Die rolling around a 6x6 grid

Question

Each side of a standard 6-sided die is painted with a different color. A 6x6 grid is drawn on paper and the die is placed in one of its corners. At each turn the die can be rolled to an adjacent cell (horizontally or vertically). As the die is rolled, the color of its bottom side that touches the paper is imprinted on the paper. The die cannot be rolled outside the grid or to an already painted cell. Is it possible to roll the die around the grid such that it paints exactly 6 cells with each color?

Bonus question: Can you do it such that no two adjacent cells get the same color?

Answer 1

The first and simplest method I tried worked:

Roll the die left to right along the top row, and then right to left along the next row, and so on alternating direction boustrophedon style.

 5
 1 2 3 4
 6

 1 2 3 4 1 2
 1 6 3 5 1 6
 4 6 2 5 4 6
 4 3 2 1 4 3
 5 3 6 1 5 3
 5 2 6 4 5 2

Bonus Question:

This is not possible. Every time the path takes two consecutive right turns or two consecutive left turns you will always get two adjacent squares with the same colour. I.e. if you visit the squares below in numerical order then squares 1 and 4 have the same colour:

 1 2
 4 3

There is no path without such consecutive turns. Given any die path that covers the squares and consider walking along the edges between the squares that the die does not cross, starting somewhere on the outside boundary, and then going into the interior of the grid. You cannot go in a loop (any loop would mean a closed off area that the die cannot have visited), and if you don't retrace your steps you also cannot reach the outside boundary again (that would split the grid in two unconnected parts which the die cannot both have covered) so you must eventually reach a dead end. That means that the die has taken two consecutive turns at that point.