2 Grundy Numbers of chocolate bar

In this paper we study Grundy numbers of chocolate bar. For a general bar, the strategies seem complicated. We focus on bars that grow regularly in height. Let Difinition 2.1. $f$ be a function that satisfies the following two conditions:
(i) $f(t)\in Z_{\geq 0}$ for $t \in Z_{\geq 0}$ .
(ii) $f$ is monotonically increasing,i.e., we have $f(u) \leq f(v)$ for $u,v \in Z_{\geq 0}$ with $u \leq v$ .

Definition 2.2. Let $f$ be the function that satisfies the conditions in Definition 2.1.
For $y,z \in Z_{\geq 0}$ the chocolate bar will consist of $z+1$ columns where the 0th column is the bitter square and the height of the $i$ -th column is $t(i) = \min (f(i),y) +1$ for i = 0,1,...,z. We will denote this by $CB(f,y,z)$ .
Thus the height of the $i$ -th column is determined by the value of $\min (f(i),y) +1$ that is determined by $f$ , $i$ and $y$ .

Example 2.1. Here are examples of chocolate bar games $CB(f,y,z)$ .

fig2,1

Figure 2.1 $CB(f,3,13)$ $f(t)$ $= \lfloor \frac{t}{4}\rfloor$

fig2.1

Figure 2.2 $CB(f,4,9)$ $f(t)$ $= \lfloor \frac{t}{2}\rfloor$

fig2.3

Figure 2.3 $CB(f,2,9)$ $f(t)$ $= \lfloor \frac{t}{2}\rfloor$

fig2.4

Figure 2.4 $CB(f,8,31)$ $f(0)=f(1)=0$ and $f(t)$ $=2^{\lfloor log_2t \rfloor -1}$ for $t > 1$ .

For a fixed function $f$ , we denote the position of $CB(f,y,z)$ by coordinates $\{ y,z\}$ without mentioning $f$ .

Example 2.4.

Here, we present four examples of coordinates of positions of chocolate bars when $f(t)$ $= \lfloor \frac{t}{2}\rfloor$ .

fig2.5

Figure 2.5. $CB(f,2,5)$ $\{ 2,5\}$

Figure 2.6. $CB(f,1,3)$ $\{ 1,3\}$

Figure 2.7. $CB(f,0,5)$ $\{ 0,5\}$

Figure 2.8. $CB(f,1,5)$ $\{ 1,5\}$

For a fixed function $f$ , we define $move_ f$ for each position $\{ y,z\}$ of the chocolate bar $CB(f,y,z)$ . This $move_ f$ is a special case of $move$ defined in Definition 1.4.

Definition 2.3.

For $y,z \in Z_{\ge 0}$ we define
$move_ f(\{ y,z\} )=\{ \{ v,z \} :v<y \} \cup \{ \{ \min (y, f(w)),w \} :w<z \}$ , where $v,w \in Z_{\ge 0}$ .

Example 2.3. Here, we explain about move when $f(t)$ $= \lfloor \frac{t}{2}\rfloor$ . If we start with the position $\{ y,z\} =\{ 2,5\}$ and reduce $z=5$ to $z=3$ , then the y-coordinate (the first coordinate) will be $\min (2, \lfloor 3/2 \rfloor )=\min (2,1)=1$ .
Therefore we have $\{ 1,3\} \in move_ f(\{ 2,5 \} )$ . It is easy to see that $\{ 1,5\} , \{ 0,5\} \in move_ f(\{ 2,5\} )$ , $\{ 1,3\} \in move_ f(\{ 1,5\} )$ and $\{ 0,5\} \notin move_ f(\{ 1,3\} )$ .

3 A Chocolate Game Bar $CB(f,y,z)$ whose Grundy Number is $G(\{ y,z\} )=y \oplus z$