Behold the solution to Doctor E. Strikes Back #4.  If you have forgotten the question, click here to see it again.

The prices for the four items were $1.20, $1.25, $1.50 and $3.16 .  Ah, but how to find them?  There were several different methods sent in, including a scary-brilliant algebraic analysis by Eric P. involving a three dimensional graph and the equation:

One way to do the problem is to convert everything to pennies.  So you have four amounts, such that w + x + y + z = 711.  Since (w/100)(x/100)(y/100)(z/100)=711/100, we have wxyz = 711,000,000.

So we can find all the ways to split 711,000,000 into four factors, and then find which combinations add up to 711.  This isn't as bad as you would think, because we know that all of the factors we use have to be less than 711 (or else their sum would be too big).  You can also make things easier by first thinking about the factor containing 79.  This gives you the following cases to explore: w = 474, 395, 316, 237, 158, 79.  It is logical to check the big cases first, because there are less subcases to check. 

There are other ways to simplify your search.  For example, since w + x + y + z is an odd number, you know that exactly 1 or 3 of them are odd.  If 3 of them are odd, then one of your prices has to be a multiple of 64 (and that will NOT be the multiple of 79), and you can eliminate the possible combinations of multiples of 64 and multiples of 79 pretty quickly.  Another way to quickly eliminate choices is to look at last digits.  w + x + y + z has to end in a "1".  Any multiple of 5 will end in a 5 or 0, and you can quickly see that there will be at least two multiples of five in the final answer, probably more.

This problem comes from the branch of number theory called Diophantine Equations.  This branch deals with solving equations where the solutions have to be integers.  A very famous theorem, that took centuries to prove, is that x^n + y^n = z^n has no integer solutions for n>2.  This is called Fermat's Last Theorem, and was only proved recently.

Here is the deal with a "Hoop - La:"  Doctor E.'s girlfriend, the Fayre Lady E, was reading Dead Man's Folly by Agatha Christie.  One important scene was at a party, where people were going to see the "Hoop - La."  This was not being used in the sense of a general festival or to-do, but more like a particular thing that one goes to see and participate in, like a specific game that was common when Ms. Christie was writing mysteries.  She asked Dr. E, and was shocked to find that there was a question that even the Great Dr. E. couldn't answer.  So Dr. E. agreed to ask his contest entrants, figuring that since they are the smartest and most dedicated students on the planet, they would know. And this way Dr. E. didn't have to look it up himself.

Well, there were many definitions of "Hooplah" submitted, but none that would work in this context.  Credit will be given as appropriate, but the real definiton of "Hoop - La" is still a mystery.

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