## Welcome to BrainDen.com - Brain Teasers Forum

 Welcome to BrainDen.com - Brain Teasers Forum. Like most online communities you must register to post in our community, but don't worry this is a simple free process. To be a part of BrainDen Forums you may create a new account or sign in if you already have an account. As a member you could start new topics, reply to others, subscribe to topics/forums to get automatic updates, get your own profile and make new friends. Of course, you can also enjoy our collection of amazing optical illusions and cool math games. If you like our site, you may support us by simply clicking Google "+1" or Facebook "Like" buttons at the top. If you have a website, we would appreciate a little link to BrainDen. Thanks and enjoy the Den :-)
Guest Message by DevFuse

# The King takes a long walk

Best Answer bonanova, 21 June 2014 - 05:22 AM

Spoiler for Putting this puzzle to bed

Go to the full post

14 replies to this topic

### #1 bonanova

bonanova

bonanova

• Moderator
• 6153 posts
• Gender:Male
• Location:New York

Posted 04 May 2014 - 06:57 AM

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.

• 0

Vidi vici veni.

### #2 Perhaps check it again

Perhaps check it again

Junior Member

• Members
• 95 posts
• Gender:Male

Posted 04 May 2014 - 06:18 PM

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?

• 0

### #3 Perhaps check it again

Perhaps check it again

Junior Member

• Members
• 95 posts
• Gender:Male

Posted 04 May 2014 - 07:30 PM

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

• 0

### #4 bonanova

bonanova

bonanova

• Moderator
• 6153 posts
• Gender:Male
• Location:New York

Posted 04 May 2014 - 10:34 PM

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

• 0

Vidi vici veni.

### #5 Perhaps check it again

Perhaps check it again

Junior Member

• Members
• 95 posts
• Gender:Male

Posted 05 May 2014 - 02:53 AM

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 by Perhaps check it again, 05 May 2014 - 02:59 AM.

• -2

### #6 bonanova

bonanova

bonanova

• Moderator
• 6153 posts
• Gender:Male
• Location:New York

Posted 05 May 2014 - 10:38 AM

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

• 0

Vidi vici veni.

### #7 k-man

k-man

Senior Member

• Members
• 536 posts
• Gender:Male

Posted 05 May 2014 - 11:04 PM

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

Spoiler for the longest path I could find so far

• 0

### #8 bonanova

bonanova

bonanova

• Moderator
• 6153 posts
• Gender:Male
• Location:New York

Posted 05 May 2014 - 11:24 PM

Cool. Best so far. There is a slightly longer path.
What's the fraction of diagonal moves if the dimensions are 8x2?
• 0

Vidi vici veni.

### #9 plasmid

plasmid

Senior Lolcat

• VIP
• 1484 posts
• Gender:Male

Posted 09 May 2014 - 05:12 AM

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?
• 0

### #10 bonanova

bonanova

bonanova

• Moderator
• 6153 posts
• Gender:Male
• Location:New York

Posted 09 May 2014 - 08:46 AM

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.

• 0

Vidi vici veni.

#### 0 user(s) are reading this topic

0 members, 0 guests, 0 anonymous users