| From the introduction, the drive north from Summerset to Fallbrook is a distance
of 10.0 miles, with the five scenic spots along the route marked by posts
designating distance from Summerset to the tenth of a mile. By clue 1,
no distances between any two consecutive senic spots are the same; the
shortest distance from one to another is .4 and the longest is 3.6 miles.
By clue 3, the Amish Covered Bridge isn't the last spot and is 1.0 miles
from Fallbrook. If the Amish Covered Bridge were the third-last spot, by
clue 1 it would have to be at least 1.5 miles from Fallbrook; so the
Amish Covered Bridge is the second-last scenic spot. By clue 1, the
shortest distance between spots is .4 miles. The distance from the Amish
Bridge to the last spot can't be .5 miles, since the distance from the
last spot to Fallbrook would also be .5 miles. The distances from the
bridge to the last stop and from the last stop to Fallbrook must then be
.4 and .6 miles or vice versa. By clue 2, the drive from Summerset to
Cucumber Creek is the same distance as the drive north from Old Man
Mountain to the White Oak Inn. Cucumber Creek can't be the last spot,
because then the drive from Old Man Mountain to the White Oak Inn would be
a subsection of the Summerset-to-Cucumber Creek one, and they couldn't be
the same distance. If Cucumber Creek were the third scenic spot, the
White Oak Inn couldn't be an earlier spot or again the Old Man Mountain
to White Oak Inn distance would be part of the Summerset to Cucumber Creek
drive. If Cucumber Creek were the third scenic spot, the White Oak Inn
would be the last spot before Fallbrook. However, the distance between
the two spots would be 4.2 miles at most (the high of 3.6 (1) and the
.6 from clue 3--contradicting clue 5. If Cucumber Creek were the first
scenic spot and therefore 1.8 miles from Summerset (6), Old Man Mountain
and the White Oak Inn couldn't be consecutive spots or they would also be
1.8 miles apart (2), making two legs of the Trail equidistant--no (1). If
Old Man Mountain were the third spot and the White Oak Inn then were the
fifth, Sugar Maple Farm would be the second; but this would make clue 4
an impossibility. If Old Man Mountain were the second spot and the White
Oak Inn the fifth, Sugar Maple Farm would be the third spot. However, by
clue 4, the distances from Cucumber Creek to Old Man Mountain and from
Old Man Mountain to Sugar Maple Farm would have to be the same,
contradicting clue 1. Therefore, Cucumber Creek can't be the first scenic
spot and must be the second. By clues 5 and 1, the White Oak Inn can't
be the third spot and must be the fifth (2). If Old Man Mountain were the
first spot and Sugar Maple Farm the third, the distance from Cucumber Creek
to Sugar Maple Farm would be 2X and from Old Man Mountain to Cucumber Creek
X (4). The Cucumber Creek to Sugar Maple Farm distance can't be the high
of 3.6 (1), or the Old Man Mountain to Cucumber Creek distance would be
1.8, same as the first leg and conflicting with clue 1. So, the 3.6 miles
would be from Sugar Maple Farm to the Amish Covered Bridge. By clue 2,
then, we have the distance from Summerset to Cucumber Creek equalling X + 1.8
miles and the distance from Old Man Mountain to the White Creek Inn being
3X + 3.6 + (.4 or .6)--clearly not equal as required by clue 2. So,
Old Man Mountain must be the third spot and the Sugar Maple Farm the first.
By clue 4, the distance from Sugar Maple Farm to Cucumber Creek would be 2X
and from Cucumber Creek to Old Man Mountain X (4). The Sugar Maple Farm
to Cucumber Creek distance can't be the high of 3.6 (1), or the Cucumber
Creek to Old Man Mountain distance would be 1.8, same as the first leg and
conflicting with clue 1. Therefore, the 3.6 mile leg must be the one
from Old Man Mountain to the Amish Covered Bridge. Letting the distance
from Sugar Maple Farm to Cucumber Creek be 2X and from Cucumber Creek to
Old Man Mountain X (4), we have 3X + 6.4 = 10.0, so that X = 1.2 miles.
The distance from Sugar Maple Farm to Cucumber Creek is 2.4 miles and
from Cucumber Creek to Old Man Mountain 1.2 miles. From Summerset to
Cucumber Creek is 4.2 miles, so by clue 2, the leg from the Amish Covered
Bridge to the White Oak Inn is .6 miles. The last leg is .4 miles. In
sum, the drive along the Autumn Leaves Trail has scenic spots as follows:
- Milepost 1.8, Sugar Maple Farm
- Milepost 4.2, Cucumber Creek
- Milepost 5.4, Old Man Mountain
- Milepost 9.0, Amish Covered Bridge
- Milepost 9.6, White Oak Inn
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