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In Ultra Sun/Ultra Moon, is there a way to increase the chance of finding an Ultra Wormhole Legendary? Because I almost caught every single one (missing: Heatran, Giratina, and Rayquaza), and it's been very hard for me to even find a Legendary.

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Go in the more colorful hoops, with more rings.
I know about the colorful hoops with more rings, but is there actually a way to increase it without the special Ultra Wormholes?
Other than advancing further into Ultra Space to get rarer wormholes, then no, there isn't another way to increase the odds.

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The deeper into Ultra Space you ride the rarer the wormholes. They come in four rarities, pictured below:

The rate of encountering legendaries in a wormhole, in percentage, is written in the upper portion of this image (note that distance traveled is measured in lightyears, basically the deeper into Ultra Space you go):

So rarer wormholes yield a higher result of finding legendaries. You can find rarer wormholes by going deeper into Ultra Space, the probabilities of encountering each type of wormhole in how many "lightyears" you've traveled through goes as follows:

Sorry for all the images but hopefully this helped. I thought you may also want to know what color wormholes to find the legendaries you're after:
- Heatran is found in yellow wormholes in US only.
- Giratina is found in yellow wormholes if Dialga (exclusive to US in green wormholes) and Palkia (exclusive to UM in yellow wormholes) are in your party.
- Rayquaza is found in red wormholes if Groudon (exclusive to US in yellow wormholes) and Kyogre (exclusive to UM in blue wormholes) are in your party.

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Sorry to burst your bubble, but he wants to boost chances WITHOUT the wormhoe colours.
I forgot that I don't have Mesprit could you add what Wormhole you can find Mesprit? I have Dialga, Palkia, and Groudon, but I'm still waiting for the Legendary event with Kyogre (since I have Ultra Sun) so I can hunt for Rayquaza. If it measures in lightyears, how far would you have to play the Ultra Wormhole minigame thingy to travel 2,500 lightyears, 5 minutes? 10 minutes? Maybe even 30 minutes?
Mesprit can be found in blue wormholes. Unfortunately there's no set time on how far you need to travel to reach a certain amount of lightyears, and even if there was it would vary greatly seeing as you can travel through Ultra Space at different speeds, if you gather more energy orbs you can go further faster and the light-year counter will go up the farther you travel. I know that it sure as Hell doesn't take 30 minutes, though.
Ok. Thanks for your help. I'll start hunting tomorrow at 12:30 p.m.