Chance for Encounters up to Falkner in Pokémon Crystal Manipless

Encounters	Chance		Sum

0		2,92503E-06	2,92503E-06
1		5,09471E-05	5,38721E-05
2		0,000421687	0,000475559
3		0,00220738	0,002682939
4		0,008204991	0,010887929
5		0,023052117	0,033940046
6		0,050878491	0,084818538
7		0,0905211	0,175339638
8		0,132194162	0,3075338
9		0,160542197	0,468075997
10		0,163692036	0,631768033
11		0,14111164	0,772879673
12		0,103367322	0,876246995
13		0,064566421	0,940813416
14		0,034466923	0,975280339
15		0,015742603	0,991022941
16		0,006153621	0,997176562
17		0,002057331	0,999233893
18		0,000587489	0,999821382
19		0,000142988	0,99996437
20		2,95783E-05	0,999993948
21		5,1817E-06	0,99999913
22		7,65467E-07	0,999999895
23		9,48695E-08	0,99999999
24		9,80624E-09	1
25		8,39644E-10	1
26		5,90902E-11	1
27		3,38756E-12	1
28		1,56586E-13	1
29		5,76683E-15	1
30		1,66842E-16	1
31		3,72707E-18	1
32		6,2893E-20	1
33		7,78429E-22	1
34		6,77375E-24	1
35		3,87732E-26	1
36		1,29467E-28	1
37		1,88972	E-31	1

Average		9,731988215



Some stuff I used to calculate These numbers
I assumed that the Chance of encountering a Pokemon is 25/256 per step and after every Encounter you get 4 steps of Encounter immunity.
I divided the Path into 11 Disconnected* Patches of grass. *Disconnected means, more than 4 steps apart (assuming movement that minimizes encouter tiles)

for These Patches I got:


		Patch 1		Patch 2		Patch 3		Patch 7		Patch 4		Patch 5		Patch 8		Patch 6		Patch 9		Patch 10	Patch 11
Tiles		11		7		6		3		7		6		3		9*		9		9		8*/9*
Max Encounters	3		2		2		1		2		2		1		3		2		2		3
Encounterless	0,32291793	0,487084267	0,539799015	0,734710157	0,487084267	0,539799015	0,734710157	0,396595819	0,396595819	0,396595819	0,439517444
1 Encounter	0,533169569	0,486168149	0,450664242	0,265289843	0,486168149	0,450664242	0,265289843	0,454900891	0,525334486	0,525334486	0,491405538
2 Encounter	0,142981178	0,026747584	0,009536743	0,026747584	0,009536743	0,137153808	0,078069695	0,078069695	0,068145695
3 Encounter	0,000931323							0,011349482			0,000931323
Average		0,821925892	0,539663317	0,469737728	0,265289843	0,539663317	0,469737728	0,265289843	0,763256953	0,681473876	0,681473876	0,630490896

Patch 1, Patch 2 and Patch 3 are the grass Patches of R29, 4,5 and 6 are the Patches you have to pass on R30 to get to Mr Pokemon.
7,8 and 9 are the remaining Patches on R30 Right before and after Fighting Mickey.
10 and 11 are the final Patches before violet.
Average is additiv. For example: The Average amount of Encounters from the start until Mr Pokemon is 0,821925892+0,539663317+0,469737728+0,539663317+0,469737728+0,763256953=3,603984935 (Sum of averages from Patch 1 to 6).

Mostly useless Information, but might be interesting for some :)

R32 stuff ^-^

4 different ways to go through the route:
Bottom repel, Leftside
Bottom Repel, Rightside
Top Repel, Leftside
Top Repel, Rightside

fastest possible route is getting repel from the top and taking the left path
lowest encounter chance is bottom repel and rightside (Spinner pass!!!)

Likely optimal to change approach based on pace/Encounters.
Getting the super potion for safety in exchange for extra spinner pass can be a factor as well.

Patch 1,2b,3L,4		
Encounters	Chance			Sum
0		0,030384248		0,030384248
1		0,144399591		0,17478384
2		0,282393044		0,457176883
3		0,293897647		0,751074531
4		0,175666144		0,926740675
5		0,060655807		0,987396482
6		0,011521441		0,998917923
7		0,001050162		0,999968085
8		3,16183E-05		0,999999703
9		2,96701E-07		1
		
Average	2,673557631	


Patch 1,2b,3R		
Encounters	Chance			Sum
0		0,094092788		0,094092788
1		0,291814642		0,38590743
2		0,34965093		0,73555836
3		0,20170208		0,937260439
4		0,056249815		0,993510254
5		0,006375781		0,999886035
6		0,000113965		1
		
Average	1,853784694	


Patch 1, 2t,3L,4		
Encounters	Chance			sum
0		0,024739592		0,024739592
1		0,125650879		0,15039047
2		0,263954148		0,414344618
3		0,297130874		0,711475492
4		0,194393824		0,905869316
5		0,075201536		0,981070852
6		0,016804958		0,99787581
7		0,002014413		0,999890222
8		0,000108221		0,999998443
9		1,5569E-06		1
		
Average	2,814345185	


Patch 1,2t,3R		
Encounters	Chance			sum
0		0,06237986		0,06237986
1		0,23419457		0,29657443
2		0,34214159		0,638716019
3		0,248123944		0,886839964
4		0,094097126		0,98093709
5		0,017752196		0,998689285
6		0,001310715		1
		
Average	2,135863352	


Overall average Encounters Start to Union cave is between 11,58577291 and 12,5463334 depending on R32 strat.