Let's suppose I have a bike with a flip-flop rear wheel--fixed on one side, free on the other--with a single chainring up front. The fixed sprocket and freewheel are both identically sized; that is, they have the same number of teeth (and neither one is missing teeth ).

On a windless day, I climb a hill, from base to summit, with the bike in freewheel mode. After walking the bike back down to the base of the hill--and taking enough time to recover completely--I flip the wheel over. I then climb the same hill, from base to summit, in fixed-gear mode. A glance at the bike computer shows both rides are done with the same average speed.

My question, then, is as follows....

Assuming my strength remains constant, which ride--if any--requires less energy to reach the summit?

ECB