Let's kill two birds with one stone here, and I'll use the keywords NICK, BATHROOM, and PILL so hopefully people will find it when they use the search feature. People do use the search, right? Right...? Because this information has been posted many, many times now. But anyway...
In v10, to advance the Special Chemistry event, you need to have completed ALL of the events with Kate, Julia, and Nick that are found under the Progress (spoilers) menu. Only when all of those are completed can Kate take the pill before bed and have some fun with Nick.
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I posted my questions, then problems, then complaints, many pages back about achieving the Nick-walks-in-on-Kate events. I'm no expert and was stuck for a long, long time. The truth is, it's just a brutal grind. It can easily take over 100 in-game days to achieve, and it's random. Here is how to maximize your chances:
On days Nick comes over to the apartment, before midnight,
choose Sleep for six hours so that Kate wakes up before 6am. Go into the bathroom and wait until exactly 5:59, then have Kate take a shower or bath. From information on Patreon page comments, there is a 1-in-4 chance Nick will walk in on any given morning. I don't know if that's true, but it sure seemed liked it--22 fucking times Nick has to walk in on Kate with only a 1 in 4 chance each time!!!! smfh. Since he visits two to four times per week (also random), you can go weeks without advancing progress, and that's using the 5:59 trick. If you choose to sleep overnight, there could be as high as a 1-in-9 chance Kate will wake up in time to get into the shower/bath before Nick (she wakes up between 6:00 and 6:09, Nick almost always gets into the bathroom before 6:03.
So anyway, how do you get the special chemistry 3event? The answer, my friend, is grind. Grind, grind, then grind some more.