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?