that “he who plays in second place wins”. For example, suppose that this game is played with piles of 2, 3, 8 and 9 toothpicks. We see that G(2) = 0, G(3) = *1, G(8) = *1 and G(9) = 0, so that the sum is zero. Therefore, no player will want to be the first to play, for whatever the first player does, the sum afterwards will no longer be zero and the second to play can certainly move so that the sum returns to value zero after his move.

It can be proven that the mex sequence of any subtraction game is periodic.

Kayles

Fri, 23 Dec 2011 00:07:08 -0200

Kayles

i)a player may remove one or two toothpicks from any pile with that number or more of toothpicks, or
ii)remove 1 or 2 toothpicks from a pile, dividing the remaining toothpicks in two arbitrar piles

The analysis is done just as in Nim. Kayles has in fact the following mex sequence

n = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
G (n) = 0. 1 2 3 1 4 3 2 1 4 2 6 4 1 2 7 1 ...

What is the value of the figures above? Let's calculate the value of the figure on the right first: it may be transformed into any of the five figures below. Their values are illustrated below..

*1 + *1

*1 + *1 + *1

Original