I made a post earlier today talking about how the crown sizes are calculated and after spending some time playing around with Cheat Engine and the save editor, I was able to figure out what kind of monster sizes you can get from each quest.
Investigations have a hidden size modifier, but that only seems to increase the chances of getting larger/smaller monsters within its own size distribution table by an unknown amount. The size table is chosen solely based on the investigation rewards (number of purple, gold, and silver) and has nothing to do with the difficulty or any other investigation settings.
All possible quest types can be broken into a few groups when it comes to the chances of getting crowns:
Assignment quests, optionals, LR investigations, and HR/tempered investigations
None of these can give you crowns. Assignments/special assignments will always give you the default monster size, except for the Lunastra special assignment, which has a large crown Teostra, and the Kula-Ya-Ku special assignment, which gives you a ridiculously large gold crown. Optionals have random monster sizes and HR optionals will give you up to the largest silver crown and the smallest possible size before a tiny crown but will never give you a gold crown of any kind.
HR investigations without a gold and a silver box or tempered investigations with only 2 purple boxes cannot give you any gold crowns. An investigation with 2 gold and 2 bronze boxes might seem better than one with 1 gold, 1 silver, and 1 bronze, but it will still not give you crowns (possibly because these usually come with 8 attempts).
HR investigations with at least 1 gold and 1 silver + tempered investigations with 3 purple boxes
These have a decent chance of giving you a crown. You can see the distribution table for these investigations in the first column of this spreadsheet (gold crown sizes are highlighted in yellow). If I'm reading the size table right, it's 6% chance for a tiny crown and 3% for a large crown. The table for monsters that get large crowns at 120% of their default size instead of 123% looks slightly different, but the chances of getting crowns are identical.
Investigations with 4 or 5 gold or purple boxes
Yes, you can make an investigation with 4-5 gold boxes, but I have no idea how unlikely it is to find one of these without cheating. You can see the full distribution table in the second column of the spreadsheet. These have a slightly better chance of giving you gold crowns than the previous group (6% for each instead of 6% for tiny and 3% for large).
Crown event quests
The chance of getting a crown from these quests is so high that you probably shouldn't even bother spamming investigations when hunting for crowns unless it's an elder dragon, Black Diablos, or Deviljho (this one isn't needed for the achievement, though). The size rates for these quests are in the third column of the spreadsheet.
IMPORTANT NOTE: if you have a quest that meets one of the requirements above, all monsters (including those that you aren't required to kill) will use the same size table. This means that if you have a few awful T1/T2 investigations with three monsters and 4-5 boxes that you think you'll never do, you could use those to look for crowns instead of just deleting them.
TL;DR: some investigations literally can't give you crowns. Only do investigations with at least 1 gold and 1 silver box, but tempered investigations with 4-5 boxes are better
© Post "Investigations and possible monster sizes/crowns" for game Monster Hunter World.
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