What is the game kernel?
Once, still when I studied at school, showed me one simple game. On a usual chessboard three white and three black checkers as follows settled down:
It was possible to go only forward, to any number of cages, without jumping through the opponent`s checkers. The one who made the last course was considered as the winner.
Game is quite simple if not to tell primitive. But when I decided to dig it more deeply, my shovel suddenly tinkled about something metal. It appeared a game kernel.
What is it - a game kernel? This game, as well as many widely known games, has a logical condition of the termination - the player either wins, or loses. And, in this case, the third is not given. As well as any consecutive (step-by-step) game (for example, chess, checkers, go), it has intermediate states - positions. Positions in which there is such sequence of the courses which leads to a prize at any actions of the opponent form a kernel (advantageous strategy). For example, everyone who is fond of chess, knows chess etudes “ to find a mat in 3 courses “ or “ white begin and win “. It is also tasks on search of a kernel of a position.
However today nobody can tell whether there is a kernel for an initial position in chess or checkers. As soon as someone finds this kernel, game will die.
Coming back to the game described by me, I am forced to admit: I killed her. I found a kernel of an initial position and even any any position on a board. And even for any number of couples of checkers.
So, we will make investigative experiment - I will tell you as I killed this game. I hope, frank recognition will facilitate my fate. :) Any position on a board I will designate
the three of numbers A - B - C where A - number of cages between the first couple of checkers, In - between the second, and With - between the third couple. Thus, the initial position registers in the form of 2 - 4 - 6. The sequence of numbers has no value.
At first I paid attention that the position x - x - 0 is advantageous for the player going the second. Really, in a position 1 - 1 - 0 it does the last course with all evidence, and in any other position it just levels number of cages between not close checkers.
Further, I found the first piece of a kernel - a position 1 - 2 - 3. It is won besides by the player going the second - he parries any course of the opponent, putting a position x - x - 0. Check!
The following piece of a kernel was in a position 1 - 4 - 5. And again the second player wins, parrying any course of the opponent in a position 1 - 2 - 3 or x - x - 0.
And, at last, the initial position 2 - 4 - 6 was belonging to a kernel too! Any course “ white “ it is parried by black, leading to one of the positions described above.
By the way, now it is possible to allow to go not only forward, but also back: the course is parried by the course forward on the same number of cages with deduction of distance between them back.
Further, I considered a new initial position - 6 - 6 - 6. All checkers stand on the first and last line. And it was belonging to a kernel. There is an intermediate piece of a kernel 3 - 5 - 6, inevitably bringing to 1 - 4 - 5 or 1 - 2 - 3 or x - x - 0. Or game is reduced to previous 2 - 4 - 6 - everything depends on the first course “ white “.
So, we see that black win always if the initial position belongs to a kernel. Otherwise, always win white - the first course they put suitable “ nuclear “ position.
Now it is possible to expand game on bigger number of checkers. If it multiple to three, game breaks into the independent three - i.e. the same game, but in several copies going in parallel turns out. If it multiple to two, then game breaks into couples, in each of which it is necessary just to level number of cages. All other options of game are divided into the three and couples.
That`s all - game is killed on the spot. I ask indulgences - I pled guilty and frankly admitted everything.]