If a U.S. patent was re-issued in 1931, what was the original date?

Probably just a few years before the reissue date, but there is no way to tell from your question. You may be able to tell from the front of the reissue patent. If you tell us the patent number, title, and applicant (inventor) I can look it up and give you further information. And what do you mean by "the original date"? Do you mean the date the patent was originally issued?

The previous two answerers are apparently under the mistaken impression that a reissue patent extends a patent term after the first patent term is ended. This does not happen; when the patent term expires the invention passes into the public domain. A reissue patent is generally an amended patent that a patent holder applies for when he realizes that he made a mistake writing the first version, usually because he claimed too much or not enough. It doesn't change the patent expiration date. Reissue patents are usually applied for within the first two years after the original patent issues. I'm a patent attorney, by the way.

1911

"Generally, the term of a new patent is 20 years from the date on which the application for the patent was filed in the United States or, in special cases, from the date an earlier related application was filed, subject to the payment of maintenance fees."

Not sure, but probably not 1911

patents are issued for 20 years TODAY, but it was a different period then (and i can't remember off the top of my head - if i think of it tonight ill look it up)