Some definitions for lists of Grundy numbers.

We need definitions for lists of Grundy numbers to study the mathematical structures of Grundy numbers.

Definition 5.2   For a fixed coordinates $\{y,z\}$ we define four lists $L11(\{y,z\})$ , $L12(\{y,z\})$ , $L13(\{y,z\})$ and $L1(\{y,z\})$ of Grundy numbers.
$(1)$ $L11(\{y,z\})$ $=\{G1R(\{w,w\}),0 \leq w \leq y, w < z\}.$
These Grundy numbers are on the diagonal part of the table of Grundy numbers.
See numbers on blue rectangles in Example 5.2.
$(2)$ $L12(\{y,z\})$ $=\{G1R(\{y,w\}),y < w < z\}.$
These Grundy numbers are above $G1R(\{y,z\})$ in the table of Grundy numbers.
See numbers on red rectangles in Example 5.2.
$(3)$ $L13(\{y,z\})$ $= \{G1R(\{v,z\}), 0 \leq v < y\}.$
Grundy numbers in $L11(\{y,z\})$ are on the left side of $G1R(\{y,z\})$ in the table of Grundy numbers.
See numbers on yellow rectangles in yellow Example 5.2.
Let $L1(\{y,z\}) = L11(\{y,z\}) \cup L12(\{y,z\}) \cup L13(\{y,z\})$ .

Lemma 5.1   $G1R(\{y,z\})= Mex(L1(\{y,z\}))$ , in other words $G1R(\{y,z\})$ is the smallest number that does not belong to $L1(\{y,z\})$ .

Proof   By Definition 4.8 $G1R(\{y,z\}) = Mex(G1R(\{v,w\});\{v,w\} \in move1R(\{y,z\}))$ and by Definition 5.2 $movekR(\{y,z\}) = \{\{v,z \};0 \leq v<y \} \cup \{ \{\min(y, w),w \};0\leq w<z \}$ $=\{\{v,z \};0 \leq v<y \} \cup \{ \{y,w \};y < w < z \}$ $\cup \{w,w \};0\leq w\leq y, w < z \}$ , where $v,w \in Z_{\ge 0}$ . This lemma is direct from these facts.

Example 5.2   Let $y = 8, z = 23$ . Then $G1R(\{y,z\})$ $=G1R(\{8,23\}) = 19$ is on the rectangle colored in green.
Grundy numbers of $L11(\{y,z\})$ are in blue rectangles.
Grundy numbers of $L12(\{y,z\})$ are in red rectangles.
Grundy numbers of $L13(\{y,z\})$ are in yellow rectangles.

Figure 5.14