Chess board combinatorics

STATEMENT: A dolphin is a special chess piece that can move one square up, OR one square right, OR one square diagonally down and to the left. Can a dolphin, starting at the bottom-left square of a chessboard, visit each square exactly once?

QUESTION: How would one approach this type of problem. It seems that whatever way the dolphin moves the maximum moves always involve the dolphin to traverse an 6x8 square.

Solution 1:

Hint: Instead of the usual black/white pattern, color the square $(a,b)$ either green, blue, or yellow, according to the remainder of $a+b$ (mod $3$). Each color forms a diagonal stripe going from the upper left to the lower right, with the stripe pattern going green, blue, yellow, $\dots$ as you move up or to the right. Think about how this coloring relates to a Dolphin's movement, and what a tour of the board would look like in terms of these colors.

Addendum: For your viewing pleasure:

Solution 2:

In response to Mike Earnest's question regarding the existence of a Dolphin Tour from alternate starting points: If we started at a blue square, say for example the bottom right corner instead,

The punchline of the proof is that there are in fact 21 yellow, 21 green, and 22 blue squares. Any sequence of moves will be (...,green,yellow,blue,...) repeated. Having started from a green square, after reaching the 21st blue, you will no longer have another green square to go to, and so you cannot continue until reaching the 22nd blue.

In a similar fashion, it is then impossible to reach every square having started at a yellow square either.

Having started at a blue however, as shown above, can be possible (depending on the square), and if is impossible for a specific starting square, would require a different argument.