2021 Fall AMC 12B Problem 17

A bug starts at a vertex of a grid made of equilateral triangles of side length 1. At each step the bug moves in one of the 6 possible directions along the grid lines randomly and independently with equal probability. What is the probability that after 5 moves the bug never will have been more than 1 unit away from the starting position?

\textbf{(A)}\ \frac{13}{108} \qquad\textbf{(B)}\ \frac{7}{54} \qquad\textbf{(C)}\ \frac{29}{216} \qquad\textbf{(D)}\ \frac{4}{27} \qquad\textbf{(E)}\ \frac{1}{16}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.