Here’s a 5 move game from C, the complex numbers game, annotated by Hendrik Lenstra in Nim multiplication.
WhiteBlack1.3−2i3R2.3R(22/7)Q3.(−44Z,−14Z)?−44Z4.−44Z(0N,44N)!5.Resigns
He writes : “The following 5 comments will make the rules clear.
1 : White selected a complex numbers. Black knows that C=R×R by a+bi=(a,b), and remembers Kuratowski’s definition of an ordered pair: (x,y)=x,x,y. Thus black must choose an element of 3R,3R,−2R. The index R here, and later Q,Z and N, serve to distinguish between real numbers, rational numbers, integers and natural numbers usually denoted by the same symbol. Black’s move leaves White a minimum of choice, but it is not the best one.
2 : White has no choice. The Dedekind definition of R which the players agreed upon identifies a real number with the set of all strictly larger rational numbers; so Black’s move is legal.
3 : A rational number is an equivalence class of pairs of integers (a,b) with b≠0; here (a,b) represents the rational number a/b. The question mark denotes that White’s move is a bad one.
4 : The pair (a,b) of natural numbers represents the integer a−b. Black’s move is the only winning one.
5 : White resigns, since he can choose between 0N and 0N,44N. In both cases Black will reply by 0N, which is the empty set” (and so wins because White has no move left).
These rules make it clear what we mean by the natural numbers N game, the Z-game and the Q and R games. A sum of games is defined as usual (players are allowed to move in exactly one of the component games).
Here’s a 5 term exercise from Lenstra’s paper : Determine the unique winning move in the game N+Z+Q+R+C
It will take you less than 5 minutes to solve this riddle. Some of the other ‘exercises’ in Lenstra’s paper may take you a lot longer, if not forever…
Exactly 5 years ago I wrote : “As it is probably better to run years behind than to stand eternally still, I’ll try out how much of a blogger I am in 2004.”
5 months ago this became : “from january 1st 2009, I’ll be moving out of here. I will leave the neverendingbooks-site intact for some time to come, so there is no need for you to start archiving it en masse, yet.”
5 minutes before the deadline, this will be my last post….
of 2008
less entropy in 2009!
5 Comments