### Case 5

Next we study the case that and is odd.
An example of this case is Example 5.6.

In Fig. 5.23 we suppose that Predictions 5.1, 5.2, 5.3,5.4 and 5.5 are valid for and for . We also suppose that these predictions are valid for and . Under these assumptions we prove that .

Remark 5.4   In this case we assume similar conditions that are written in Remark 5.3. Note that some Grundy numbers with 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 in the table of Grundy numbers.
Here we assume that , which is the last number of the sequence that is an arithmetic sequence with common difference of and its length is .
Therefore the position of is in the midst of the the sequence , and this condition make Case 5 very different from Case 1, Case 2 and Case 3.

 (5.61)

 (5.62)

Clearly

 (5.63)

 (5.64)

By using Prediction 5.3 for and

 (5.65)

 (5.66)

By (5.61) and (5.63) we have

 (5.67)

By (5.62) and the fact that we have

 (5.68)

By (5.65) and (5.66) we have

 (5.69)

By using predictions for and
we know that the sequence of (5.69) starts as an arithmetic sequence with common difference of whose length is , and hence the last term of this arithmetic sequence is . The next term in (5.69) is , and after that we have an arithmetic sequence with common difference of up until it reaches . Therefore by (5.68), (5.67) and (5.69) we have . It is clear that there is no in , and hence we have .

Lemma 5.2   Predictions 5.1, 5.2, 5.3,5.4 and 5.5 are valid.

Proof   The outline of the proof is almost clear from Case 1, Case 2, Case 3, Case 4 and Case 5.

Theorem 5.1   The list of L-states of the game with inequality is the union of the following lists , , , , and .
. ,     and
.     and
. ,
. ,     and
. ,     and
. ,     and  .

Proof   This is direct from Prediction 5.5.