We’ve been looking at simple problems that can be made even more simple by either guessing an answer and verifying it or by finding some parameter other than what is desired first and then using that to finally solve the problem itself.
Here’s an interesting puzzle that was sent to me by a reader. It had no source listed. That is common with short puzzles of this type, but someone invented it, and I would like to credit the proper person, so if anyone knows the origin of this little gem, please let me know, and I will post it.
Consider the following and please overlook any implied sexism or religious overtones. This is an innocent number puzzle:
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?
Think about this for a bit before going on to see at least one way of doing it. And remember, this puzzle came to me without a solution, so what you will see below only represents my way of looking at it. You might have another way. Whatever works is okay.
With that in mind, I first sought to simplify the repeated fraction simply because working with an infinite number of digits can be time-consuming. From high school introductory algebra, we can use an old trick. Let
X = 0.TALKTALKTALK…
Multiply both sides by 10,000
10,000X = TALK.TALKTALKTALK…
But the right side can be rewritten
10,000X = TALK + X
since the fractional part of the right side is just the same as before the multiplication.
Then obviously X = TALK/9999 = EXE/DID
That simplifies matters immensely, but we do not look much nearer to a solution. Is there something else we can do that looks simple? Since all the letters represent digits, multiplying both sides of the last equality by 9999*DID shows that – well, I’m sure it shows something.
Let’s interrupt that line of thought for a moment to consider that by the initial conditions, EVE < DID. Does that help?
Gosh, it’s getting late. I’ve got to go now. Just time enough to warn you not to stop when you get an answer. To solve the problem correctly, you need to show if your answer is unique. If it is not, then you need to find the other answer(s). To be continued…
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.