Awake During A Bake?
Several weeks ago, I woke up in the middle of the night because I had gotten too hot. Usually, this involves a throwing off of covers and going back to sleep, but I had done that several times all ready, and so I had an additional problem of being annoyed as well as being hot. I decided to get up instead, even though it was around 2:00 am. I was of course, browsing the internet in a semi-groggy state when I happened across one of those “think carefully about this” calculation sort of problems. This also, somewhat annoyed me, since I was trying to get my brain to go to sleep and not think about some problem that would not necessarily allow for a deeper sleep state. What was the problem? Well, allow me to present it here:
Billy The Whizz, a 19th century cowboy, carries an 1847 Colt single action 6 shooter revolver. So proficient is he with this weapon that when he fires all 6 shots in a row, the time between the first bullet and the last is 60 seconds. How long would it take him to fire 3 shots?
In my sleepy state, I knew the question was not an obvious answer and I vaguely remembered something about how degrees of freedom in statistics as a concept overlapped this one. If you haven’t taken time to think about it yourself, you ought to do it for a few moments before you go on. I’ll just wait here.
Did you think about it? Good. Let me just go ahead and hit you with the answer that was supplied that my groggy mind also found rather annoying. The answer, as provided by the puzzle, is 12 seconds. The part that my groggy brain got quite irritated by was there was a little disclaimer saying that no matter how much you might want it to be, the interval between shots was 12 seconds, not 10. Something about this that I could not immediately say with a clearly defined “Why” nagged at the back of my brain. So, I decided “To hell with this not sleeping thing and this damn problem,” and went back to bed. My brain, though, was going to churn on this matter for the night. Its equilibrium had been disturbed.
Somewhere in the middle of the night, I had the vague thought that most of this problem concerned how one defined an interval. The answer provided gave a clear visual which I will reproduce here:
Image Credits - http://puzzles.nigelcoldwell.co.uk/sixtythree.htm
As can be seen, it all makes perfect mathematical sense. So what was my brain putting up a fuss about? Well, the problem proceeds on the basis of giving one an interval by definition in the problem. It tells you to consider the first and last bullets and the space between them. However, we are accustomed to dealing with time as being a rate. Therefore, I tell you that a bullet is fired say, once every ten seconds. That means, then, that our little picture above has ten seconds that exists OUTSIDE that first bullet and ENDS when the last shot is fired which is the sixth bullet. So, a rate is a “rational”–by definition–mathematical construct. An interval does not tell us when something begins in time. It only tells us how long it lasted. Therefore, when the first shot is fired, the first bullet is at “zero” because the rate is measured from that moment forward. We tend not to think very much in quality-from-our-senses derived intervals. We travel, for instance in miles per hour. If I give you a number of miles, and a unit of time, you determine the rate of speed at which I traveled per unit of time. If I tell you I covered X number of miles in an hour and that I was traveling at a constant speed, you can figure that out as well. If I tell you my first mile began at 5:33 and my last began at 7:00 and the distance was 500 kilometers or something, you might look at me a little strangely. Certainly you can express distance in this manner, but something about it feels not quite right–like you are measuring a coastline with a ruler.
So Why The Annoyance?
So having dealt with that problem and then basically being able to articulate that the interval changes depending on whether one has defined a ratio, I felt some better. To put it more clearly, the perception of the time of the interval is different and therefore when it starts is different in time depending solely upon whether one gives a rate or whether one gives the “first instance of a thing occurring”. Note the second definition is caught up specifically in perception. I have to “See” the first shot, or “hear” the first shot. At that moment, zero is defined. In a rate, my units start at the zero rate of the beginning of the ratio. 10 miles per hour means at time 0, I haven’t gone anywhere yet and at 60 minutes I have gone 10 miles. The happening of the thing is moving along with the occurrence of motion. The zeros match. In the bullet example, something has all ready happened, and that moment is defined as zero. Too little, too late–especially if you happen to be on the receiving end of the bullet being fired. Something about that feels “wrong” to define as a zero. It’s like saying, “The moment that I fell off the building and hit the ground is the beginning of the interval of a lot of pain.” The truth is, that interval started well before that–when you tripped off the steps or when you did not sleep well the night before due to being hot and having puzzles like these thrust before your eyes. A ratio is more “predictive” and a quality-sensory-based-interval is more descriptive after-the-fact or more akin to something being a priori or a posteriori respectively. Put differently, a ratio of normal speed/time is a bit like a Prophet, and the description of the interval traveled is a bit like a dead man. The second kind of reasoning is handy academically, but is useless for purposes of predicting when that first bullet is going to fly. Because mathematics is “Bad” at distinguishing this kind of knowledge from one another, it has a kind of “I knew it all along” that comes along with it. However, it cannot tell you when the bullet is fired. It might not be able to tell you when the journey will start with the car either, but it can tell you where it will be at some point in the future given a starting point. That, is useful information since one can “plan” with that information. The shooting begins when you hear it, though, is not especially useful information for the purposes of planning. At that point, everything is reactive should you happen not to be dead, and otherwise the interval of anything doesn’t matter to you if you have left this mortal coil. (12 seconds between shots probably won’t be in your calculations either from the experience of a firefight)
So really, the problem is that the question is asking a life and death question where the first occurrence of something makes the rest of the question potentially useless, since if you are dead, it is irrelevant. It “feels smart”, but it really isn’t all that useful, since you can “feel smart all you like” and still be stone dead. Hence, the description of YOUR EXPERIENCE of the question is potentially the first bullet flies at time zero, and the rest of the rate collapses for you since you are no longer around to “measure” anything else beyond that first occurrence.