Lucky Seven

Congratulations to Joel Brown, a 26 year old graduate of the University of Illinois, the first person to solve this tricky puzzle septulet. Joel has a B.S. in Psychology, and thanks his girlfriend Paulette. Good show Joel!

Ok, now for the answers:

1. The rope is 60 feet long. The wording makes it tricky, but the algebraic equation would be 30 + x/2 = x. Doubling both sides of the equation gives us 60 + x = 2x. Subtract x from both sides, and there you go.
2. Well this was certainly tricky. The pattern had absolutely nothing to do with any kind of mathematical progression. Write out the numbers in English: thousand, billion, octillion, hundred, zero, four, eight, three. The elements correspond to the letters of the alphabet. The value of an element is the first integer to contain that letter its name. So the next element is five.
3. This apparently was the hardest one of the lot. As I mentioned in the discussion forum hint, "Seven of Nine" is a red herring. Star Trek had nothing to do with the solution, I threw Seven of Nine in as a false lead.
   In Kit Williams's classic puzzle book, "Masquerade" (see links page for details) one of the clues to the treasure's location was "One of Six to Eight". The solution to this clue was the first wife of six wives to Henry VIII, Catherine of Aragon.
4. The answer is 2. To see this, square both sides of the equation. Notice that x + the original series now equals four.
5. There are 7 1's 3 2's 2 3's 1 4's 1 5's 1 6's 2 7's 1 8's 1 9's and 1 0's in this sentence. There are a few key pieces of insight that help solve the problem. First, there can only be 1 zero, because there aren't zero of any number. Next there can't be more than 1 nine. Then one can recognize that most of the larger numbers will have value of 1. The only except will be whatever number fills 1's slot. Trial and error can solve the rest quickly.
6. The next number is 132. This series is known as the Catalan series.
7. The answer is about 3.9 x 10^-14. This is, of course, a trivial problem in unit conversion. Trivial if you happen to know what c, petasmoots and femtotribulations are. The symbol "c" is the speed of light, about 3 x 10⁸ meters per second. Most people probably figured out that "peta" and "femto" prefixes are power of ten modifiers, like "mega" and "micro". Peta means "a quadrillion times", femto means "one quadrillionth".
   A tribulation is a biblical term referring to the Book of Revelations. Depending on your reading it was either seven years, or half that. I was prepared to accept either answer.
   Finally, a smoot is unit of measurement used only for the length of the Harvard Bridge in Boston, Massachusetts. You can read about the recent origin of this curious unit here.

I hope everyone had fun with these, and I'll try to do another Potpourri-class puzzle where people can send in answers soon.