2017 AMC 8 Problems/Problem 15
Contents
Problem
In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path only allows moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture.
Solution 1 (Not Very Efficient)
Notice that the upper-most section contains a 3 by 3 square that looks like:
It has 6 paths in which you can spell out AMC8. You will find four identical copies of this square in the figure, so multiply to get total paths.
Solution 2
There are three different kinds of paths that are on this diagram. The first kind is when you directly count , , in a straight line. The second is when you count , turn left or right to get , then go up or down to count and . The third is the one where you start with , move up or down to count , turn left or right to count , then move straight again to get .
There are 8 paths for each kind of path, making for paths.
Solution 3
Notice that the is adjacent to s, each is adjacent to s, and each is adjacent to 's. So for each , there are s, and for each , there are s, and for each , there are s. Thus, the answer is
Video Solution
See Also
2017 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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