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## 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 we define four lists , , and of Grundy numbers.

These Grundy numbers are on the diagonal part of the table of Grundy numbers.
See numbers on blue rectangles in Example 5.2.

These Grundy numbers are above in the table of Grundy numbers.
See numbers on red rectangles in Example 5.2.

Grundy numbers in are on the left side of in the table of Grundy numbers.
See numbers on yellow rectangles in yellow Example 5.2.
Let .

Lemma 5.1   , in other words is the smallest number that does not belong to .

Proof   By Definition 4.8 and by Definition 5.2 , where . This lemma is direct from these facts.

Example 5.2   Let . Then is on the rectangle colored in green.
Grundy numbers of are in blue rectangles.
Grundy numbers of are in red rectangles.
Grundy numbers of are in yellow rectangles.

Next: The structure of Grundy numbers Up:Abstract and the table of contents Previous: The structure of columns