This is the home page for Section 05 of Honors 391A.
HONORS 391A-05 is a one-credit seminar, intended for upper-level Commonweath
College honors students, on J. H. Conway's theory of games. In his book
*On Numbers and Games*, Conway develops a theory that assigns values to
a particular class of **two-player games**. Some of these values are
called **numbers**, and in fact every **real number** is the value of
some game. But there are also various kinds of **surreal numbers**, some
**infinite** (bigger than any real number) and some **infinitesimal**
(positive, but smaller than any positive real number).
The course is intended for students with or without formal mathematical
training. Even students with such training are likely to be unfamiliar with
the formal definitions of the conventional number systems, and it is not likely
that any students will have previously studied Conway's theory.

**Instructor Contact Info:**
David Mix Barrington, 210 CMPSCI
building, 545-4329, office hours TBA
I generally answer my email fairly
reliably.

**Co-Instructor Contact Info:**
Dan Stubbs, an advanced
computer science undergraduate.

The course will meet for one lecture meeting each week, on Mondays from 1:25-2:15 p.m. in Elm Room 212 in the new Honors College complex. Attendance is an important component of the course.

- Course Requirements and Grading
- Assignment Directory (NEW)
- Questions and Answers on Assignments (none yet)
- Syllabus (Lecture-by-Lecture Schedule) (NEW)

- We will play and analyze a lot of games. The first, which some of you may
have seen, is
**Nim**. In this game a position consists of a number of piles, each with a positive number of stones. A move consists of taking one or more stones from one of the piles. The first player who cannot move*loses*. (This rule is called**normal play**in the theory. There is another game called**misere nim**where the first player who cannot move*wins*.)**Analyzing**a game like this means determining which player will win if both players play optimally. Since the starting position is variable, a full analysis tells us which sets of pile are**winning positions**for the first player and which are**losing positions**. If you know this, you also know how to make a winning move from any winning position. - We will learn some of Conway's theory of numbers and games. For example,
we can classify games as
**positive**,**negative**,**zero**, or none of the above. We'll be able to**add**two games, determine which of two games is**larger**, and so forth. - We'll connect Conway's games to the
**definition of numbers**in conventional mathematics. We'll learn (or review) the formal definition of the**natural numbers**in terms of the**Peano axioms**, then look at the definitions of**integers**,**rational numbers**, and**real numbers**in terms of natural numbers. We'll look at different notions of infinity, such as infinite**ordinals**and**cardinals**. Finally, we'll look at Conway's definition of numbers as presented in Donald Knuth's book*Surreal Numbers*, where two people on a desert island work out the properties of numbers from a set of axioms, without reference to games.

**Textbooks:** There are three books we will use to varying degrees
(detailed information is on the course SPIRE page). Only the Knuth text
is required (and it is available for free download as well as being relatively
cheap):

- Conway,
*On Numbers and Games (ONAG)*. The "bible" of the theory, written largely for mathematicians. - Berlekamp, Conway, and Guy,
*Winning Ways for Your Mathematical Plays*, a four-volume more popular presentation of the theory with lots of examples. We'll be dealing primarily with Volume I. - Knuth,
*Surreal Numbers*, a brief presentation of the theory of Conway numbers as an example of axiomatic reasoning, in the form of a dialogue between two people on a desert island.

**Announcements (16 September 2013):**

(16 Sept) I have started an assignment page which includes the assignment for next Monday.

(9 Sept) Here are the rules for **Division Nim** as presented
in class today. As in ordinary Nim, we have one or more piles, each
with
one or more stones. On her turn, Left may take any even-sized pile
and take half of its stones, thus dividing its size by two. On her
turn,
right may take any pile whose size is divisible by an odd number,
divide it into that odd number of subpiles, and take away all but
one
of the subpiles. (Thus she divides the size by the odd number.)

Your assignment (not to be written up) is to play this game and figure out how to tell who has a winning strategy, given an initial set of piles. (It may or may not depend on which player moves first.)

In class today we saw how to play optimally and determine the winner in any Nim game with 0, 1, or 2 piles. Please think about (or research) the correct strategy with three or more piles. (In particular, what is the best move to make with starting piles of size (3, 5, 7) as we considered today?) If you write up this argument we will look at it.

(8 Sept) We'll see you all tomorrow in Elm Room 212 at 1:25. There's now a syllabus and a requirements and grading page linked above.

(15 August) This is just a preliminary web site so far, but more is coming soon!

(19 August) Course Overview and textbook information added.

Last modified 16 September 2013