In the last article, I presented a puzzle of unknown origin sent to me by a reader, who seems to have had difficulty solving it and thought it would be suitable to discuss:

EVE/DID = .TALKTALKTALK….

which should really be written as

EVE/DID = 0.TALKTALKTALK….

to emphasize that the right-hand side is a continued fraction. That is, the “…” means the TALK symbols repeat an infinite number of times.

This equation can be solved by replacing each letter with a unique digit. Can you find the appropriate replacements?

Since it came to me without a solution, I suggested the best way to start would be to eliminate the infinite repetition on the right side by a simple trick from high school math to get

EVE/DID = TALK/9999

Next I hinted that noting EVE is obviously less than DID might help. But first is you multiply both sides such that

9999 * EVE = DID * TALK,

then remember that the letters stand for digits. This suggests looking at the factors of 9999 which have the first and last digits the same (like DID).

Doing this shows that 101, 303, and 909 are likely candidates for DID, but 101 is immediately ruled out by the inequality since starting EVE with a zero would not be proper notation. That leaves only 303 and 909 as possibilities. At this point, I tried direct substitution of the other possible digits. Luckily I started with 303 as the denominator because only 1?1 and 2?2 needed to be checked and bingo:

242/303 = 0.79867986…

However, as I have emphasized, solving problems in this way does not guarantee that the solution is unique. There is an assumption in my solution that hides the possibility of other solutions begin possible. Can you identify the assumption?

Having seen how I slipped that one in, can you prove either that my solution is unique or that others exist? If you conclude that others exist, what are they? Did identifying the assumption help to answer the other questions?

Does anyone have a method of solving this puzzle that is significantly different?

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.