Entei hunting math

UPDATE: The reload strat sucks because the code doesn't allow Entei to go back to the route that I was on 2 transitions ago (weird).

Entei always goes to Route 37 after being released from Burned Tower. However, in order for the player to reach Route 37, they must perform two area transitions: 1) Escape Rope out of Burned Tower, and 2) enter Route 37 from Ecruteak City. Each time an area transition occurs, Entei will change routes. To determine which route, the game does the following:

1) Determine how many routes are adjacent to Entei's current route. For Route 37, there are three (Route 36, Route 38, Route 42).
2) At a random chance of 1/8n, where n is the number of adjacent routes, have the dog jump to a random route other than its current route.
3) If the dog did not jump to a random route, have it jump to a random adjacent route.

Using this information, we can determine that Entei has a 26.1% chance of being on Route 37 after two area transitions. (I'll call this the "2-transition rate.") However, when the player actually searches for Entei, that rate is constantly changing. This is because each step on which the player DOES not encounter Entei gives us information through probability. The more you search, the more likely Entei was never on the route to begin with.

In mathematical terms, the odds of Entei appearing on any given step are 1250/65536. So the odds of Entei being on Route 37 after n tiles are (0.261)*((64286/65536)^n). That means after 50 steps, it's only 10% likely that Entei's there.

Now comes the tricky part. If the player were to perform two ADDITIONAL area transitions (enter Ecruteak, enter Route 37) without gaining any search information, Entei's chance of appearing on Route 37 would be 15.4%. (I'll call this the "4-transition rate.") That's higher than the 2-transition rate becomes after 28 steps. However, the 4-transition rate decreases as the 2-transition rate decreases. This is because a decent chunk of the 4-transition rate is Entei being on Route 37, then leaving for Route 36/38/42 as you enter Ecruteak, then returning to the adjacent Route 37 along with you. If we assumed Entei was on another route, the 4-transition rate would only be 11.7%.

The 4-transition rate exceeds the 2-transition rate after about 35 steps. However, reloading the route takes time and precious Repel steps, so it's probably optimal to stay for 60 steps or so. The math at this stage becomes much more complex, and I'd be interested to see people tackle it.