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avatar+621 

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?

 May 16, 2024
 #1
avatar+29 
+1

As shown in the image, Franklin could be at (0, 1), (0, -1), (1, 0), or (-1, 0) after 1 step, as shown in the blue points in the image.

We add further possibilities to those on the 2nd step, giving the orange points in the image.

Note that we do not count the blue dots in our final answer. So, we count only the orange dots, in which there are 9. Therefore, there are 9 different points that Franklin could possibly be at after 2 steps.

 May 17, 2024
 #2
avatar+129514 
+1

After n steps, he has visited (n + 1)^2  lattice points (points with integer coordinates)

 

So

(2 + 1)^2  =  3^2  =  9

 

 

cool cool cool

 May 17, 2024

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