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Scott Cram
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Recently, I was working with the fifteen puzzle (click the text if you're not familiar with it), and I was thinking about magic squares.

If you think of the empty space as a "0", I wondered if it was possible to arrange a fifteen puzzle into a magic square, preferably with the empty space in the lower right-hand corner (as in the traditional solution).

It's important to note that, in a fifteen puzzle, not every permutation of the numbers 0-15 is possible. This fact was exploited by Sam Loyd, who released a version in which the 14 and 15 were switched, thus making the puzzle impossible to solve (Until you got down to the last few pieces, you couldn't be sure whether your particular one was solvable or not).

All the 4 x 4 squares using the numbers 0-15 that I've found on the web so far aren't able to be created on a fifteen puzzle.

Does anyone know of a magic square that can be created on a fifteen puzzle?
Nir Dahan
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Scott,
if you have a copy of "the moscow puzzles" by Boris Kordemsky - edited by Martin Gardner (who else) you can find a similar puzzle on page 151 (puzzle 334 - the magic game of 15).
it is stated in the book that when exchanging the 14 and 15 blocks and counting the empty space as 0. a magic square of constant 30 can be reached from the initial position in exactly 50 moves. it says that it is unknown if there exists a shorter solution.
given the fact that the 14 and 15 blocks MUST be exchanged in order to reach a magic square, it seems that in analogy to the well known 14-15 puzzle, it is impossible to reach a magic square from the initial position - at least not the one in the book.
hope this helps.
Nir
p.s. if you are a puzzle fan, you should get the book. most puzzles are fairly basic but there are some gems inside. I am sure it can be found for very little money in abebooks.com
Scott Cram
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Doug Dyment was able to provide a working solution (albeit not with the 0 in the lower left hand corner, but it is a perfect magic square!).
stanalger
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Scott,

If you want the space (zero) in lower left corner, use:

14...03...08...05
09...04...15...02
07...10...01...12
00...13...06...11

Still perfect!

Stan Alger
Scott Cram
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Actually, I mis-typed. I meant to say the lower RIGHT corner. Interestingly, rotating your square by 90 degrees counterclockwise effectively involves an odd number of swaps, and is therefore impossible.

Thanks for the lower left 0 "15" magic square though, Stan!
stanalger
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Scott,

I agree with you (and Kordemsky): if you're going to assign a number value
to the space, it should be zero.

However, Doug Dyment has suggested in a parallel thread
(over in Magical Equations) that the space should represent 16.

This solves the parity problem. You CAN find numerous solutions with the
(value 16) space in the lower-RIGHT corner. For example, it's easy to
reach this position:

04...14...07...09
15...01...12...06
10...08...13...03
05...11...02...XX

So which (if any) trade-off are you willing to settle for?

Stan Alger

P.S. Kordemsky, in case you haven't looked it up, presents a 50 move
solution that takes you from

01...02...03...04
05...06...07...08
09...10...11...12
13...15...14...XX

to

13...01...06...10
14...02...05...09
XX...12...11...07
03...15...08...04

Note that his space doesn't finish in
ANY corner. This was the shortest path he was aware of from either of
the start positions (even or odd) to ANY magic square. He didn't finish
with a very magical magic square either. Many of the 2x2s don't sum to
30...nor do all of the 2x4s or 4x2s or broken rows or broken columns or
pyramids, etc.

Nir Dahan, thanks for the Kordemsky reference.
stanalger
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Has anyone else looked up problem #334 in Kordemsky's THE MOSCOW PROBLEMS ?
On the solution page, the destination magic square appears below the 50 move
solution. To the square's right appears another 4x4 arrangement...but it
certainly isn't magic. What is it? Why is it there?

Stan Alger
Scott Cram
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Since I first posted this question, I've settled on using Stanalger's first arrangement.

You can now see me solving the fifteen puzzle as a magic square on YouTube.
airship
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Scott. That is totally sick. And people say I'm a geek. Smile
'The central secret of conjuring is a manipulation of interest.' - Henry Hay
Scott Cram
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Thank you!

You may be A geek, but I'm shooting for the title of THE geek!
Smile Smile
stanalger
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I'm sure most of the regulars here are already aware of
Jerry Slocum's new book on the 15 Puzzle,
but here's a "heads-up" for the rest:

http://www.slocumpuzzles.com/

If you're reading this post long after July 2006, go to
your favorite book site and do a search on "15 Puzzle"
by Jerry Slocum.
Scott Cram
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It's too bad that the version in which the 15 and the 14 are switched isn't made (at least, it isn't to my knwoledge). I know the method behind the ingenious approach to solving this version, but I don't think I could have ever come up with it on my own.

Maybe someone needs to contact Thinkfun and suggest they make it.
stanalger
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Scott,

I just now got around to watching your YouTube video.
(I had to wait until I got to my office:
Old, slow computer at home. New, fast computer at work.)
Very cool! I loved the way you highlighted the
various patterns. Excellent use of the medium.

Stan
stanalger
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Quote:
On 2006-07-03 12:40, Scott Cram wrote:
It's too bad that the version in which the 15 and the 14 are switched isn't made (at least, it isn't to my knwoledge). I know the method behind the ingenious approach to solving this version, but I don't think I could have ever come up with it on my own.

Maybe someone needs to contact Thinkfun and suggest they make it.


If you could find such a version (with the 14 and 15 tiles switched), you
could finally achieve your original goal: Shift the tiles and end with a
magic square with the blank (zero) in the lower right-hand corner.

The following related Serhiy Grabarchuk puzzle CAN BE SOLVED!!!
(I've yet to find the minimum-number-of-moves solution.)

http://www.puzzleworld.org/SlidingBlockPuzzles/digits.htm

Stan
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