Summary of combined games-basic game problems

It has been several days since I learned the game. The headache is that all things need to be searched by myself. What I want to learn is that I can only find good information through constant searches and in-depth searches. Fortunately, I have done a lot of work now. I 'd like to share some of my materials. I hope it will be useful to you. First, I will share my papers on the national training team of Zhang Yifei, Wang Xiaoke, and Jia Zhihao, then there is the PPT of ppt "several variants of the issue of stone game and Their Solutions", which can be easily found on the Internet, and a deep understanding of the SG value !! The idea is flexible and you have to think more about it. Without reading these materials, I do not recommend that you look at the problem-solving report to answer questions !! There are also a few English websites that are quite good at speaking the game. I don't know much about English, and I won't talk about anything else, as long as Baidu is the information on the road.

Below is a very good article I see, address: http://www.cppblog.com/sdfond/archive/2010/02/06/107364.html

I will take a look at it myself later !! I have not understood the game models mentioned here. Continue !!

[Basic game model]
Classic Model 1: Nim variant.Including:
(1) stair Nim. Take the stones of odd steps as Nim. The two are equivalent because the winning strategy is the same.
(2) K heap can be obtained each time. This is the classic Moore Nim. It is a general Nim game.
(3) You can fetch one or two piles of stones each time. The number of stones obtained from the two stacks must be the same. This is the famous witv game, which is related to the Golden split number. The details are not clear.
(4) subtraction games. A general-purpose Nim game that selects the next state from the available State set each time, with many variants. The core idea is to calculate the SG function value.
(5) take-and-break game. Each time the situation is divided into multiple Nim sub-games, the corresponding SG value is obtained using the SG decomposition theorem. Pay attention to the sub-game and sub-status !!!
Classic Model 2: coin game)
(1) One-dimensional coin game, each time you can flip one or two. By separately considering each coin that can be flipped, we can find that the SG value of coin turning game is equivalent to that of Nim, so the two models are equivalent. It should be noted that many coin flip games generally start from 0 according to the requirements of the subject.
(2) A one-dimensional coin flip game can flip one or two at a time, which limits the range of the second coin, which is equivalent to subtraction game.
(3) one-dimensional coin flip games can be divided into one, two, or three each time. This game is called mockturtles and has a magical rule that is the sequence of odious numbers.
(4) For a high-dimensional coin game, Nim products and the tartan theorem are required.
The coin flip model has more changes, and many models have some wonderful rules. You need to make a table to find out.
Classic model 3: Green Hackenbush)
(1) tree edge deletion game: the colon principle proves that this model and Nim are still equivalent. The SG values of Multiple forks are different or the SG values of the corresponding root node.
(2) undirected graph edge deletion game: The fursion theorem is used to compress the circle and then convert it into a tree edge deletion game. However, this theorem does not prove.

Supplement:

I have never understood coin game. I understand it today. Let's talk about it here.

The only information I see is Jia Zhihao's article.

First, I was inspired by one sentence, because one of the two front-facing coins is equivalent to the one without the front-facing coins -- the SG value is equivalent, what does winning or losing determination mean ?? That is to say, the SG value of the situation is the single existence of each front-up chess piece in the situation. The explanation is that it is equivalent to creating another game, I cannot understand. For example,

This game, after turning one, the game is over, and all is the opposite, but after turning the two, it is a game that disappears (all is the opposite)

In this way, all games can be viewed as take-and-break.
For example, if there are two coins between the second coin and the rightmost coin, and there are 0 coins between the third coin and the second coin, the second coin will become

OK. It seems that it is still not described.