Next: Case 5
Up:Abstract and the table of contents Previous: Case 3
Case 4
Next we study the case that
and
is odd.
An example of this case is Example 5.5.
In Fig. 5.22 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 5.3
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 4 very different from Case 1, Case 2 and Case 3.

(5.54) 

(5.55) 
Clearly

(5.56) 

(5.57) 
By using Prediction 5.3 for
and

(5.58) 

(5.59) 
By (5.54) and (5.56) we have

(5.60) 
By (5.55), (5.57), (5.58) and (5.59)
The lists
,
and
do not contain
, and hence by
(5.60) we have
.
Next: Case 5
Up:Abstract and the table of contents Previous: Case 3