Sorry that I dropped that riddle and then left. I meant to look up the source before I posted the answer, but I just haven’t had the time. Maybe someone here can post the source after seeing the answer.
The Halacha says that a person is required to bow down at the words Modim Anachnu Loch. (The gemara says that one who does not bow down by Modim loses the chance to be part of techiyas hamaisim.) The Halacha also states other specific points in Shemona Esrai where one bows down. But one is not allowed to bow at extra points in the Shemona Esrai because doing so will create confusion.
There is a shittah that if one is about to start the silent Shemona Esrai while the Chazzan is repeating it out loud, and it is clear that the chazzan will reach Modim before one is finished with his own Shemona Esrai, then one may not daven Shemona Esrai because he will not be able to bow down. He cannot bow down when the Chazzan reaches Modim because one can only bow at specific points in Shemona Esrai and therefore one may not daven until after the Chazzan passes Modim.
P.S. I am virtually certain that this is not how we pasken lehalacha. It is just a riddle, and there IS a source (which can be found in a Shulchan Aruch). Can someone else find it and post?