Thanks for sharing, I had almost the exact same experience it seems like. I interviewed at one of the Big 5 (from your explanation it sounds like it may have been the same one actually...), where I had 4 hour-long interviews back to back in one day.
I tried to prepare and review my algorithms etc. before going (it was a cross-country trip for me, so I had time...). I felt like I did pretty well on the first 3 interviews, and I'm sure I did a crappy job on the last one (which was purely algorithmic in nature).
And it was mostly because of the same things you described...I just did a bad job taking my time, explaining my approach, asking questions for clarification, etc. Literally as soon as I got back to my hotel room after that interview I banged out the solution in no time, but I just kind of choked in the room during the interview. I think it was nerves and being a little exhausted from the multiple interviews and probably just not being fully prepared with enough algo stuff. I don't know how old you are, but I've been out of college for almost 15 years, so it had been awhile since I had focused on that stuff.
Anyway, all this is to say that I have some idea of how you feel, and while it was pretty disappointing at the time, life goes on :) You sound like you learned a lot from the experience, so at least you got something out of it ;) Some of the feedback I got was to brush up on some of that stuff, and re-apply after some time. I have no idea if that would actually work (I haven't tried...) but it's probably worth a shot.
Thank you for sharing your experience! I'm not alone :)
As you said it's been a learning experience so I'm fine :) Thank you!
And in your case, of course it's worth a shot, re-apply!
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