Imagine a six-sided die, D6, the right size to exactly occupy a square on a chessboard.
The die can move to any adjacent square, but does so by rolling rather than sliding, so the topmost side of the die will show a different value.
Now suppose the chessboard is infinite in every direction: north, south, east & west. And there is a constraint: the die must at no point show a 6 on top. 1, 2, 3, 4 & 5 are all ok.
Subject to this constraint, can you define a sequence of moves for the rolling die across the infinite chessboard, so that each square of the board is occupied exactly once?
EDIT: have added my own suggested solution below.
EDIT 2: have awarded the legendary green tick to what I consider to be the best-explained answer. A clear picture that didn’t confuse people is a big part of this. I excluded my own (very different) answer from contention, although I do like it! Thanks all!