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The structure of Grundy number

We can use mathematical induction to prove Predictions 5.1, 5.2, 5.3,5.4 and 5.5.
As you see in Subsections 5.1 and 5.2, the mathematical structure of the table of Grundy numbers seems to be clear enough, but it is difficult to prove that these mathematical structures do exist, since there are many cases that should be covered in the proof.