HONORS 391A: Seminar: On Numbers and Games

David Mix Barrington and Dan Stubbs

Fall, 2013

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)

1. 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.

2. 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.

3. 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):

1. Conway, On Numbers and Games (ONAG). The "bible" of the theory, written largely for mathematicians.

2. 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.

3. 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