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octal game 0.177 |
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Octal Game 0.177
The nim sequence has a period of length 20, but doesn't settle down completely until after about 500 values. Normal Play Nim Sequence
000+ 0 1 1 2 2 3 1 1 4 4 3 2 2 1 1 4 2 2 6 4
020+ 4 1 1 2 2 7 1 1 4 4 3 2 2 1 1 4 8 2 7 4
040+ 4 1 1 2 2 7 8 1 4 4 7 2 2 1 1 4 4 2 7 7
060+ 4 1 1 2 2 7 7 1 1 4 7 2 2 1 1 4 4 6 2 7
080+ 4 1 1 2 2 8 8 5 1 4 7 2 2 1 1 8 8 6 7 7
100+ 4 1 1 2 2 3 1 1 4 4 7 2 2 1 1 8 2 2 7 4
120+ 4 1 1 2 2 8 1 1 4 4 7 2 2 1 1 4 8 2 7 4
140+ 4 1 1 2 2 7 8 1 4 4 7 2 2 1 1 4 8 2 7 4
160+ 4 1 1 2 2 7 8 1 1 4 7 2 2 1 1 4 4 2 2 7
180+ 4 1 1 2 2 8 8 1 1 4 7 2 2 1 1 8 8 6 7 7
200+ 4 1 1 2 2 8 1 1 4 4 7 2 2 1 1 8 2 2 7 4
220+ 4 1 1 2 2 8 1 1 4 4 7 2 2 1 1 4 2 2 7 4
240+ 4 1 1 2 2 7 1 1 4 4 7 2 2 1 1 4 8 2 7 4
260+ 4 1 1 2 2 7 8 1 4 4 7 2 2 1 1 4 8 2 7 4
280+ 4 1 1 2 2 8 8 1 1 4 7 2 2 1 1 8 8 2 7 4
300+ 4 1 1 2 2 8 1 1 4 4 7 2 2 1 1 8 2 2 7 4
320+ 4 1 1 2 2 8 1 1 4 4 7 2 2 1 1 8 2 2 7 4
340+ 4 1 1 2 2 8 1 1 4 4 7 2 2 1 1 4 2 2 7 4
360+ 4 1 1 2 2 7 1 1 4 4 7 2 2 1 1 4 8 2 7 4
380+ 4 1 1 2 2 8 8 1 4 4 7 2 2 1 1 8 8 2 7 4
400+ 4 1 1 2 2 8 1 1 4 4 7 2 2 1 1 8 2 2 7 4
420+ 4 1 1 2 2 8 1 1 4 4 7 2 2 1 1 8 2 2 7 4
440+ 4 1 1 2 2 8 1 1 4 4 7 2 2 1 1 8 2 2 7 4
460+ 4 1 1 2 2 8 1 1 4 4 7 2 2 1 1 4 2 2 7 4
480+ 4 1 1 2 2 8 1 1 4 4 7 2 2 1 1 8 8 2 7 4
500+ 4 1 1 2 2 8 1 1 4 4 7 2 2 1 1 8 2 2 7 4
520+ 4 1 1 2 2 8 1 1 4 4 7 2 2 1 1 8 2 2 7 4
The values in the final (two) rows repeat themselves indefinitely.
Misère Play I haven't been able to solve this one completely. I've been using it as a test case for investigating whether ONAG- and WW-style genus calculations can be usefully married to the Sibert-Conway decomposition attack. Here are some facts about this game from various sources, collected together. The genus sequence to heap size 16, as printed in WW, Table 5, Chapter 13, pg 425 (see the row for the game in the two digit "first cousin" .45), looks like this: 11223 114146443 222211104057 22 Note added 4 March 2003: Here's the output of a Mathematica program that computed single-heap genera to heap size 30. The output is shifted by one term because it was calculated using the .45 code rather than .177:
heap genus
1 {0,{1,2,0}}
2 {1,{0,3,1}}
3 {1,{0,3,1}}
4 {2,{2,0}}
5 {2,{2,0}}
6 {3,{3,1}}
7 {1,{0,3,1}}
8 {1,{0,3,1}}
9 {4,{1,4,6}}
10 {4,{4,6}}
11 {3,{3,1}}
12 {2,{2,0}}
13 {2,{2,0}}
14 {1,{1,3}}
15 {1,{0,3,1}}
16 {4,{0,5,7}}
17 {2,{2,0}}
18 {2,{2,0}}
19 {6,{4,6}}
20 {4,{4,6}}
21 {4,{6,5,7}}
22 {1,{1,3}}
23 {1,{8,3,1}}
24 {2,{2,0}}
25 {2,{2,0}}
26 {7,{4,6}}
27 {1,{1,8,10}}
28 {1,{1,3}}
29 {4,{8,5,7}}
30 {4,{4,6}}}
The games for the first two non-nim-heap sizes (8 and 9)
are also indentified in WW. They are the games 22321 and 223210, respectively. Allemang gives a partial analysis with the following equivalents to heap size 13 :1 :1 A B :3 :2 :2 :5 and he says that "special games [ie, A and B, or heap sizes 8 and 9] can be reduced according to the following rules": B+B+B+B = B+B Now, in Allemang's notation, the games :n are defined like this: :2 = nim heap of size 2 :3 = :2 + :1 :4 = :2 + :2 :5 = :2 + :2 + :1 :6 = :2 + :2 + :2 :7 = :2 + :2 + :2 + :1 etc The game :3 is also equal to the nim heap of size 3 in misère play. However, this is a nonobvious fact that requires proof (for example, via applications of the "Misère Mex Rule" and "Misère Nim Rule" described in WW, pg 398). The game :4 is not the nim heap of size four. The WW and Allemang computations agree through the first ten heap sizes. Here's a Sibert-Conway decomposition to heap size 10 that I computed:
PN Positions
{} E{{1,2,6,7}} D{}
{8,8} E{{1,2,6,7}} D{}
{9,8} E{{5,10},{3,4},{1,2,6,7}} D{}
{9,8} E{} D{{5,10},{3,4},{1,2,6,7}}
NP Positions
{} E{} D{{1,2,6,7}}
{8} E{} D{{1,2,6,7}}
{8,8} E{} D{{1,2,6,7}}
{9,8} E{{5,10},{1,2,6,7}} D{{3,4}}
{9,8} E{{3,4}} D{{5,10},{1,2,6,7}}
Now, the heapsizes 1 , 2, 6, and 7 are all
equal to the nim heap of size 1 (this is already recorded
in the genus sequence calculations, above).
The heap sizes 5 and 10 are both nim heaps of size 3. The heap sizes 3 and 4 are both nim heaps of size 2. |