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
.