WARNING: Before reading this you need to fully understand 13mjvr's patterns theory, which can be read here
Introduction:
Hi, everybody, I hope you all have a nice 2009. So, I'd first like to say what is this text you're reading: In the last days, I've been thinking and researching about chaining mechanics and in this time, I ended up proving why people say that the patches in zone 4 are safer to go into while chaining, if there aren't any patterns. When you finish reading this introduction, you'll see how I proved it and, obviously, why the 4-zone patches are safer. So, here it is:
The theory:
What the colors mean on the number charts:
Black = grass patches that didn't shake.
Red = grass patches that shook.
Orange = grass patches that would be associated with patterns if they shook.
Pink = grass patches that wouldn't be associated with patterns if they shook.
Situations:
- 1-zone patches:
4 4 4 4 4 4 4 4 4
4 3 3 3 3 3 3 3 4
4 3 2 2 2 2 2 3 4
4 3 2 1 1 1 2 3 4
4 3 2 1 0 1 2 3 4
4 3 2 1 1 1 2 3 4
4 3 2 2 2 2 2 3 4
4 3 3 3 3 3 3 3 4
4 4 4 4 4 4 4 4 4
Comments: The 1-zone patch that shook in this case isn't at any of the corners of zone 1. When a patch like this, that isn't at any corner of zone 1, shakes, out of all the 16 patches on zone 2, 6 of them would be associated with patterns if they shook, and 10 of them wouldn't.
4 4 4 4 4 4 4 4 4
4 3 3 3 3 3 3 3 4
4 3 2 2 2 2 2 3 4
4 3 2 1 1 1 2 3 4
4 3 2 1 0 1 2 3 4
4 3 2 1 1 1 2 3 4
4 3 2 2 2 2 2 3 4
4 3 3 3 3 3 3 3 4
4 4 4 4 4 4 4 4 4
Comments: The 1-zone patch that shook in this case is a patch that is at one of the corners of zone 1. When a patch like this, which is at one of the corners of zone 1, shakes, out of all the 16 patches on zone 2, 7 of them would be associated with patterns if they shook, and 9 of them wouldn't. Therefore, in the first situation, the chances of patches that aren't associated with patterns being generated are bigger than the chances on this situation.
4 4 4 4 4 4 4 4 4
4 3 3 3 3 3 3 3 4
4 3 2 2 2 2 2 3 4
4 3 2 1 1 1 2 3 4
4 3 2 1 0 1 2 3 4
4 3 2 1 1 1 2 3 4
4 3 2 2 2 2 2 3 4
4 3 3 3 3 3 3 3 4
4 4 4 4 4 4 4 4 4
Comments: This situation has been simulated just to prove that the idea on the first situation about patches that aren't at the corner of the zone is correct.
4 4 4 4 4 4 4 4 4
4 3 3 3 3 3 3 3 4
4 3 2 2 2 2 2 3 4
4 3 2 1 1 1 2 3 4
4 3 2 1 0 1 2 3 4
4 3 2 1 1 1 2 3 4
4 3 2 2 2 2 2 3 4
4 3 3 3 3 3 3 3 4
4 4 4 4 4 4 4 4 4
Comments: This situation has been simulated just to prove that the idea on the first situation about patches that are at the corner of the zone is correct.
- 2-zone patches:
4 4 4 4 4 4 4 4 4
4 3 3 3 3 3 3 3 4
4 3 2 2 2 2 2 3 4
4 3 2 1 1 1 2 3 4
4 3 2 1 0 1 2 3 4
4 3 2 1 1 1 2 3 4
4 3 2 2 2 2 2 3 4
4 3 3 3 3 3 3 3 4
4 4 4 4 4 4 4 4 4
Comments: The 2-zone patch that shook in this case is a patch that is at one of the corners of zone 2. When a patch like this, which is at one of the corners of zone 2, shakes, out of all the 24 patches on zone 3, 7 of them would be associated with patterns if they shook, and 17 of them wouldn't.
4 4 4 4 4 4 4 4 4
4 3 3 3 3 3 3 3 4
4 3 2 2 2 2 2 3 4
4 3 2 1 1 1 2 3 4
4 3 2 1 0 1 2 3 4
4 3 2 1 1 1 2 3 4
4 3 2 2 2 2 2 3 4
4 3 3 3 3 3 3 3 4
4 4 4 4 4 4 4 4 4
Comments: The 2-zone patch that shook in this case is a patch that isn't at one of the corners of zone 2. When a patch like this, that isn't at one of the corners of zone 2, shakes, out of all the 24 patches on zone 3, 6 of them would be associated with patterns if they shook, and 18 of them wouldn't. Therefore, on this situation, the chances of patches that aren't associated with patterns being generated are bigger than the chances on the previous situation.
- 3-zone patches:
4 4 4 4 4 4 4 4 4
4 3 3 3 3 3 3 3 4
4 3 2 2 2 2 2 3 4
4 3 2 1 1 1 2 3 4
4 3 2 1 0 1 2 3 4
4 3 2 1 1 1 2 3 4
4 3 2 2 2 2 2 3 4
4 3 3 3 3 3 3 3 4
4 4 4 4 4 4 4 4 4
Comments: The 3-zone patch that shook on this case isn't at any corner of zone 3. When a patch like this, that isn't at any corner of zone 3, shakes, out of all the 32 patches on zone 4, 6 of them would be associated with patterns if they shook, and 26 wouldn't.
4 4 4 4 4 4 4 4 4
4 3 3 3 3 3 3 3 4
4 3 2 2 2 2 2 3 4
4 3 2 1 1 1 2 3 4
4 3 2 1 0 1 2 3 4
4 3 2 1 1 1 2 3 4
4 3 2 2 2 2 2 3 4
4 3 3 3 3 3 3 3 4
4 4 4 4 4 4 4 4 4
Comments: The 3-zone patch that shook on this case is at one of the corners of zone 3. When a patch like this, which is at one of the corners of zone 3, shakes, out of all the 32 patches on zone 4, 7 of them would be associated with patterns if they shook, and 25 wouldn't.
Conclusion:
After studying these hypothetical situations, I concluded that I had proved why people say (and they're right to do so) that going to 4-zone patches is the best option if you want your chain to continue when there aren't any patterns (the four patches aren't connected in any means, neither lined or twin nor diagonally). Here is what I concluded that proves what people say: The 4-zone patches are the best ones to go into if there isn't any pattern and you want to continue your chain because it doesn't matter what 3-zone patch shook, the chances of the 4-zone patch generated being a patch unassociated with patterns are bigger than the chances of the 2 or 3-zone patches being unassociated with patterns when any 1-zone or 2-zone patch, respectively, shakes.
When a 1-zone patch that isn't at one of the corners of zone 1 shakes, the chances of a patch that shakes on zone 2 being unassociated with patterns are 10 in 16 (62.5%) and, if the patch is at one of the corners of zone 1, 9 in 16 (56.25%). When a 2-zone patch that isn't at one of the corners of zone 2 shakes, the chances of a patch that shakes on zone 3 being unassociated with patterns are 18 in 24 (75%) and, if the patch is at one of the corners of zone 2, 17 in 24 (70.833…%). However, when a 3-zone patch that isn't at one of the corners of zone 3 shakes, the chances of a patch that shakes on zone 4 being unassociated with patterns are 26 in 32 (81.25%) and, if the patch is at one of the corners of zone 3, 25 in 32 (78.125%). So, we can see clearly that the chances of the 4-zone patches being unassociated with patterns are bigger than the other zones' ones and that's why people allways say that 4-zone patches are safer. Also, with these conclusions, I have proved that the patches at the corners of the zones have a higher chance of forming patterns than the ones that aren't at zone's corners. In case you were confused with the percentages, here are them in a more simple way:
Chances of patches unassociated with patterns being generated:
Zone----Corner----Not-corner
2--------56.25%-------62.5%
3--------70.833…%----75%
4--------78.125%------81.25%
If you see anything you think is a mistake, leave a message telling what is it on my profile.
By: Mega Weavile
Special thanks: to ninjascyther for reading this before I posted and giving his opinion and suggestions.