2020 AMC 8 Problems/Problem 21
Problem 21
A game board consists of squares that alternate in color between black and white. The figure below shows square in the bottom row and square in the top row. A marker is placed at A step consists of moving the marker onto one of the adjoining white squares in the row above. How many -step paths are there from to (The figure shows a sample path.)
Solution
We count paths. Noticing that we can only go along white squares, to get to a white square we can only go from the two whites beneath it. Here is a diagram: So the answer is ~yofro (Diagram credits to franzliszt)
Solution 2
Suppose we "extend" the chessboard indefinitely to the right:
The total number of paths from to (including invalid paths which cross over the red line) is . We subtract the number of invalid paths that pass through or . The number of paths that pass through is and the number of paths that pass through is . However, we overcounted the invalid paths which pass through both and , of which there are 2 paths. Hence, the number of invalid paths is and the number of valid paths from to is . -scrabbler94
Solution 3
The easiest way to solve this problem is just writing out the steps to get to each square as follows:
The answer is thus . Interestingly, the final answer is also
-franzliszt
See also
2020 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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