Considering the office setting and that you weren't able to see any pieces of his missing tail in the tank anywhere, I'm going with the tail-biting bet. Your betta probably got stressed and attacked his own tail, ripped it off and ate it. What caused the stress attack could have been one of many things, or a combination of them. The cold water, the sudden bright lights going on and off when people show up and leave the office, and/or someone bumbping/tapping the tank ... any of it could have been the cause.
As Kim suggested, get a heater and bump that temp up, a cold betta is an unhappy betta and they are much more prone to disease and anxious behavior (aka tail-biting). Keeping the temp around 78-82 will probably alleviate much of his stress.
Also, dirty water makes a betta much more susceptible to disease, which is something you really need to watch for now that he's missing a tail. So keep up on the water changes, that is the best way to prevent infection.