The structure of each column

Here we study the structure of each column.

Example 5.1   Here we study two columns as examples.

Figure 5.13  

In Fig. 5.13 we study the mathematical structures of the columns of Grundy numbers.
The mathematical structure of each column is more simple than that of each row.
For example, when $y = 8$ , then the number at the top of the column is 12. If we start with 12 and move down picking up every other number, then we have $12,14,16,18,...,26$ that is an arithmetic sequence with common difference of $2$ .
On the other hand if we start with 1 that is the second number from the top of the column and move down picking up every other number, then we have $1,4,7,10,13$ , and these numbers forms an arithmetic sequence with common differences of $3$ . This number $13$ is the first number of a sequence $13, 15, 17, 19$ that is an arithmetic sequence with common difference of $2$ . In this sense this column is a combination of two arithmetic sequences.
Similarly, when $y = 7$ , the column is a combination of two arithmetic sequences.

Next we study the mathematical structure of a column generally.

Prediction 5.5   When $y = n$ , then the number at the top of the column is $\frac{3n}{2}$ $($ if $n$ is even $)$ and $\frac{3n+1}{2}$ $($ if $n$ is odd $)$ .
If we start with the number on the top of the column and move down picking up every other number, then we have an arithmetic sequence with common difference of $2$ .
On the other hand if we start with 1 that is the second number from the top of the column and move down picking up every other number, then we have $1,4,7,10,13,...$ that is an arithmetic sequences with common difference of $3$ whose length is $\left\lfloor \frac{n}{2} \right\rfloor +1$ , and after that the sequence becomes an arithmetic sequences with common difference of $2$ .