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A Simplified Game

This is the newest result of our research, and hence it is in the last section of this article.
Here we are going to study the game in Section 5 again. Since the game was not simple enough, we are going to study a simpler version in this section.

Definition 9.1   There are 3 coins and in the strip there are 10 squares, and they are numbered with 1,2,3,...,10. The shape of the strip is not a straight line here.
The strip consists of sub-strips {1,2,3,4,5} and {6,7,8,9,10}. See Graph 9.1
The rule of the game is very similar to that of the previous game on a strip.
(1) The difference is that you cannot make a coin jump over another coin when they are at the corner.

Graph 9.1  

By the same method we used in Section 5 we can find all the P-positions.
The following is the list of all the P-positions.

{{1, 2, 3}, {1, 3, 5}, {1, 3, 7}, {1, 3, 9}, {1, 4, 6}, {1, 4, 8}, {1, 4, 10}, {1, 5, 7}, {1, 5, 9}, {1, 6, 8}, {1, 6, 10}, {1, 7, 9}, {1, 8, 10}, {2, 3, 4}, {2, 3, 6}, {2, 3, 8}, {2, 3, 10}, {2, 4, 7}, {2, 4, 9}, {2, 5, 6}, {2, 5, 8}, {2, 5, 10}, {2, 6, 9}, {2, 7, 8}, {2, 7, 10}, {2, 9, 10}, {3, 4, 5}, {3, 5, 7}, {3, 5, 9}, {3, 6, 8}, {3, 6, 10}, {3, 7, 9}, {3, 8, 10}, {4, 5, 6}, {4, 5, 8}, {4, 5, 10}, {4, 6, 9}, {4, 7, 8}, {4, 7, 10}, {4, 9, 10}, {5, 6, 8}, {5, 6, 10}, {5, 7, 9}, {5, 8, 10}, {6, 7, 8}, {6, 7, 10}, {6, 9, 10}, {7, 8, 9}, {8, 9, 10}}

On the other hand we can get the list of all the N-positons.
{{1, 2, 4}, {1, 2, 5}, {1, 2, 6}, {1, 2, 7}, {1, 2, 8}, {1, 2, 9}, {1, 2, 10}, {1, 3, 4}, {1, 3, 6}, {1, 3, 8}, {1, 3, 10}, {1, 4, 5}, {1, 4, 7}, {1, 4, 9}, {1, 5, 6}, {1, 5, 8}, {1, 5, 10}, {1, 6, 7}, {1, 6, 9}, {1, 7, 8}, {1, 7, 10}, {1, 8, 9}, {1, 9, 10}, {2, 3, 5}, {2, 3, 7}, {2, 3, 9}, {2, 4, 5}, {2, 4, 6}, {2, 4, 8}, {2, 4, 10}, {2, 5, 7}, {2, 5, 9}, {2, 6, 7}, {2, 6, 8}, {2, 6, 10}, {2, 7, 9}, {2, 8, 9}, {2, 8, 10}, {3, 4, 6}, {3, 4, 7}, {3, 4, 8}, {3, 4, 9}, {3, 4, 10}, {3, 5, 6}, {3, 5, 8}, {3, 5, 10}, {3, 6, 7}, {3, 6, 9}, {3, 7, 8}, {3, 7, 10}, {3, 8, 9}, {3, 9, 10}, {4, 5, 7}, {4, 5, 9}, {4, 6, 7}, {4, 6, 8}, {4, 6, 10}, {4, 7, 9}, {4, 8, 9}, {4, 8, 10}, {5, 6, 7}, {5, 6, 9}, {5, 7, 8}, {5, 7, 10}, {5, 8, 9}, {5, 9, 10}, {6, 7, 9}, {6, 8, 9}, {6, 8, 10}, {7, 8, 10}, {7, 9, 10}}

From the list of P-positions and the list of N-positions we can make the following chart.

In this chart P-positions are colored in yellow and N-positions are colored in purple.

{1,2,3} is a P-position, and hence you can win the game by using this chart if you start with {1,2,3}.

gameforcanada : Variants of Nim. Part