4. Some interesting graphs produced by the set of L states

In this section we study 3D and 2D graphs produced by L states of chocolate games that satisfy inequalities $y \le \lfloor {z/k} \rfloor$ for $k = 1,2,3,4,5$ .

Example 4.1   Here we study graphs made by the set of L states for the chocolate game of Definition 3.1 which satisfies the inequality $y \leq z$ .
We denote by $LS[1]$ the set of L state of this chocolate.

Fig 4.1 is a 3D graph of $LS[1]$ .

$LS[1]$ .
Figure 4.1 

By $LS[1]$ we define a sequence of sets $d[1,n]$ $=\{\{y,z\};\{n,y,z\} \in LS[1]\}$ , $n = 0,1,2,...$ . By plotting $d[1,n]$ for $n = 0,1,2,...,30$ we get Fig 4.2.
We use the following Mathematica program. Note that we join points with curved segments by Mathematica command " $Joined -> True$ ".

LS[1]=ppo[50];
Clear[d]; Do[
 d[1,k] = Map[{#[[2]], #[[3]]} &, 
   Select[LS[1], #[[1]] == k &]], {k, 0, 50}];
ListPlot[Table[d[1,k], {k, 1, 50}], Joined -> True]

Figure 4.2  

By plotting $d[1,n]$ for $n = 0,1,2,...,50$ we get Fig 4.3.

Figure 4.3  

Example 4.2   Here we study graphs made by the set of L states for the chocolate game that satisfies the inequality $y \le \lfloor {z/2} \rfloor$ .
We denote by $LS[2]$ the set of L state of this chocolate.

Fig 4.4 is a 3D graph of $LS[2]$ .

Figure 4.4  

By $LS[2]$ we define a sequence of sets $d[2,n]$ $=\{\{y,z\};\{n,y,z\} \in LS[2]\}$ for $n = 0,1,2,...$ . By plotting $d[2,n]$ for $n = 0,1,2,...,30$ we get Fig 4.5.

Figure 4.5  

By plotting $d[2,n]$ for $n = 0,1,2,...,50$ we get Fig 4.6.

Figure 4.6  

Example 4.3   Here we study graphs made by the set of L states for the chocolate game that satisfies the inequality $y \le \lfloor {z/3} \rfloor$ .
We denote by $LS[3]$ the set of L state of this chocolate.

Fig 4.7 is a 3D graph of $LS[3]$ .

Figure 4.7  

By $LS[3]$ we define a sequence of sets $d[3,n]$ $=\{\{y,z\};\{n,y,z\} \in LS[3]\}$ . By plotting $d[3,n]$ for $n = 0,1,2,...,30$ we get Fig 4.8.

Figure 4.8  

By plotting $d[3,n]$ for $n = 0,1,2,...,50$ we get Fig 4.9.

Figure 4.9  

Example 4.4   Here we study graphs made by the set of L states for the chocolate game that satisfies the inequality $y \le \lfloor {z/4} \rfloor$ .
We denote by $LS[4]$ the set of L state of this chocolate.

Fig 4.10 is a 3D graph of $LS[4]$ .

Figure 4.10  

By $LS[4]$ we define a sequence of sets $d[4,n]$ $=\{\{y,z\};\{n,y,z\} \in LS[4]\}$ for $n = 0,1,2,...$ . By plotting $d[4,n]$ for $n = 0,1,2,...,30$ we get Fig 4.11.

Figure 4.11  

By plotting $d[4,n]$ for $n = 0,1,2,...,50$ we get Fig 4.12.

Figure 4.12  

Example 4.5   Here we study graphs made by the set of L states for the chocolate game that satisfies the inequality $y \le \lfloor {z/5} \rfloor$ .
We denote by $LS[5]$ the set of L state of this chocolate.

Fig 4.13 is a 3D graph of $LS[5]$ .

Figure 4.13  

By $LS[5]$ we define a sequence of sets $d[5,n]$ $=\{\{y,z\};\{n,y,z\} \in LS[5]\}$ for $n = 0,1,2,...$ . By plotting $d[5,n]$ for $n = 0,1,2,...,30$ we get Fig 4.14.

Figure 4.14  

By plotting $d[5,n]$ for $n = 0,1,2,...,50$ we get Fig 4.15.

Figure 4.15  

For $k = 2,4$ the 3D graphs are beautiful, but 2D graphs are not beautiful, however, for $k=1,3,5$ the 3D graphs and 2D graphs are both beautiful.