How much do the books differ from their first editions? I can't give a
comprehensive answer, but there are changes. One of the first pages in
WW has a one paragraph introduction for each author accompanied by
a reasonably sized likeness. So some changes are discernible to the naked
eye, provided you have the first edition on hand to compare. The authors of
WW claim to have expanded the Extras at the ends of the chapters,
to have inserted many references for further reading, and to have corrected
some of the one hundred sixty three mistakes (by the authors' count) in the
first edition.
whose proof is said to be obvious. As before, the book begins with
Chapter 0 and, as before, the book is in two = {zero, one | }
parts. Part 0 covers the notion of number, part one = {zero
| } covers games. Some games are numbers, some are not. The
games in question are played by two players — usually Left
and Right — who make alternate moves. (WW also has
material on one person games — puzzles.) Games are described by
positions. For each position P, rules of the game determine positions
PL available to Left (Left options) and positions
PR available to Right (Right options). Conversely,
every position is completely defined by the options available to the
players, which is concisely expressed as P = {PL|
PR}.
Naturally, 2 = {0,1|}, 3 = {0,1,2|}, and so on, and similarly for
negative integers: -2 = {|0,-1}, etc. In general, a game is negative,
positive, or zero, depending on whether Left, Right, or the second player
has a winning strategy. Some games are neither. For example, the game
* = {0|0} is a clear win for the first player. It's said to be
fuzzy or confused with 0, but should not be confused with
the subject of Fuzzy Set Theory.
The theory and practice of mathematical games often force the players to
play several games simultaneously. This can be done in several ways
(ONAG, Chapter 14). The most common is taking their
disjunctive sum: the player to move selects a component game, and
then makes a move legal in that game. For any game P, P + 0 =
P. We may also check that * + * = 0. Indeed, a move by
one of the players leads to * = {0|0}, which is a win for the
other player. As another example, 1 + (-1) = {0|} +
{|0} = 0, since a move by either player leaves the other player with
a single winning move. In general, negation is defined
inductively as
-P = {-PR| -PL},
which effectively reverses Left's and Right's options. Further, consider
the game {0|1}, which is a clear win for Left and is thus positive. Now
let's try to compute the sum {0|1} + {0|1} + {|0}.
Any move by Left leads to {0|1} + {|0} with two options for
Right. The winning option is to play the first game in the sum which leads
to {0|} + {|0}, which is 0.
A first move by Right leaves two options for Left: {0|1} +
{0|1} and {0|1} + {|0}. The winning strategy is to
choose the first option, which leads to a positive (a win for Left) game
{0|1}.
This means that {0|1} + {0|1} + {|0} = 0, and thus {0|1} =
1/2. In a similar fashion, 1/4 = {0|1/2} and 3/4
= {1/2|1}. Other integers and dyadic fractions are
similarly defined in a finite number of steps.
Games can be compared. P > Q, provided the sum
P + (-Q) is positive. Numbers are defined
inductively as those games P whose options are numbers with an added
condition that no left option is greater than or equal to any right
option. As there are no options to compare, 0 is a number, as are all
integers defined above. The adopted notations
reflect on the value associated with a game. Thus, for example,
the game 1 is greater than the game 0.
A left option in a game is said to be dominated by any
greater (or equal) left option. A right option is
dominated by any smaller (or equal) right option. It can
be shown that dominated options can be removed without affecting the game's
value. In particular, the integer definitions are simplified to
n + 1 = {n|} and -n-1 = {|-n}.
It also can be shown that (2p + 1)/2n+1 =
{p/2n| (p + 1)/2n}.
Other numbers (and games) are constructed out of dyadic rationals in an
infinite number of steps similarly to Dedekind's sections. For example,
since 1/3 = 1/4 + 1/16 + 1/64 + ... and also 1/3 = 1/2 -
1/8 - 1/32 - 1/128 - ..., 1/3 is defined as
1/3 = {1/4, 1/4 + 1/16,1/4 + 1/16 + 1/64, ... | 1/2, 1/2 - 1/8, 1/2 - 1/8 -
1/32, ...}
In the same breath, we also get w = {0, 1, 2,
3, ...|}. And, with this, more numbers like w +
1 = {0, 1, 2, 3, ..., w|}, and so
on. At the other end of the spectrum, 1/w
= {0|1, 1/2, 1/4, 1/8, ...}, {0|1/w} = 1/2w, ...
Further on are obtained marvels like w -
1 = {0, 1, 2, 3, ...|w} and
w - 2 = {0, 1, 2, 3, ...|w, w - 1}, etc. and
w/2 = {0, 1, 2, 3, ...|w, w - 1, w - 2, ...}, Öw = {0, 1, 2,
...|w, w/2, w/4, ...}.
Little wonder the term surreal numbers, coined by D. Knuth, stuck around. Surreal numbers have
been written about by M. Gardner and the term has
been officially adopted in the second edition of ONAG.
So some games are numbers, while others are not. An additional
classification distinguishes between partisan and
impartial games. In the latter, exactly the same options are
available to the two players. Examples of impartial games include Nim and its many
dependents and incarnations: Scoring, Nimble, Northcott's
game, Plainim, and
more. Appearances notwithstanding, numbers (as defined
above) do not appear as values of impartial games. Furthermore, numbers
are also looked at condescendingly as options in the partisan games. The
reason is that, in general, a number (in its simplified form) lies between
its left and right options. So each player makes himself worse off by
moving into a number. In WW, this fact appears as The Number
Avoidance Theorem: DON'T MOVE IN A NUMBER UNLESS THERE'S NOTHING ELSE
TO DO. In ONAG, numbers are simply regarded as the stopping
positions. Once there, simply award the payoff to Left if the number is
positive and to Right if its negative. Naturally, if a number is to be
reached, Left will try to get its value as large as possible, Right will
try to have it small.
But there is a value in trying. Practice! Beat your friends until they learned
the game's secrets. Here's one example.
Contorted Fractions
You are given several fractions that can be modified according to
certain rules:
- The number replacing a given one must have strictly smaller
denominator, or,
- If the denominator is already 1, it must have numerator strictly
smaller in absolute value than that of the original number.
There are two players: Left and Right. A replacement
is only legal for Left if it decreases the number, legal for
Right only if it increases the number. You choose to be either
Right or Left and may force your computer to go first by
pressing You move button. All numbers in the game are modified by
clicking a little off their center line. Clicking on the right increases
the number, clicking on the left decreases it. To perform a move, select a
fraction, adjust it to the desired value and press the I move
button.
The game has two parameters. #Numbers is the number of
fractions in a game. Max is the absolute maximum number that may
appear in the numerator or denominator of a game.
This game has a curious theory. The numbers that appear in the game are
related to the game's value somewhat indirectly. First of all, it's clear
that the game is the disjunctive sum of simpler games each consisting of a
single fraction. Each of these, depending on the sign, are a one move win
to either Left or Right. When several games are played simultaneously, Left
will try to decrease the value of a fraction as little as possible, Right
will try to increase its value as little as possible.
From position '2/5' Left can move to positions '1/3', '1/4', '-1/2',
'0', '-2', etc., and Right can move to positions '1/2', '2/3', '3/4', '1',
'10', etc. According to the above piece of wisdom, Left should choose
'1/3', Right should choose '1/2'. Symbolically:
'2/5' = {'1/3'| '1/2'}.
What the players are looking for in that game, is the nearest
approximation to a rational number by fractions with smaller
denominators. These are given by the number's
parents on the Stern-Brocot
tree. Truly, solving this problem provides a student of fractions with
plenty of practice.
A more sophisticated student may surmise that, as the parents are
replaced by their parents, the binary
encoding and continued fractions come into the play. Each Contorted
Fractions game has only finite number of positions. Therefore, its value is
necessarily a dyadic number. For a single fraction x, what is the value of
the position 'x'? The answer is given by juxtaposing the binary encoding of
x on the Stern-Brocot tree and what is known as Berlekamp's rule for
interpreting Hackenbush strings (WW, pp. 77-78) or
sign-expansions (ONAG, p. 31).
As an example, '(1 + Ö5)/2' = 5/3. Now a
note for those who would like to pursue the matter further. In the first
edition of ONAG, there is a remark on page 84 that refers to a
note on page 96:
|
Note to p. 84: Some analytic properties of the function discussed on
pp. 83-86 are established by J. G. Mauldon in Quart. J. Math. Oxford
Ser., 17 (1966), p.261.
|
In the second edition, we read instead (p. 84) that the function is
traditionally called "Minkowski's Question-Mark Function," and has
interesting analytic properties. Talk about changes.
Writing about M. C. Escher [ Mathematical
Carnival, p. 102]
M. Gardner remarked:
|
When I first reproduced an Escher picture for my column for April, 1961
(and Scientific American ran one of his bird tessellations for
cover), I purchased from Escher only one print, a woodcut. For mere $40
to $60 each I could have bought scores of pictures that now would each
be worth thousands. But who could then have anticipated the astonishing
growth of Escher's fame?"
|
In recent years I had similar regrets about several missed opportunities
to purchase the first editions of WW and ONAG in
1970-80s. I am happy to own the books now. I am sure that, by supplying the
demand, A. and K. Peters will see to
it that the books will not fetch much more than their present
cost. However, their worth lies elsewhere. The books show a side of
mathematics usually missed by math textbooks. As the WW's authors
wrote in the preface:
|
It is not a book on recreational mathematics because there's too much
serious mathematics in it. On the other
hand, for us, as for our predecessors Rouse Ball, Dudeney, Martin Gardner,
Kraitchik, Sam Loyd, Lucas, Tom O'Beirne and Fred. Schuh, mathematics
itself is a recreation.
|
I believe that to become a successful subject, this is how mathematics
must be taught to young children — as a recreation. For a curious and
an attentive teacher, the books, especially WW, go a long way to
suggest how is this possible. It's about time that mathematical games be
given a serious consideration by curriculum developers.
References
- E. R. Berlekamp, J. H. Conway, R. K. Guy, Winning Ways for Your Mathematical Plays, Volume 1, A K Peters, 2001
- J. H. Conway, On Numbers And Games, A K Peters, 2001
- M. Gardner, Mathematical Carnival, Vintage Books, 1977
- M. Gardner, Penrose Tiles to Trapdoor Ciphers, W. H. Freeman and Company, 1989
- D. E. Knuth, Surreal Numbers, Addison-Wesley, 1974
Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in
Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. He can be reached at alexb@cut-the-knot.com
Copyright © 1996-2001 Alexander Bogomolny
