諸悪の根源は物理的

I recently found an interesting blog entry "Finding the most elegant solution possible to reduce careless mistakes." Here are the example questions in it.

Q1. The short hand and the long hand of the clock overlap each other at 12 o'clock. What time do they overlap again?

Q2. You throw two dice one by one. Calculate the probability that the first die shows a larger number than the second.

The following are my answers and thoughts.

A1. In 12 hours (the time the short hand rotates one lap), the hands overlap each other 11 times. Therefore the answer is 12/11 hours later.

A2. The probability of the two dice showing different numbers is 5/6. The first die has a bigger number for half of the cases. Therefore the answer is 5/12.

There is a commonality in the questions. It is symmetry.

In Q1, there are 11 intervals between overlapping states packed into 12 hours. In Q2, there are two states of "the first die has a bigger number" and "the second die has a bigger number" packed into the 5/6 probability. The necessary condition here is that each component exists with the same size.

Instead of considering only the component in question, you consider the whole at first and then divide it. The number of steps to reach the answer is increased, but this can be more elegant (simpler, easier and/or shorter) than fewer steps if each step is less difficult.

In general, it seems that one solution tends to be called "elegant" if it is different from a trivial one and simpler than it. There are lots of problems that have the most trivial solutions as their simplest solutions (actually most problems are like that). They are never considered "elegant." "Elegance" relies on trivialness, and trivialness is purely a subjective matter, so I personally do not understand what really can be the standard for "elegance," but I still use the phrase "elegant solution" when I give my friends puzzles like the ones above. It is certainly a bit uncomfortable for me.