Volatile Yard 挥洒自如 Write anything I want to write...

Puzzle

I got a 'mathematical dream' last night, which involves an original mathematical puzzle. In the dream, I was in a book fair looking for some inexpensive second-hand books. Then, I saw a thick book with a title like "Adfferv...", well, which should not be a proper English word. The book covers a lot of things, and is written like "Sophie's World". The main character in the book somehow met a wise person, who taught him or her some good knowledge in literatures, physics, mathematics, etc. I browsed the book and saw a page with an eye-catching mathematical puzzle. It looked something like this:

The question is: "These people need to pay the money. But how much?" I somehow figured out the answer, and verified it with that listed at the back of the book. Are you able to figure it out? Read on for the solution.

Notes:

• From left to right: A half-burnt candle on a table; A fire hydrant [en.wikipedia.org] (also known as a fire plug or a johnny bump); Several people;
• Disclaimer: Don't try too hard! While the puzzle and the answer do make sense, they are not really rigorous and can be controversial. After all, all these things including the 'correct' answer are just from my dream!
• Hint: Trying to think in English may help here.

Solution

Clearly, the following two parts need to be tackled:

• (a) The interpretation of the three pictures; and
• (b) The solution of two simple equations.

Consider part (b) first. Let the values for the three boxes from the upper-left to the lower-right as x, y, and z respectively. There is a set of simultaneous equations here:

• (i) 1.4 / x = y
• (ii) 7y = z

The amount of money these people need to pay should be z, as the x \$7.00 part in the diagram obviously means that somehow each person has to pay \$7.00.

Now, consider part (a):

• A half-burnt candle on a table. It corresponds to 1.4. Why? The candle is almost half-burnt, and there is one whole table. So 1.4 seems to mean the number of complete things here! In other words, 1.4 should mean 1 table plus 0.4 (or 40%) of a full candle.
• Several people with a question mark (i.e. the value of y). From the question, the phrase "these people" implies that the picture means "How many people here?" Let's stick to this for the moment. With this interpretation, we know that y must be an integer that is at least 0 (zero).
• A fire hydrant with a question mark (i.e. the value of x). From the above, we can probably interpret this as "How many fire hydrants here?" With this interpretation, we know that x must be an integer that is at least 0 (zero).

Now the crucial part: If we stick to such interpretations, we soon find that the equation 1.4/x=y, or xy=1.4, is unsolvable! Since, if x and y are both integers, their product must also be an integer. This can only mean that we must have interpreted some picture(s) wrongly.

Which picture? Let's reconsider. "How many people are here?", and we have a question mark beside several people. "How many fire hydrants are here?", and we have a question mark beside some fire hydrants... wait! But there is only one fire hydrant in the picture! We might have interpreted this picture wrongly. (In other words, if the picture really means "How many fire hydrants here?", then at least two fire hydrants must be drawn.) Since the book is in English, it probably wants me to think about the interpretation in English.

What can we get from a fire hydrant? Hmm... water... but water is uncountable. Oh... how about the number of water molecules we can get from a fire hydrant? Ha, a lot, so many that it looks infinite for the things we are dealing with in daily life. Think in another way, we can always reuse water, so it's in fact possible to get infinite number of water molecules from a fire hydrant! (Well, provided that we have infinite time with a fire hydrant that never breaks!) Great, with this interpretation, we have x=∞, and we easily get y=1.4/∞=0 and z=0:

I agree that the way of solving this puzzle is a bit (or probably quite) insane... but I've told you not to try too hard! It's just from a dream anyway!

After figuring out the answer of this puzzle in my dream, I was excited and browsed to other pages to find more puzzles. However, the content of the book kept changing mysteriously! I was a bit confused and put the book back to the shelf. I walked away from the shelf, woke up suddenly, and soon realized that it's just a dream after all.

By the way, I drew the diagrams above with OpenOffice.org Draw.