You are given a magic carpet and a hoverboard. The hoverboard can only move in the direction of the vector [3;1] and the magic carpet can only move in the direction of the vector [1;2]. Visually:

magic_carpet_ride.png
the site for carpets: http://floor-carpet.blogspot.com/

The origin (0,0) is your home.

TASK 1: Old Man Gauss lives lives in a cabin 107 miles East and 64 miles North from your home. Can you get there with the board and carpet?

TASK 2: Are there some locations where Gauss can hide and you cannot reach him from your home with these two modes of transportation?

(We might stop with Tasks 1 and 2 for February 14 and save the remaining ones for later.)

TASK 3: Having made it to Old Man Gauss' cabin, he swaps out your hoverboard and magic carpet while you aren't looking with a new hoverboard and magic carpet, and transports you home. After arriving at home, Old Man Gauss calls you up to inform you that with your new means of transportation, you cannot get everywhere. What are the possible vectors for the movement of the NEW hover board and magic carpet?