id: 06513085
dt: j
an: 2016a.00861
au: Crans, Alissa S.; Rovetti, Robert J.; Vega, Jessica
ti: Solving the KO labyrinth.
so: Math. Mag. 88, No. 1, 27-36 (2015).
py: 2015
pu: Mathematical Association of America (MAA), Washington, DC
la: EN
cc: K30 A20
ut: KO Labyrinth; spherical puzzles; KO graph; Dijkstra’s algorithm; shortest
path; random walk
ci:
li: doi:10.4169/math.mag.88.1.27
ab: Summary: The KO Labyrinth is a colorful spherical puzzle with 26 chambers,
some of which can be connected via holes through which a small ball can
pass when the chambers are aligned correctly. The puzzle can be
realigned by performing physical rotations of the sphere in the same
way one manipulates a Rubik’s Cube, which alters the configuration of
the puzzle. The goal is to navigate the ball from the entrance chamber
to the exit chamber. We find the shortest path through the puzzle using
Dijkstra’s algorithm and explore questions related to connectivity of
puzzle with the adjacency matrix, distance matrix, and first passage
time analysis. We show that the shortest path through the maze takes
only 10 moves, whereas a random walk through the maze requires, on
average, about 340 moves before reaching the end. We pose an analogue
of the gambler’s ruin problem and separately consider whether we are
able to solve the puzzle if certain chambers are off limits. We
conclude with comments and questions for future investigation.
rv: