# 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.  be a function that satisfies the following two conditions:
(i)    for .
(ii)   is monotonically increasing,i.e., we have for with .

Definition 2.2. Let be the function that satisfies the conditions in Definition 2.1.
For the chocolate bar will consist of columns where the 0th column is the bitter square and the height of the -th column is for i = 0,1,...,z. We will denote this by .
Thus the height of the -th column is determined by the value of that is determined by , and .

Example 2.1.  Here are examples of chocolate bar games .

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4   and for .

For a fixed function , we denote the position of by coordinates without mentioning .

Example 2.4.

Here, we present four examples of coordinates of positions of chocolate bars when .

Figure 2.5.

Figure 2.6.

Figure 2.7.

Figure 2.8.

For a fixed function , we define for each position of the chocolate bar . This is a special case of defined in Definition 1.4.

Definition 2.3.

For we define
, where .

Example 2.3. Here, we explain about move when . If we start with the position and reduce to , then the y-coordinate (the first coordinate) will be .
Therefore we have . It is easy to see that , and .

3 A Chocolate Game Bar whose Grundy Number is