This story begins …. Oops! Wait a second. I almost forgot.
LOCK THE GATES!
OK. Now we can start… The story begins around November of 1992. That’s a long time ago. About that time, as with every Christmas season, there were a lot of watch commercials. The kind of “watch” I’m talking about is the kind you wear on your wrist. Apparently watches are a popular Christmas gift. This year, however, I’ve noticed that there seem to be much fewer “watch commercials”. Lately the only watches worth a TV ad are the Apple Watches. But I digress. Back to the story…
I happened to notice that all of the watches in the commercials displayed the same time! The time on the watch always seemed to be around 10:10. And it didn’t seem to matter whether the watch was digital or analog (just numbers vs having the old-time hands which seem to be in vogue these days). I sez to myself, “self… this is definitely not a coincidence!” So, one day in a staff meeting at work I pointed this out. Some derided me for spending time on such foolishness. But there were a few that suspected there was more here than meets the eye. In fact, a fellow named David spoke up and said (paraphrasing) “I have a book that talks about just that subject!” The next day he brought the book in. The title was “The Straight Dope” by Cecil Adams. An article in this book indicated that there was a historical reason for 10:10 and/or 8:20 being prevalent times in watch commercials and print ads. According to Cecil, when watches first became cheap enough for the masses the manufacturers started advertising them in newspapers and magazines. This was around the turn of the century. Each watch typically had the manufacturers name or logo just above or below the center of the watch and the times 10:10 and 8:20 (or thereabouts) made an attractive symmetrical framing of it. So, the “ad agencies” used these “times” in the watch ads. This apparently became a tradition that even survived the transition to digital watches. In fact, the ads for the new Apple Watch (pictured above) also do this!
Well, the question was finally answered! Or was it? It seemed that it wasn’t enough to know that the times were “about” 10:10 or 8:20. A complete understanding of how the universe worked demanded knowledge of the exact times such that the hour and minute hands both had the same angle about the vertical axis of a watch. For example, these times are approximately 9:15, 10:10, 11:05, 1:50, 2:45, 7:25, 8:20, etc. Well, another fellow who worked in the cubicle adjacent to mine also had a hankering for such enlightenment. He was (and still is) Tim. After all… it’s only by understanding these things that we get closer to seeing the face of God. Well…. maybe not. In any event, we worked on this problem and in a short time we came up with formulas that gave us all of the times. It appeared we had beaten this “watch problem” to death. But appearances are deceiving as we’ll find out in more ways than one.
The 3-Handed Watch Problem
We soon graduated from two-handed watch problems to the three-handed variety. We kept relatively quiet about this because we were unsure whether solving such problems could shake the very foundations of the time-space continuum. At the very least it might appear foolish.
At the time the most important question about time was..
What times are there (and what are they) such that the angle between the hour, minute, and second hands are the same. That is, the angle between h and m, h and s, s and m are 120o or 240o (2/3 * Pi or 4/3 * Pi). Or put another way, the hour, minute, and second hands cut the face of the clock into 3 equal pieces/areas. The location of the hands must represent a legitimate time. That is, they can not be simply placed to arbitrary positions such as 12, 4, and 8. Also, the clock doesn’t “tick” but rather it runs “smoothly”. In other words, picture a pie cut into 3 equal pieces. And note that this is NOT a trick question and 12:00:00 is not the answer!
We played with this question for awhile and concluded that there probably were not any such times. This was based on calculating the position of the second hand based on a few valid hour and minute hand positions. As it turned out, the second hand was not even close to where it had to be. That’s how we left it around January of 1993.
Now we jump forward in time to November of 1995. I move to a new project at work with another fellow named “KZ”. Somehow, and I don’t know how, he and I got started on this “clock quest” again. This time we pursued it in more depth.
We attacked the problem on two fronts: Analytical and brute-force.
Since we are both Computer Jocks we decided that the “brute-force” attack was a good place to start. So I wrote a Visual C++ program to step thru time with great accuracy using extended precision floating point calculations (i.e. double). That is, let the clock tick in extremely fine increments (user settable) and see if all of the hands got reasonably close to the required positions. Running this program was a sight to behold! There were dozens of floating point numbers flashing and changing, and a ListBox with details of the “closest” candidates. Jordy would have thought we were calculating the ETA to Ferris Minor via the Slauson Cutoff. As it turned out, there were cases where the hands of the clock nearly cut the pie into 3 equal pieces but by “nearly” I mean that the closest we could get was 1/6 of a degree. That is, if the hands had to be 120 degrees apart, the closest we could get was 1/6th of a degree off. Considering the fine precision of the calculations, this seemed to be experimental proof that there was no time such that the hands of the clock cut the pie into 3 equal pieces.
It is indeed true that there are NOT any times when
the hour, minute, and second hands divide the face of the clock into 3 equal pieces/areas.
Proofs follow below.
Or better yet, you may want to develop your own brute-force and analytical proofs. It may be “time” well-spent.
The 3 proofs are as follows (just click on them and they will open in a new tab).
The MO’ Proof is, I think, the most clear and easiest to follow but they all work equally well. They are all of the “Reductio ad absurdum” type.
OPEN THE GATES!
************ End of Famous Watch Story ************