## Hippopotamus Hexagon Hopping, Part 2July 7, 2011

Posted by Alex in : bonus , trackback

There exists a two-row hexagonal grid, shown below, that extends infinitely to the right, continuing the numbering pattern shown. Albert is on hexagon $1$. A hippopotamus is on hexagon $12345$. Every second, the hippopotamus jumps and lands on, with equal probability, any adjacent hexagon that is not further away* from Albert than the current hexagon (there is a chance the hippopotamus will not move at all). If the expected value of the number of seconds it will take for the hippopotamus to reach Albert is $t$, compute $1000 t$.

*Note: the distance between two hexagons is defined as the length of the shortest sequence of adjacent hexagons that goes between them.