Case 4

Next we study the case that $z = n = 2$ $(mod \ 4)$ and $y$ is odd.
An example of this case is Example 5.5.

Figure 5.22  

In Fig. 5.22 we suppose that Predictions 5.1, 5.2, 5.3,5.4 and 5.5 are valid for $z=0,1,2,...,4k+4s-3$ and for $y = 0,1,2,...,z$ . We also suppose that these predictions are valid for $z=4k+4s-2$ and $y=0,1,...,6k+4s-4$ . Under these assumptions we prove that $X= G1R(\{4k-3,4k+4s-2\}) = 6s+1$ .

Remark 5.3   Note that some Grundy numbers $G1R(\{y,z\})$ with $z > 4k+4s-1$ are presented in the table. We do not assume the values of these Grundy numbers to prove this prediction, but the authors present these numbers to show the relative positions of $X= G1R(\{4k-3,4k+4s-2\})$ in the table of Grundy numbers.
Here we assume that $G1R(\{4k-3,8k-6\})=6k-5$ , which is the last number of the sequence $\{1,4,7,...,6s-2,...,6k-8,6k-5\}$ that is an arithmetic sequence with common difference of $3$ and its length is $\lceil (4k-3)/4 \rceil$ .
Therefore the position of $X= G1R(\{4k-3,4k+4s-2\})$ is in the midst of the the sequence $\{1,4,7,...,6s-2,...,6k-8,6k-5\}$ , and this condition make Case 4 very different from Case 1, Case 2 and Case 3.

$L11(\{4k-3,4k+4s-2\})$ $=\{0,2,3,5,6,...,6s-1,6s,6s+2,6s+3,...6k-7,6k-6,6k-4 \}$

$$= \{0,2,3,5,6,...,6s-1,6s\}$$ (5.54)

$$\cup \{6s+2,6s+3,6s+5,6s+6,...,6k-7,6k-6,6k-4\}.$$ (5.55)

Clearly $L12(\{4k-3,4k+4s-2\})$

$$= \{1,4,7,...,6s-2\}$$ (5.56)

$$\cup \{6k-2,...,6k+4s-4\}.$$ (5.57)

By using Prediction 5.3 for $z=4k+4s-2$ and $y=0,1,...,4k-4$
$L13(\{4k-3,4k+4s-2\})$ $=\{4k+4s-2,4k+4s-3,4k+4s-1,4k+4s-4,...,6s+7,6k+4s-5,6s+4,6k+4s-4\}$

$$= \{4k+4s-2,4k+4s-1,...,6k+4s-5,6k+4s-4\}$$ (5.58)

$$\cup \{4k+4s-3,4k+4s-4,...,6s+7,6s+4\}.$$ (5.59)

By (5.54) and (5.56) we have

$$L11(\{4k-3,4k+4s-2\}) \cup L12(\{4k-3,4k+4s-2\}) \supset \{0,1,2,3,4,5,6,...,6s-1,6s\}.$$ (5.60)

By (5.55), (5.57), (5.58) and (5.59) The lists $L11(\{4k-3,4k+4s-1\})$ , $L12(\{4k-3,4k+4s-2\})$ and $L13(\{4k-3,4k+4s-2\})$ do not contain $6s+1$ , and hence by (5.60) we have $X= G1R(\{4k-3,4k+4s-2\}) = 6s+1$ .