Right, so I don't know how many people already know this but for those who don't, I thought I'd share here these tips I've learned and tested for myself.
So first tip was how to get a specific Kapp'n tour island to pop up. I found this video while googling for tips
Basically the video creator says do a soft reset before the autosave kicks in if Kapp'n doesn't bring you to a desirable island. The catch is if you're looking for a rare seasonal material island (snowflakes, summer shells, star fragments etc.) the first thing to do is go get your fortune told by Katrina and get the improved luck with belongings (LwB) fortune (please note I don't know if you can soft reset Katrina's fortunes in case she gives you something different; I haven't tried and I don't know if anyone else has).
If you get the LwB, then the islands are locked to the moss and vines island and the seasonal items islands.
I got the LwB and cycled through multiple moss and vine, summer shell, and mushroom islands + one cherry blossom while trying to get star fragments before giving up and settling on summer shells. No clue why I didn't get snow or star fragment islands as I've encountered both before, and I don't know if any of them were spring bamboo islands as stopping to check would've given autosave time to kick in and used up my trip for the day. The next two days when my fortune was luck with money, I got pretty much just regular islands and a few vine ones before giving up.
There's a definite correlation between the LwB fortune and the seasonal item mystery islands. I don't know if LwB locks the islands to just that group or if the game's code just makes it so that a LwB always coincides with a guaranteed seasonal item island. Testing would require going to Kapp'n before Katrina and seeing if I get a regular island, soft resetting, then going to Katrina and hope for the LwB before going back to Kapp'n.
If anyone wants to try testing this yourself with the variable factors I haven't explored yet, I welcome hearing your results.
For the Katrina fortunes.
This was my source: https://www.ign.com/wikis/animal-crossing-new-horizons/Katrina_Fortune_Meanings
For LwB improves, the effects are increased likelihood of villagers selling and giving you furniture, seashells on the beach more frequently, and tools not breaking. The easiest measurable effect would be tools not breaking. I haven't deliberately tested this myself, however the last time I got this fortune I caught quite a few bugs on the summer shell island I was on after having used that particular net for a few days prior, so I suspect it's true. The best test I can think of for this would be to construct a new flimsy net- which normally has 10 uses - and after getting the fortune, catching a bunch of bugs, and seeing if it outlasts 10 catches.
For LwM (money) improves, the effects are that you get more money from shaking trees, money rocks, balloons, and the glowing spot. This only works if you get the fortune first. The first time I tested it, I definitely got more money from the glowing spot (5k bells) and the money rock, but didn't pay attention to how much I got from trees and barely shot down any balloons. The second time, I dug up the glowing spot before I got the fortune and got the normal 1k, but found the money rock after the fortune and got more bells than usual, signified by five bags instead of four. So safe to say the fortune causes the effects there.
I think to test the balloons effect you would have to shoot down a set number of balloons, let's say 20 as that's supposedly the number of uses you get out of a slingshot, count how many bells you get after getting the LwM fortune, then get bells from at minimum 3 days of shooting 20 balloons each without a fortune and average them out, and see what the difference is. Note: according to the source, if you get a bad luck with money fortune you won't get any money at all, so stick with no fortune reading for your control group if you want to test this.
Beyond that, I haven't done any testing with the luck with friendships or health fortunes. I suspect that in any instance where you get a fortune where your luck diminishes, getting purified at most cancels out the negative effects and makes it so your luck goes back to neutral, i.e. pre-fortune levels. So if you get bad luck with money and get purified, you get the same amount of bells you would as if you never got your fortune told at all.
And that's all I've got. If anyone has anything to add or can corroborate these, please let us know what you've found out.
So first tip was how to get a specific Kapp'n tour island to pop up. I found this video while googling for tips
Basically the video creator says do a soft reset before the autosave kicks in if Kapp'n doesn't bring you to a desirable island. The catch is if you're looking for a rare seasonal material island (snowflakes, summer shells, star fragments etc.) the first thing to do is go get your fortune told by Katrina and get the improved luck with belongings (LwB) fortune (please note I don't know if you can soft reset Katrina's fortunes in case she gives you something different; I haven't tried and I don't know if anyone else has).
If you get the LwB, then the islands are locked to the moss and vines island and the seasonal items islands.
I got the LwB and cycled through multiple moss and vine, summer shell, and mushroom islands + one cherry blossom while trying to get star fragments before giving up and settling on summer shells. No clue why I didn't get snow or star fragment islands as I've encountered both before, and I don't know if any of them were spring bamboo islands as stopping to check would've given autosave time to kick in and used up my trip for the day. The next two days when my fortune was luck with money, I got pretty much just regular islands and a few vine ones before giving up.
There's a definite correlation between the LwB fortune and the seasonal item mystery islands. I don't know if LwB locks the islands to just that group or if the game's code just makes it so that a LwB always coincides with a guaranteed seasonal item island. Testing would require going to Kapp'n before Katrina and seeing if I get a regular island, soft resetting, then going to Katrina and hope for the LwB before going back to Kapp'n.
If anyone wants to try testing this yourself with the variable factors I haven't explored yet, I welcome hearing your results.
For the Katrina fortunes.
This was my source: https://www.ign.com/wikis/animal-crossing-new-horizons/Katrina_Fortune_Meanings
For LwB improves, the effects are increased likelihood of villagers selling and giving you furniture, seashells on the beach more frequently, and tools not breaking. The easiest measurable effect would be tools not breaking. I haven't deliberately tested this myself, however the last time I got this fortune I caught quite a few bugs on the summer shell island I was on after having used that particular net for a few days prior, so I suspect it's true. The best test I can think of for this would be to construct a new flimsy net- which normally has 10 uses - and after getting the fortune, catching a bunch of bugs, and seeing if it outlasts 10 catches.
For LwM (money) improves, the effects are that you get more money from shaking trees, money rocks, balloons, and the glowing spot. This only works if you get the fortune first. The first time I tested it, I definitely got more money from the glowing spot (5k bells) and the money rock, but didn't pay attention to how much I got from trees and barely shot down any balloons. The second time, I dug up the glowing spot before I got the fortune and got the normal 1k, but found the money rock after the fortune and got more bells than usual, signified by five bags instead of four. So safe to say the fortune causes the effects there.
I think to test the balloons effect you would have to shoot down a set number of balloons, let's say 20 as that's supposedly the number of uses you get out of a slingshot, count how many bells you get after getting the LwM fortune, then get bells from at minimum 3 days of shooting 20 balloons each without a fortune and average them out, and see what the difference is. Note: according to the source, if you get a bad luck with money fortune you won't get any money at all, so stick with no fortune reading for your control group if you want to test this.
Beyond that, I haven't done any testing with the luck with friendships or health fortunes. I suspect that in any instance where you get a fortune where your luck diminishes, getting purified at most cancels out the negative effects and makes it so your luck goes back to neutral, i.e. pre-fortune levels. So if you get bad luck with money and get purified, you get the same amount of bells you would as if you never got your fortune told at all.
And that's all I've got. If anyone has anything to add or can corroborate these, please let us know what you've found out.