cross that bridge when you come to it

I was running on the Main river in Frankfurt and was trying to think of an alternative to running one side or the other, or up one side and back on the other, and discovered a topological problem.  What if I ran down one side of the river, crossed the next bridge I came to, continued down the other side to cross the next bridge, and just continued doing that?  Eventually of course I should cross a bridge and head back, but continue to cross each bridge when I come to it on my way home.  I was thinking it through and discovered that as I crossed a bridge on the 'out' and 'back' part of the run I always crossed any given bridge running the same way each time.  I first thought this depended on whether there were an odd or even number of bridges, but I was jogging along trying to mentally draw these paths out and couldn't find a way to "cross each bridge when you come to it" and cross each bridge both ways.

my Garmins view of this run

I won't bother with you my observations on the first and last bridges crossed ...

p.s.  After getting home, I found these kinds of problems have gotten plenty of attention - see for example Seven Bridges of Königsberg