2017 AMC 8 Problems/Problem 15
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.
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(easy and recommended)
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
We can do this problem by computing how many ways there are to get to each letter (in order). There is way to get to the in the center. We can only get to each of the other s by going there from the , so there is way to get to each of the four s. For the s, we notice that four s are surrounded by one , and four s are surrounded by two s. Finally, each of the s is surrounded by one with one way to get there and one with two ways to get there. Therefore, there are three paths to any of the s. Since there are eight s, the answer is
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