Important calculations like these ought be done in at least two independent ways. For example, a detailed version accounting for every nook and cranny and a re

Important calculations like these ought be done in at least two independent ways. For example, a detailed version accounting for every nook and cranny and a reality check version that takes things in the large just to make sure both results pretty close. If they're not, then have someone else try again. Having said that, there's the famous (pretty simple) SAT question that all test takers got wrong, even the test setters. Oh, except there was that one guy who wrote "the right answer is not here." (It was a circle rolling around a bigger circle.)