Walking study. A “self-avoiding walk” describes a path in which you never retrace your steps or cross your own path. An “unrooted walk” is a path in which it is impossible to distinguish between the starting point and ending point of the path. The American Scientist (July–Aug. 1998) investigated the relationship between self-avoiding and unrooted walks. The table gives the number of unrooted walks and possible number of self-avoiding walks of various lengths, where length is measured as number of steps.
a. Construct a plot to investigate the relationship between total possible number of self-avoiding walks and walk length. What pattern (if any) do you observe?
b. Repeat part a for unrooted walks.
WALK
Walk Length (Number of Steps) | Unrooted Walks | Self-Avoiding Walks |
1 | 1 | 4 |
2 | 2 | 12 |
3 | 4 | 36 |
4 | 9 | 100 |
5 | 22 | 284 |
6 | 56 | 780 |
7 | 147 | 2,172 |
8 | 388 | 5,916 |
Source: Hayes, B. “How to avoid yourself.” American Scientist, Vol. 86, No. 4, July–Aug. 1998, p. 317 (Figure 5).
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