Let me provide a couple more details:

The engine was hot - had just been flown for 1 hour, and had only been
off for maybe 15 minutes. The starter was malfunctioning in some way
but the battery was good. When I tried to start it with the starter,
it made a noise as though the starter motor was spinning but the
engine didn't crank.

A couple of hours later, we tried again: I turned the key, and it
started instantly. I am aware of potential problems with hot starts
(vaporized fuel in the lines) but this was clearly a starter motor
problem.

When trying to get it going the first time, we tried a few different
variations on hand-propping it, including trying to after priming it
using the electric fuel pump, and trying without this step. Every
time, the hand-propper pulled through 2-3 blades before the 'contact'
start attempt. We tried with the mixture on idle cut-off (which is the
technique for normal starts - as soon as it kicks in, you go full
rich). We tried with full rich. We tried with 1/3 throttle. We tried
after pumping the throttle.

I didn't provide all these details before because I wanted to focus
the discussion on how to hand-prop a 172SP (or similar fuel-injected
spam can) rather than what might have been wrong with it in the first
place. I should have known that pilots would question the question
rather than answer it! Not saying that's a bad thing ;-)

Any further insights appreciated, especially with regard to the hand-
propping, or anything else I could have done differently to get it to
start without waiting 2 hours.

Thanks
Tom







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Still, Peter, would you not agree a dead battery would prevent a hand
propped injected engine start? You'd not be able to charge the fuel
system with the electric fuel pump, nor would the injection computer
have power to run.

If the OP was trying to hand prop an engine that just had a failed
starter motor my questions are rendered moot: he didn't tell us the
specifics, but maybe he will.