Why this is the more common real-world question
Most people's wake-up time is fixed by an external constraint — a work start time, school, an appointment — while bedtime is the flexible variable. This makes "what time should I go to bed, given I need to wake up at X" the more commonly useful direction in practice, compared to the forward bedtime-to-wake-up calculation covered in a companion article.
The calculation, step by step
Starting from the required wake-up time, subtract a whole number of 90-minute sleep cycles, then subtract the estimated fall-asleep buffer (commonly 15 minutes) on top — since the buffer accounts for time spent awake in bed before cycles actually begin, it needs to be subtracted in addition to the cycle time, not folded into it.
A worked example
Needing to wake at 06:30, with a 15-minute fall-asleep buffer: the four suggested bedtimes are 01:45 (working back 3 cycles), 00:15 (4 cycles), 22:45 (5 cycles), and 21:15 (6 cycles) — note that more sleep cycles means an EARLIER bedtime is needed, the opposite direction from the forward calculation, since you're now counting backward from a fixed end point instead of forward from a fixed start point.
Why the options span such a wide range
The four options here span from 21:15 to 01:45 — a 4.5-hour range — which reflects the full spread from a minimal 4.5-hour night (3 cycles) to a generous 9-hour night (6 cycles), all ending at the identical 06:30 wake-up time. Picking among them is a genuine trade-off between going to bed earlier for more sleep cycles versus staying up later and accepting fewer cycles before the fixed wake-up time arrives.
Using this to plan backward from tomorrow's schedule
This calculation is most useful the evening before a day with a known, fixed start time — rather than picking an arbitrary bedtime and hoping it works out, working backward from the actual required wake-up time gives a set of concrete bedtime options that each land on a full-cycle boundary, letting you choose based on how early you're able to get into bed that specific night rather than by guesswork.