Question

Charlotte the spider is walking along a web shaped like a $5$ -pointed star, shown in the figure below. The web has $5$ outer points and $5$ inner points. Each time Charlotte reaches a point, she randomly chooses a neighboring point and moves to that point. Charlotte starts at one of the outer points and makes $3$ moves (re-visiting points is allowed). What is the probability she is now at one of the outer points of the star?

$\dfrac{1}{5}$

$\dfrac{1}{4}$

$\dfrac{2}{5}$

$\dfrac{1}{2}$

$\dfrac{3}{5}$

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Written solution:

From an outer point, Charlotte must move to an inner point. From an inner point, $2$ of the $4$ neighboring points are outer, so the probability of moving to an outer point is $1/2$ . After one move she is inner. After two moves she is outer with probability $1/2$ and inner with probability $1/2$ . After the third move, only the inner case can move to outer, with probability $1/2$ , so the final probability is $\frac12\cdot\frac12=\frac14$ .

2026 AMC 8 Problem 21

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Problem 21 in Other Years