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This question was posed to me, and I was surprised that I could not find a solution (as I thought that all rook tours [open or closed] were possible). Starting from a8, could a rook visit every square on the board once, ending on f3?
I tried a few times, with a few different strategies, but I always ended up missing one square.
It's really easy to burn pairs of rows or columns, so the problem space could be reduced...
...but at some point (4x4), I was able to convince myself that it is impossible (at least at this size and state):
...but it might be possible that shaving off column or row pairs is also discarding a solution?
Uhhhh yeah?
I found this to be really easy but I don't play chess so maybe I'm misunderstanding or perhaps the question was worded wrong?
Using these rules from your post
"Starting from a8, could a rook visit every square on the board once, ending on f3?"
And standard rook movement
I came up with this solution.
Follow the numbers up or down from the start or finish. Highlighted squares show moves to squares that aren't adjacent.
I'd imagine there's lots of solutions with different patterns, this is just the one that came to me on the fly.
The rule is better said that you can visit each square "once and only once." Every time you move to a non-adjacent square, you're crossing a square for a second time and breaking the rule.
If you can only move to black squares from white squares then for a sequence with an even number of terms if you start on a white square you have to end on a black square. Because the problem states you have to start and end on white it's impossible to do without crossing over squares.
Did you start from the square labeled โendโ?
Don't think of it as the number being the order of the steps, think of it as being how many are left lol
The other puzzles in your post are also solvable btw but I won't spoil them