You and your team of cyclists are climbing a mountain with a constant gradient in the Tour de FiveThirtyEight. You make the climb single file, with one rider in front of the other. The first rider sets the tempo for the rest to follow — that is, until that first rider runs out of energy, or “cracks.” Upon cracking, a rider can no longer maintain the tempo and falls well behind any riders who still have some remaining energy.
When you begin the climb, there are eight riders on your team, and you start at the very back. The seven riders in front of you have exactly enough energy to make it up the entire climb, as long as they are not the first rider. Setting the tempo is hard work, and riding first in line is twice as exhausting as being one of the other riders.
At some point up the mountain, the first rider cracks. Then the next rider cracks, then the next. Eventually, the last rider in front of you cracks, leaving you to contend with the remaining portion of the mountain all on your own. What fraction of the mountain do you climb alone?
The distance each rider travels depends on the amount they travel behind the front and in the front. The following table summarizes the riders.
| Rider | Distance Not in Front | Distance in Front |
|---|---|---|
| 1 | 0 | $\dfrac{1}{2}$ |
| 2 | $\dfrac{1}{2}$ | $\dfrac{1}{4}$ |
| 3 | $\dfrac{3}{4}$ | $\dfrac{1}{8}$ |
| 4 | $\dfrac{7}{8}$ | $\dfrac{1}{16}$ |
| 5 | $\dfrac{15}{16}$ | $\dfrac{1}{32}$ |
| 6 | $\dfrac{31}{32}$ | $\dfrac{1}{64}$ |
| 7 | $\dfrac{63}{64}$ | $\dfrac{1}{128}$ |
| You | $\dfrac{127}{128}$ | $\dfrac{1}{256}$ |
You travel $\dfrac{1}{256}$ of the mountain alone.