The King takes a long walk

[bonanova]

The Black King sets out one day to tour his kindom, a standard 8x8 chessboard.

He's not feeling well, though, and he wants to return by the shortest path to his starting square.

We'll assume all squares are one unit on a side and ask, what is the length of such a trip?

Wait. This is Brainden. You all are geniuses. Let's add a wrinkle.

The King is actually feeling fine, and he wants his walk to provide him the maximum possiblle workout.

Diagonal moves now come into play. To avoid radical complications , we'll count their length as two N-S or E-W moves.

So here's the actual question: what is the maximal length of a complete King's tour of an 8x8 chessboard?

Bonus points if a proof is given.

[bonanova]

[Perhaps check it again]

In one paragraph you are asking for the shortest path for the complete tour, but in another tour,

you are asking for the maximal length of a complete tour? How are these two plans not contradicting

each other?

[Perhaps check it again]

It should be: "..., but in another paragraph, you are asking for..."

[bonanova]

If you like, warm up with "What's 64 times 1?" before getting to the actual question.

[Perhaps check it again]

The statement of your problem would have been much clearer (and leaving no ambiguity that you wanted only

one question answered) if you had phrased part of it along these lines:

[insert the first original two sentences as given.]

We'll assume all squares are one unit on a side, and we might ask, "What is the length of such a trip?"

Wait. This is Brainden. You are all geniuses. Let's alter the problem.**

[insert the remaining original text.]

** You're not just "adding a wrinkle." You're substantially changing the problem, mainly going from

shortest path to maximal length path.

Edited May 5, 2014 by Perhaps check it again

[bonanova]

It seems you understand the OP. Now about the maximal path length ...

[k-man]

Assuming that

1) The king starts from e8 (the normal starting position for the black King) and

2) Every square is to be visited only once

is 116 with 12 lateral moves and 52 diagonal moves.

Here is the picture of my solution:

[bonanova]

Cool. Best so far. There is a slightly longer path.

What's the fraction of diagonal moves if the dimensions are 8x2?

[plasmid]

Cool. Best so far. There is a slightly longer path.

What's the fraction of diagonal moves if the dimensions are 8x2?

If I'm drawing correct inferences from this hint... does this mean the king doesn't have to finish the tour adjacent to his starting square so he can take one more step to get back home?

[bonanova]

A tour of the chessboard is technically a path that visits all the squares.

I mean to ask for a tour that comes back to the original square - a closed path.

There will be an even number (64) of moves.

And the answer, no secret here, maximizes the number of diagonal moves.

[plasmid]

Ok, asking for the solution for an 8x2 grid seemed that you were suggesting he repeat this four times, minus one move at the end, to visit every square with only eight non-diagonal moves. But since I have yet to encounter a cylindrical chess board, this won't fly.

[bonanova]

yeah, everything you said. the comment was to suggest the obtainable degree of diagonal moves.

[bonanova]

Assuming that

1) The king starts from e8 (the normal starting position for the black King) and

2) Every square is to be visited only once

is 116 with 12 lateral moves and 52 diagonal moves.

Here is the picture of my solution:

Black King.gif

Four fewer lateral moves is possible.

[bonanova]

Ok, asking for the solution for an 8x2 grid seemed that you were suggesting he repeat this four times, minus one move at the end, to visit every square with only eight non-diagonal moves. But since I have yet to encounter a cylindrical chess board, this won't fly.

King moves.jpg

But it is the right idea.

The minimum number (8) of adjacent steps can be maintained as small parts of

the four rectangles are minimally changed to effectively stitch them together.