MathDB

Consider a hexagon with vertices labeled

M

M

A

T

H

S

in that order. Clayton starts at the

M

adjacent to

M

and

A

, and writes the letter down. Each second, Clayton moves to an adjacent vertex, each with probability

\frac{1}{2}

, and writes down the corresponding letter. Clayton stops moving when the string he's written down contains the letters

M, A, T

, and

H

in that order, not necessarily consecutively (for example, one valid string might be

MAMMSHTH

.) What is the expected length of the string Clayton wrote?
Proposed by Andrew Milas and Andrew Wu

8