Brain Teasers
Rectangular Path
The rectangle shown consists of eight squares. The length of each side of each
square is 1 unit. The length of the shortest path from A to C using the lines
shown is 6 units.
How
many different six-unit paths are there from A to C?
Solution: 15.
Any
of the six‑unit paths will consist of two vertical segments and four horizontal
segments, so this problem reduces to determining the number of ways that two
vertical segments can be chosen. However, the vertical segment selected in the
upper half must be to the left of the vertical segment chosen in the lower
half. Otherwise, the path will be longer than 6 units.
For
instance, in the figure below, if p
is chosen as the vertical segment in the upper half, then only y or z
could be chosen as the segment in the lower half. If any of v, w,
or x were chosen, the path would
have to “backtrack” and thus be longer than 6 units.
.jpg "BT16-solution")
Thus,
we have the following:
|
If this segment is chosen
in the upper half…
|
Then only these segments
can be chosen in the lower half…
|
Yielding this many paths…
|
|
m
|
v, w, x, y,
z
|
5
|
|
n
|
w, x, y, z
|
4
|
|
o
|
x, y, z
|
3
|
|
p
|
y, z
|
2
|
|
q
|
z
|
1
|
That
gives a total of 5 + 4 + 3 + 2 + 1 = 15 paths.
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