I'm huffing and puffing along. Finally I can see Ron off in the distance.
'HEY RON!' I shout. 'HOLD UP!'
The call is relayed up the trail by the other boys. Ron, who probably heard the call the first time, turns and looks back. Herbie, seeing relief in sight, slows to a fast walk. And so do the rest of us. As we approach, all heads are turned our way.
'Ron, I thought I told you to set a moderate pace,' I say.
'But I did!' he protests.
'Well, let's just all try to stay together next time,' I tell them.
'Hey, Mr. Rogo, whadd'ya say we take five?' asks Herbie.
'Okay, let's take a break,' I tell them.
Herbie falls over beside the trail, his tongue hanging out. Everyone reaches for canteens. I find the most comfortable log in sight and sit down. After a few minutes, Davey comes over and sits down next to me.
'You're doing great, Dad,' he says.
'Thanks. How far do you think we've come?'
'About two miles,' he says.
'Is that all?' I ask. 'It feels like we ought to be there by now. We must have covered more distance than two miles.'
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'Not according to the map Ron has,' he says.
'Oh,' I say. 'Well, I guess we'd better get a move on.'
The boys are already lining up.
'All right, let's go,' I say.
We start out again. The trail is straight here, so I can see everyone. We haven't gone thirty yards before I notice it starting all over again. The line is spreading out; gaps between the boys are widening. Dammit, we're going to be running and stopping all day long if this keeps up. Half the troop is liable to get lost if we can't stay together.
I've got to put an end to this.
The first one I check is Ron. But Ron, indeed, is setting a steady, 'average' pace for the troop-a pace nobody should have any trouble with. I look back down the line, and all of the boys are walking at about the same rate as Ron. And Herbie? He's not the problem anymore. Maybe he felt responsible for the last de- lay, because now he seems to be making a special effort to keep up. He's right on the ass of the kid in front of him.
If we're all walking at about the same pace, why is the dis- tance between Ron, at the front of the line, and me, at the end of the line, increasing?
Statistical fluctuations?
Nah, couldn't be. The fluctuations should be averaging out. We're all moving at about the same speed, so that should mean the distance between any of us will vary somewhat, but will even out over a period of time. The distance between Ron and me should also expand and contract within a certain range, but should average about the same throughout the hike.
But it isn't. As long as each of us is maintaining a normal, moderate pace like Ron, the length of the column is increasing. The gaps between us are expanding.
Except between Herbie and the kid in front of him.
So how is he doing it? I watch him. Every time Herbie gets a step behind, he runs for an extra step. Which means he's actually expending more energy than Ron or the others at the front of the line in order to maintain the same relative speed. I'm wondering how long he'll be able to keep up his walk-run routine.
Yet... why can't we all just walk at the same pace as Ron and stay together?
I'm watching the line when something up ahead catches my eye. I see Davey slow down for a few seconds. He's adjusting his
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packstraps. In front of him, Ron continues onward, oblivious. A gap of ten... fifteen... twenty feet opens up. Which means the entire line has grown by 20 feet.
That's when I begin to understand what's happening.
Ron is setting the pace. Every time someone moves slower than Ron, the line lengthens. It wouldn't even have to be as obvi- ous as when Dave slowed down. If one of the boys takes a step that's half an inch shorter than the one Ron took, the length of the whole line could be affected.
But what happens when someone moves faster than Ron? Aren't the longer or faster steps supposed to make up for the spreading? Don't the differences average out?
Suppose I walk faster. Can I shorten the length of the line? Well, between me and the kid ahead of me is a gap of about five feet. If he continues walking at the same rate, and if I speed up, I can reduce the gap-and maybe reduce the total length of the column, depending upon what's happening up ahead. But I can only do that until I'm bumping the kid's rucksack (and if I did that he'd sure as hell tell his mother). So I have to slow down to his rate.
Once I've closed the gap between us, I can't go any faster than the rate at which the kid in front of me is going. And he ultimately can't go any faster than the kid in front of him. And so on up the line to Ron. Which means that, except for Ron, each of our speeds depends upon the speeds of those in front of us in the line.
It's starting to make sense. Our hike is a set of dependent events... in combination with statistical fluctuations. Each of us is fluctuating in speed, faster and slower. But the ability to go faster than average is restricted. It depends upon all the others ahead of me in the line. So even if I could walk five miles per hour, I couldn't do it if the boy in front of me could only walk two miles per hour. And even if the kid directly in front of me could walk that fast, neither of us could do it unless all the boys in the line were moving at five miles per hour at the same time.
So I've got limits on how fast I can go-both my own (I can only go so fast for so long before I fall over and pant to death) and those of the others on the hike. However, there is no limit on my ability to slow down. Or on anyone else's ability to slow down. Or stop. And if any of us did, the line would extend indefinitely. What's happening isn't an averaging out of the fluctuations
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in our various speeds, but an
Looking ahead, I can see that how much distance each of us has to make up tends to be a matter of where we are in the line. Davey only has to make up for his own slower than average fluc- tuations relative to Ron-that twenty feet or so which is the gap in front of him. But for Herbie to keep the length of the line from growing, he would have to make up for his own fluctuations plus those of all the kids in front of him. And here I am at the end of the line. To make the total length of the line contract, I have to move faster than average for a distance equal to
