The Chocolates that satisfy the inequality .

In this section we study the chocolate of Fig 5.1. The mathematical structure of this chocolate is interesting when it is compared to that of chocolate of Definition 2.4. While the chocolate of Definition 2.4 has a simple formula for L-states, this chocolate has complicated formulas for L-states.

Figure 5.1  

The list of L-states of the game with inequality $y \leq z$ is the union of the following lists $(1)$ , $(2)$ , $(3)$ , $(4)$ , $(5)$ and $(6)$ .
$(1)$ $\{\{3 n + 2 k, 2 n, 2 n + 2 k\}$ , $n \in Z_{\geq0}$    and $k \in Z_{\geq0} \}$
$(2)$ $\{\{3 k+1, 2 n, 2 n + 2 k + 1\}$ , $n \in Z_{\geq0}$    and $k =1,2,...,n \}$
$(3)$ $\{\{3 n + 2 k+3, 2 n, 4 n + 2 k + 3\}$ , $n \in Z_{\geq0}$$\text { and } k \in Z_{\geq0} \}$
$(4)$ $\{\{3 n + 2 k+2, 2 n + 1, 2 n + 2 k+1\}$ , $n \in Z_{\geq0}$    and $k \in Z_{\geq0} \}$
$(5)$ $\{\{3 k+1, 2 n + 1, 2 n + 2 k + 2\}$ , $n \in Z_{\geq0}$    and $k = 0, 1, 2, ..., n\}$
$(6)$ $\{\{3 n + 2 k+3, 2 n + 1, 4 n + 2 k+4\}$ , $n \in Z_{\geq0}$    and $k \in Z_{\geq0} \}$ . The above facts are proved in Theorem 5.1. The proofs for the case of $k = 1$ are presented in this article for the first time. The logic behind proofs is simple, but is is extremely difficult to present a complete proof. In fact there are too many cases that need to be treated. The authors present several typical cases to be proved, and omit other cases. They think that it is not difficult to treat other cases in a method that is similar to the one used in these typical cases.
We define a new type of chocolate precisely.

 

Definition 5.1   There is a blue $1 \times 1$ block, and on both side of it there are columns of brown blocks that are numbered by integers. In the chocolate in Fig. 5.1 we have the first, the second, the fourth,...,10th column on the right side of the blue block and $(-1)$ th, $(-2)$ th,...,$(-6)$ th column on the left side of the blue block.
The i-th column is $1 \times (i+1)$ block for $i =1, 2, 3,...,12$ and the i-th column is $1 \times 1$ block for
$i =-1, -2, -3,-4,-5,-6$ .
By generalizing the chocolate in Fig. 5.1 we make a chocolate that has the first, the second, the fourth,...,n-th column,... on the right side of the blue block and $(-1)$ th, $(-2)$ th,...,$(-m)$ -th column,... on the left side of the blue block.
The i-th column is $1 \times (1+ i)$ block for $i =1, 2, 3,...,n,...$ and the i-th column is $1 \times 1$ block for $i =-1, -2, -3,-4,-5,-m,...$ .
Brown blocks are sweet chocolate that can be eaten, and the blue block is the bitter chocolate that cannot be eaten.
You can cut these chocolate along the segments in three ways.
$(1)$ You cut vertically on the left side of the blue (bitter) block.
$(2)$ You cut horizontally above the blue (bitter) block.
$(3)$ You cut vertically on the right side of the blue (bitter) block.
Therefore it is proper to represent this chocolate with $\{x,y,z\}$ , where $x,y,z$ stand for the maximum numbers of times that we can cut these chocolate in each direction. For example in Fig. 5.1 we can cut 6 times at most vertically on the left side of the blue block, we can cut 10 times at most horizontally above the blue (bitter) block and we can 10 times at most vertically on the right side of the blue block. Therefore $x=6$ , $y = 10$ and $z=10$ . Therefore we represent the chocolate in Fig. k1demochoco with the coordinates $\{6,10,10\}$ .

For any $x,y,z\in Z_{\geq 0}$ we denote by $\{x,y,z\}$ the state of chocolate. We study the chocolate in Fig. 5.2 to calculate L-states of the chocolate game in Fig. 5.1. In Fig. 5.2 If you can cut horizontally above the blue (bitter) block $y$ times at most and vertically on the right side of the blue (bitter) block $z$ times at most, then we represent the chocolate with the coordinates $\{y,z\}$ .

Figure 5.2  

Fig 5.3 is the table of Grundy numbers of the chocolate in 5.2 for $y=0,1,2,...,30$ and $z=0,1,2,...,30$ .

Figure 5.3  

In Fig 5.3 there are many kinds of mathematical structures, but in this section we study the structures of each row and column of the table of Grundy numbers.