Winning Ways for Your Mathematical Plays. Volume 1, Second Edition. Elwyn R. Berlekamp, John H. Conway, Richard K. Guy. A K Peters. Wellesley. Winning Ways for your mathematical plays / Elwyn Berlekamp, John H. Conway,. Richard Distinguished Professor of Mathematics, Conway served as profes-. Winning Ways for your mathematical plays / Elwyn Berlekamp, John H. Conway,. Richard groups and the mathematical study of knots, and has written over
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Winning Ways: For Your Mathematical Plays. Home · Winning Ways: For Your Mathematical Plays Author: Elwyn R. Berlekamp. 9 downloads 96 Views 4MB. ""Winning Ways is an absolute must have for those who are interested in mathematical game theory. It is sure to please any fan of recreational mathematics or. Winning Ways for Your Mathematical Plays. Volume 2, Second Edition. Elwyn R. Berlekamp, John H. Conway, Richard K. Guy. AK Peters. Natick, Massachusetts.
When, a short time later in , it was , I saw On Numbers and Games originally published in for sale at a local science bookstore, I couldn't resist downloading a copy. What a marvelous book it turned out to be! First of all, it was fun to read. Just look at the names of things: "contorted fractions", "hackenbush unrestrained", "col" and "snort" Second, the theory it developed was fascinating.
I was only a fledgling mathematician at the time, so many of the details were beyond me, but I could still appreciate and enjoy the creativity and insight on display here. Here, 25 years later, is a new edition of the book, which has long been out of print. So let's begin by saying that even an unchanged new printing would be a great thing to have: people who missed the chance of downloading the book then can now get a copy. The new edition does not include a great number of changes: some corrections have been made, and an Epilogue discusses what progress has been made since in studying the Surreal Numbers.
Coming from Conway, it contains several interesting ideas and even suggested questions for further research. One of the more interesting things I learned when I read On Numbers and Games was that Conway had discovered a connection between numbers and combinatorial games.
In fact, a number turned out to be a certain kind of game. This theory is developed in the "first" part of ONAG the number theory in the zeroth part , but further development was promised in a forthcoming book.
This was another marvelous book. GUY the sum of previously named ones, with repetitions allowed. Whoever names 1 loses. See [Nowakowski , Section 3]. Extend the analysis of Chomp. Players alternately name divisors of N , which may not be multiples of previously named numbers. Since Chomp and the superset game [Gale and Neyman ] can be described by directed acyclic graphs but not by diforests, the proposed extension could pre- sumably throw some light on these two unsolved games.
A King and Rook versus King problem.
Can White win? If so, in how few moves? In an earlier edition of this paper I attributed this problem to Simon Norton, but it was proposed as a kriegsspiel problem, with unspecified position of the WK, and with W to win with probability 1, by Lloyd Shapley around L and M are confined to the non- negative quadrant of the plane.
They move alternately a distance of at most one unit. For which initial positions can L catch M? Conjecture: L wins if he is northeast of M.
Winning Ways for Your Mathematical Plays - Vol 3
This condition is clearly necessary, but the answer is not known even if M is at the origin and L is on the diagonal. Replace quadrant by wedge-shaped region. Cards numbered 1 through 10 are laid on the table. L chooses a card. Then R chooses cards until his total of chosen cards exceeds the card chosen by L. Then L chooses until her cumulative total exceeds that of R, etc. The first player to get 21 wins.
Who is it? Jeffery Magnoli, a student of Julian West, thought the second interpretation the more interesting, and found a first-player win in six-card Onze and in eight-card Dix-sept. Subset Take-away. Given a finite set, players alternately choose proper subsets subject to the rule that, once a subset has been chosen, none of its subsets may be chosen later by either player. Eggleton and Fraenkel ask for a theory of Cannibal Games or an analysis of special families of positions.
They are played on an arbitrary finite digraph. A move is to select a cannibal and move it along a directed edge to a neighboring vertex. If this is occupied, the incoming cannibal eats the whole population Greedy Cannibals or just one cannibal Polite Cannibals.
A player unable to move loses. Draws are possible. A partizan version can be played with cannibals of two colors, each eating only the opposite color. Place a number of monochro- matic tokens on distinct vertices of a directed acyclic graph.
A token may be moved to any unoccupied immediate follower. Make a dictio- nary of P-positions and formulate a winning strategy for other positions. See A. Restricted Positrons and Electrons. Fraenkel places a number of Posi- trons Pink tokens and Electrons Ebony tokens on distinct vertices of a Welter strip.
Any particle can be moved by either player leftward to any square u provided that u is either unoccupied or occupied by a particle of the opposite type.
In the latter case, of course, both particles become annihilated i.
Play ends when the excess particles of one type over the other are jammed in the lowest positions of the strip. Formulate a winning strategy for those positions where one exists. See Problem 47, Discrete Math. General Positrons and Electrons. Like 36, but played on an arbitrary digraph. Start with heaps of 1, 2, 3, 4, 5, 6 and 7 beans. Two players alternately transfer any number of beans from one heap to another, except that beans may not be transferred from a larger to a smaller heap.
The player who makes all the heaps have an even number of beans is the winner.
The total number of beans remains constant, and is even 28 in this case, though one is interested in even numbers in general: a similar game can be played in which the total number is odd and the object is to make all the heaps odd in size. As the moves are nondisjunctive, the Sprague—Grundy theory is not likely to be of help. GUY The number of odd heaps is even. If the number of odd heaps is two, there is an immediate win: such a position is an N-position of remoteness 1.
In normal play, P-positions have even remoteness, and N-positions have odd remoteness. If the number of odd heaps is zero, the position is terminal. All other P- positions must contain at least four odd heaps.
Take a row of consecutive coins. A move is to take one or more coins from a heap and put them on an adjacent heap of coins provided that the second heap is at least as large as the first. Assume that we start with a row of adjacent coins, that is, each heap consists of one coin. The nim-values are periodic with period seven and the values are 0 1 2 0 3 1 0 with no exceptional values. Table 2. The latter is periodic with period seven: the nim-vales are 2 1 0 0 1 3 0 with no exceptions.
The former also has period seven. The values are 5 5 4 2 5 6 4 but they do not start until the game 2.
GUY 2. Take a row of three consecutive heaps, of sizes i, j, k. Sowing or Mancala Games. Kraitchik [, p.
Bell and Cornelius [, pp. Botermans et al. Many of these games go back for thousands of years, but several should be susceptible to present day methods of analysis.
For a start, see [Erickson ] in this volume. Conway starts with a line of heaps of beans. A typical move is to take some of a heap of size N and do something with it that depends on the game and on N. In the partizan version, the position 1 a single bean has value 0, of course; the position 1.
Winning Ways for Your Mathematical Plays - Vol 3
Chess again. Find endgames with new values. See also Problems 29, 30 and Sequential compounds of games have been studied by Stromquist and Ullman . They mention a more general compound. Consider a game G P played as follows. They have no coherent theory of games G P for arbitrary posets. Compare Problem 23 above. Beanstalk is played between Jack and the Giant. The Giant chooses a positive integer, n0. Then J. The winner is the person moving to 1. Are there any drawn positions?
There are certainly drawn plays, e. What we want to know is: are there any O-positions positions of infinite remoteness? For a definition of remoteness see Problem 38 above.
There are several unanswered questions about the remotenesses of positions in these two games. Remoteness may also be the best tool we have for Problems 18 and 19 above. Inverting Hackenbush.
The game can be generalized to play on trees: a move that prunes the tree at a vertex V includes replanting the tree with V as its root. Konane [Ernst and Berlekamp]. There are various ways of playing two-dimensional Nim.
One form is dis- cussed on p. Start with a rectangular array of heaps of beans. At each move a row or column is selected and a positive number of beans is taken from some of the heaps in that row or column [Fremlin ]. A variant is where beans may be taken only from contiguous heaps. Many results are known concerning tiling rectangles with polyominoes. One can extend such problems to disconnected polyominoes.
For instance, can a rectangle be tiled by 22 22? By 22 22 2? If so, what are the smallest rectangles that can be so tiled? Find all words that can be reduced to one peg in one-dimensional Peg Soli- taire. Examples of words that can be reduced to one peg are 1, 11, , , 1 10 k 1.
If the board is cyclic, the condition is simply n even. Elwyn Berlekamp asks if there is a game that has simple, playable rules, an intricate explicit solution, and is provably NP or harder. John Selfridge asks: is Four-File a draw? Four-File is played on a chess- board with the chess pieces in their usual starting positions, but only on the a-, c-, e- and g-files a rook, a bishop, a king, a knight and four pawns on each side. The moves are normal chess moves except that play takes place only on these four files; in particular, pawns cannot capture and there is no castling.
Are there more nimber decomposition theorems? Com- pile a datebase of nim-values. Berlekamp notes that overheating operators provide a very concise way of expressing closed-form solutions to many games, and David Moews observes that monotonicity and linearity depend on the parameters and the domain.
How does one play sums of games with varied overheating operators? Find a simple, elegant way of relating the operator parameters to the game. References [Allemang ] D.
Allis, H. Bell and M. Press, Cambridge, Theory A49 , 67— Berlekamp and Y. Berlekamp and D. Peters, Wellesley, MA, Berlekamp, J.
Conway, and R. Cambridge Philos.
El berlekamp jh conway rk guy winning ways for your
Fraenkel and A. Game Theory, 16 , —The Giant chooses a positive integer, n0. The country you have selected will result in the following: Are there any drawn positions? As the moves are nondisjunctive, the Sprague—Grundy theory is not likely to be of help.
Start with a rectangular array of heaps of beans. Exclusive web offer for individuals on print book only.
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