To finish up the EVE problem, there is a second solution:

212/606 = 0.34983498…

This was not obvious at first because I looked at factors of 9999 when in fact, I should have looked at factors of 9999 * EVE. Brute force substitution can show that these two solutions are the only possible ones.

While this problem is simple enough that it can be solved in a few minutes, it represents a class of problems that can be quite complex with many potential answers. Exploring all of them would be difficult for a human armed with only a calculator. Ffinding a solution by testing all possibilities is simple a computer, but some problems become so complex so quickly that avoiding a blind brute force search for answers is necessary to find all possible solutions in a reasonable time.

So can the EVE problem teach us anything about how to simplify similar problems? The approach I used was first to see if the problem was presented was in the simplest form. There was no good reason to believe that eliminating the infinite string of letters would lead more quickly to a solution, other than the general observation that a simpler expression is often easier to deal with. (Note: this is not always true. Consider the question, “What is life?”. It is simple, but difficult to answer)

Next, before jumping into the substitution phase, we looked at limitations on the possibilities. That is, observing the consequences of the numerator being smaller than the denominator. Another property, that of the replacement of letters with digits only, not arbitrary numbers, allowed the simplification to considering only factors of 9999. As it turned out, that was too much of a simplification in that is gave the correct answer, but would have prevented us from finding another one if we had not gone back and considered carefully the assumptions made in finding the first solution.

Once the various simplifications had been made, direct substitution of candidate solutions to see if they worked was a simple matter.

The detailed method outlined here will not apply to very many puzzles, but the general approach of seeking ways to use the conditions of the problem itself to simplify the hunt for a solution is generally valuable. It is a cousin to the study of plausible inference that we discussed a few columns ago. Another value of this approach is that even if it does not lead directly to a solution, it can help an investigator become so knowledgeable about the problem, that other avenues of exploration might become apparent.

By comparison, one could write a simple program with nested loops which run through all the possible combinations of digits and check for answers. At the least, one needs four nested loops for the EVE problem. And regardless of where the answers lie in the spectrum of possibilities, the program must cycle through every possibility to be sure all answers have been found. That is, there is no simple STOP condition that can be applied. Deriving a STOP condition would essentially be equivalent to the reduction of possibilities that we have already done. The only real simplification that can be made is to add some IF statements to prevent testing of conditions with the same digit assigned to different letters. Even so, writing a little program to solve this puzzle might take less time than working it out. The value of actually solving it is to develop the skills of looking at the problem to see ways of simplifying it.

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.