Today, while reading A Practical Guide to Problem Solving in Mathematics by Carol Meyer and Tom Sallee, I came across a problem I was unsure of how to solve except by brute forcing possibilities and eliminating them if they did not fit the given criteria. I was wondering if there was possibly a diagram or something else that could explicitly show/help show the route one needed to take. It seems to be a much more efficient strategy, but I was not able to draw a diagram that was of very much use. The problem in whole is listed below:
"Most Hrunkla lived in giant, $12$ story apartment houses, and their
homes were large square rooms bounded on four sides by corridors. Each
room had a single door which opened halfway along the corridor. On
even numbered floors: the doors opened onto the east corridors; on odd
numbered floors, the doors opened onto the north corridor. At each
intersection of corridors, there was something like an elevator which
could be ridden up or down. Half of the corridors had moving belts on
the floor, and no self respecting Hrunkla would walk if they could
ride the belts. The belts were so arranged that those on floors $1$,
$5$, and $9$ ran to the east; those on floors $2$, $6$, and $10$ ran
to the south; those on floors $3$,$7$,and $11$ ran to the west; and
those on the floors $4$,$8$, and $12$ ran to the north. Describe how a
Hrunkla who lived on floor $10$ could use these moving belts and
elevators to visit a friend who lived in the room directly below his."
After quite a while, I found a solution that works:
a. Ride belt to SE corner.
b. Ride to eighth floor (or fourth or twelfth).
c. Ride North to NE corner.
d. Ride to seventh floor (or third or eleventh).
e. Ride W to NW corner.
f. Ride to ninth floor
g. Ride E to friend's floor.
I assume there are many different routes possible, but I was not sure if I found the shortest route possible (even though the question does not ask for that). I am mainly looking to see if there is an "easy" way to solve the problem via a diagram or something of the sorts, so when I come across future problems such as this, I have a better plan of attacking it.
Any help would be greatly appreciated.
Best Answer
The first step has to be riding the belt south to the southeast corner of the 10th floor, and the last step has to be riding the belt from the northwest corner of the 9th floor to the friend's room. To go from a southeast corner to a northwest corner, you need to go north (on floor 4, 8 or 12) and west (on floor 3, 7 or 11), using the elevators to go to floors that have belts heading in the proper direction. You could either do it as you did, or go west first and then north.