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Franklin the fly starts at the point $(0,0)$ in the coordinate plane. At each point, Franklin takes a step to the right, left, up, or down. After $2$ steps, how many different points could Franklin end up at?

 Jan 30, 2024
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We can imagine a coordinate grid The right-most point he can travel to is (2,0), left-most is (-2,0), top and down is (0,2) and (0,-2) respectively.

Now, in between those points, the furthest the fly can go is (1,1),(1,-1),(-1,1), and (-1,-1). Therefore, the total number of possible points Franklin can end up at is \(\fbox{12}\)

 Jan 30, 2024

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