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Coyote
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 Posted: Sun Dec 30, 2012 10:41 pm Post subject: 1 |
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Beginner:
1. Place eight knights on the chessboard so that each knight can move to seven unoccupied squares.
2. Place sixteen knights so that each knight can move to three unoccupied squares.
Intermediate:
3. Place twelve knights so that each knight can move to six unoccupied squares.
4. Place twenty-four knights so that each knight can move to four unoccupied squares.
Advanced:
5. There's no way to place any number of knights on a standard chessboard so that each knight can move to one single unoccupied square.* But it is possible when a torus-board is used. Place (unspecified number) knights on the torus-chessboard so that each knight can move to just one unoccupied square.
*Extra credit:
6. While I firmly believe there's no solution to the one-square puzzle on the standard board, I haven't been able to actually prove it. The truly ambitious might try to come up with a nice informal proof---or if you really want to embarrass me, find a counter-example! 
Oh, and since I'm here...
*waves*
Happy New Year to everyone here at the Grey Labyrinth!! Hope it's a good one for y'all! |
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Zag
Tired of his old title
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| Posted: Mon Dec 31, 2012 12:21 am Post subject: 2 |
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Well, there are lots with 8 knights that can each move to 6 squares. Here's one
And this and its reflection are the only ones with 8 knights that can each move to all 8 squares.
Here's your answer, though, which is frustratingly asymmetrical.
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Trojan Horse
Daedalian Member
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| Posted: Mon Dec 31, 2012 3:21 am Post subject: 3 |
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That's fine if you want six knights that can each move to seven unoccupied squares. But if you want eight such knights:
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Trojan Horse
Daedalian Member
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| Posted: Mon Dec 31, 2012 3:27 am Post subject: 4 |
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And for sixteen knights that can each move to three unoccupied squares:
I think I'll stop here for tonight. |
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Zag
Tired of his old title
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| Posted: Mon Dec 31, 2012 12:21 pm Post subject: 5 |
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| Trojan Horse wrote: |
That's fine if you want six knights that can each move to seven unoccupied squares. But if you want eight such knights:
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Reading comprehension fail. |
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Trojan Horse
Daedalian Member
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| Posted: Thu Jan 03, 2013 5:38 pm Post subject: 6 |
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Sigh.
Would anyone object if I posted solutions to 3-5? I've got them figured out, but I don't want to hog all the fun... |
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Zag
Tired of his old title
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| Posted: Thu Jan 03, 2013 8:47 pm Post subject: 7 |
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| Since I managed to fail on a "Beginner" problem, I'm certainly not going to be upset. I actually spent a little time on how one would go about proving #6, and I decided that just making a program to do an exhaustive search would be easiest. I didn't even bother with 3-5. |
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Coyote
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| Posted: Fri Jan 04, 2013 12:19 am Post subject: 8 |
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Yeah, you may as well post your answers, as there doesn't seem to be a lot of response so far. I'm curious to see how you present your solution for the torus-board puzzle.
I'd thought of doing a follow-up thread titled 'Rook Management' if this thread seemed popular, but I think I'll just add my three favorites here.
Sixteen rooks each have seven moves.
Sixteen rooks each have eight moves.
Twenty rooks each have three moves. |
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Trojan Horse
Daedalian Member
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| Posted: Fri Jan 04, 2013 1:08 am Post subject: 9 |
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Okay. Here's #3: twelve knights that can each move to six empty spaces.
Here's #4: twenty-four knights that can each move to four empty spaces.
#5 coming in a bit. |
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Trojan Horse
Daedalian Member
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| Posted: Fri Jan 04, 2013 1:36 am Post subject: 10 |
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| Coyote wrote: |
| I'm curious to see how you present your solution for the torus-board puzzle. |
I figured you would be! 
Okay. Here was my plan of attack:
If every knight on the torus-board can move to exactly one empty square, then clearly, most of the squares are going to be filled. So it's really about figuring out where the empty squares should go.
Note that knights on white squares can only move to black squares, and vice versa. So my goal was to do the following:
1. Pick a set of white squares, such that every black square is a knight's move away from exactly ONE of the chosen white squares.
2. Pick a set of black squares, such that every white square is a knight's move away from exactly ONE of the chosen black squares.
Then, if I make the chosen squares empty, and put knights everywhere else, that should do it. And here's what I got:
Yep, that's right. If you start on any white square, then make three 2-square diagonal hops in the same direction, those four white squares will work. Same goes for the black squares.
But then again, there are plenty of other solutions, since the white and black squares can be chosen independently. For example:
So that's it!
Oh, by the way: #6 can be done too! There is a way to place a collection of knights on a REGULAR chessboard, so that each and every knight on the board can move to exactly one empty square. Want to see it? Here it is:
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lostdummy
Daedalian Member
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| Posted: Fri Jan 04, 2013 3:16 pm Post subject: 11 |
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Another #4 (twenty-four knights that can each move to four empty spaces), this one bit more symmetrical ;p
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Coyote
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| Posted: Tue Mar 12, 2013 1:36 am Post subject: 12 |
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Apologies for bumping this, but since I stopped in to look at Zag's Fractional Squares puzzle, I thought I'd tie up a few loose ends here.
First off, thanks to the select few who participated here, and a special tip of the hat to lostdummy for finding a solution to the four-squares puzzle that I'd overlooked! (My intended solution was the same as Trojan Horse's.) An odd bit of trivia about TH's solution: while there are twenty squares on the chessboard where a knight attacks four squares, none of them are occupied in the solution!
Secondly, a couple of misconceptions should be addressed. While Zag correctly showed how a maximum of 8 eight-move knights could be placed, his solution is not unique, as there are two other distinct ways of placing those eight horses. One of them can be produced by resetting the d5 and e4 knights (in Zag's diagram) on c3 and f6. The other one I'll leave as an exercise for the reader, as it's a bit of an 'Aha!' solution.
Moving on to the torus-board puzzle: Trojan Horse used the same approach toward solving this as I did. However, while it's true that the white and black squares can be selected independently, it's somewhat inaccurate to claim 'plenty' of other solutions, as there's only a very few ways those two sets of squares can be placed relative to each other--in fact there are only three ways, of which TH showed two. Here's the third:
(Keep in mind the diagram here is just an attempt to use a 2-D board to show a torus-board solution. Settings that look very different on the 2-D board can actually be identical via translation, rotation, & reflection. For example, a nice symmetrical version of TH's second diagram is produced by leaving these squares empty: a1, h1, c3, f3, d5, e5, b7, g7.)
Finally, if anyone's interested, here's solutions to the three rook puzzles.
Seven moves each:
Eight moves each:
Three moves each:
Okay, I'm done now. Carry on! |
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