For example consider the black path one unit at a time:

You go up:

1,0,0

Up and right:

1,0,-1

Up:

2,0,-1

Up and right:

2,0,-2

etc...

2,1,-2

2,1,-3

2,2,-3

1,2,-3

So the final coordinate is 1,2,-3

Now consider the red path:

0,0,1

-1,0,1

-1,1,1

-2,1,1

-2,2,1

-2,2,0

-1,2,0

-1,2,-1

-1,3,-1

-1,3,-2

-1,4,-2

-1,4,-3

0,4,-3

0,3,-3

1,3,-3

1,2,-3

You still end up at 1,2,-3!

So it's kind of interesting to consider which coordinates are actually possible, for example -1,0,0 isn't possible to reach from the starting position's type of vertex in the above picture, but it is starting from the other type of vertex. I notice the possible coordinates are 0,0,0 and every other coordinate's components sum to an odd number for either type of starting vertex.

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