Paper 12: Computer aided discovery of a fast algorithm for testing conjugacy in braid groups by Volker Gebhardt The code for this paper takes around 5,000 seconds to run on a 64-bit 2.4GHz AMD processor. Note that the file braid.m referred to in the paper is now called Paper12.m for consistency with the other papers. Changes since book release: V2.11: To aid exposition, some of the random elements have been replaced with explicit ones instead. Where this has been done a comment has been added to indicate it.