### notes on nim-theory Dots and Boxes

5 replies. Last post: 2007-10-03

notes on nim-theory
• wccanard at 2007-09-27

Several people have asked me about where to learn nim (and its application to dots and boxes). I always suggest Winning Ways and Berlekamp's book but then again I work at a university and can just wander down to the library to look at these books. But kids these days :-/ If it's not on the internet then they're not going to shell out a tenner for Berlekamp, oh no. So I wrote some notes myself at wccanard.wetpaint.com. Check out nim.pdf. All comments, esp. constructive criticism, welcome.

wcc

• Carroll ★ at 2007-10-03

Thx WC for this nice piece of work!

No comment for 5 days does not mean funinteresting but perfect.

One point I would like you to elaborate is why for solution of previous corner puzzles you don't refer to nimbers and (maybe related) your statement that on 5x5, playing by nimbers one will be facing positions where despite the nimber value not being zero, one will lose.

This is of course related to the relatively short chains and the numerous small values of many loony finals. As the category in which the position will stand is quite obvious to 2000+ players, there must be some criteria telling you if you should better consider nimber (local) or value (global).

• wccanard at 2007-10-03

For the corner puzzle with the very long chain, knowing nimbers is very helpful in solving the puzzle. But for the others nim is basically useless. The golden rule is this: nim tells you how to win the long chain battle. In a very close game of dots and boxes, you might have to sacrifice to win the chain battle, and if you sacrifice too much then you win the chain battle but lose the game anyway. All of those corner puzzle games are very close games so nim is almost irrelevant. In fact I generated 80 corner puzzles but only posted a few—the ones where nim helped a lot were mostly quite boring :-) In the corner puzzles, it's not about getting the right number of chains, it's about making the chains as long, or as short, as possible, which is a different thing.

Here's a type of position that really happens in real life on the server here (look at e.g. 5 games that Scot played recently and I'm sure you'll find at least one that ended up looking like this after 25 or so moves):

``````+-+-+-+-+-+    |   |+-+ + + + +    |   |+-+-+-+-+-+    |   |+-+ + + + +  | |   |+ + +-+-+-+  | |   |+-+ + + + +
``````

Here P1 has won the long chain battle, with 2 chains, but the quads make

the long chain worth zero, and so P2 will win the game because the short

chain battle is worth 1. In fact it's P1 to move and his next move will probably be to sacrifice in the bottom right: that's the box that will lose him the game, nothing to do with the chain battle.

wcc

• wccanard at 2007-10-03

MISTAKE! I posted the wrong position :-/ hang on

• wccanard at 2007-10-03

Here's what the bottom of that post should say:

``````+-+-+-+-+-+    |   |+-+ + + + +    |   |+-+-+-+-+-+|   |   |+ + + + + +| | |   |+ + +-+-+-+  |     |+-+-+-+ + +
``````

Here P1 has won the long chain battle, with 2 chains, but the qauds make the long chain *battle* worth zero, and so P2 will win the game…