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The Magic Cafe Forum Index » » Magical equations » » Magic Square on Fifteen Puzzle? (0 Likes) Printer Friendly Version

Scott Cram
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Is it possible to create a magic square on the classic 15 puzzle (assuming the empty space is used as "0"), without swapping any pieces? If so, what is the arrangement?
ddyment
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Sure. Here's one solution:

.8 .5 14 .3
15 .2 .9 .4
.1 12 .7 10
.6 11 .0 13

This has the advantage that it's also a so-called "perfect square", which is only possible for 25% of "magic numbers" (in this case 30). That is, it sums to the magic number in 52 different symmetrical ways, much better than the more common 26 combinations that are true for typical magic squares.

... Doug
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Scott Cram
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Thank you, Doug! I was able to get it into that configuration with little problem!
stanalger
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Quote:
On 2005-05-01 18:50, ddyment wrote:
Sure. Here's one solution:

.8 .5 14 .3
15 .2 .9 .4
.1 12 .7 10
.6 11 .0 13

This has the advantage that it's also a so-called "perfect square", which is only possible for 25% of "magic numbers" (in this case 30). That is, it sums to the magic number in 52 different symmetrical ways, much better than the more common 26 combinations that are true for typical magic squares.

... Doug



Doug,

If "perfect magic square" means "sums to the magic number in 52 different symmetrical
ways," then you can do much better than 25%! Perfect squares can be constructed
for a full 50% of possible target numbers.

Stan Alger
ddyment
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Stan claimed:
Quote:
Perfect squares can be constructed for a full 50% of possible target numbers.

I have given (above) a perfect square for the value 30; how would one construct a perfect square for 31, 32, or 33?

... Doug
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stanalger
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Doug,

You can construct a "perfect" square for any EVEN target number.

Here's one for 32:

06... 01... 16... 09
12... 13... 02... 05
00... 07... 10... 15
14... 11... 04... 03

(Leading zeroes are included only to improve spacing.)

You can check to see that all 52 symmetric patterns that total 30 in your
square total 32 in my square.

And no repeated numbers!!!!!!
If repeats were allowed, the square
8 8 8 8
8 8 8 8
8 8 8 8
8 8 8 8
would work.
OK, so that's an EXTREME example....but if you're going to allow SOME
repeated numbers, where do you draw the line? 4x4 magic squares should
consist of 16 DIFFERENT numbers.


Stan Alger
stanalger
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But if you don't mind repeated numbers, or four numbers that are significantly
larger/smaller than the other twelve numbers, I present the
Stan Alger simplified (and NOT perfect) magic square:

000...000...key...000
key...000...000...000
000...key...000...000
000...000...000...key

I can teach this to anyone in less than an hour!

Stan Alger
ddyment
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Stan is correct in his comments. I was referring to the more formal (i.e., mathematical) definition of magic squares, which holds that the numbers must be not only unique, but also sequential. I assumed that this was implied by the nature of the topic, but I should have been more specific.

Technically, in fact, in order for a "15 puzzle" square to be a true "order 4" magic square, the blank should be treated as "16", not as zero. These are all technical subtleties that are of little consequence to the magician, though.

Lots of definitional issues arise in the formal discussion of magic squares. The type of square I referenced is actually called a most-perfect square. The simpler perfect square (more correctly, Nasik square, though they are also known as diabolic/diabolique and pandiagonal) actually has more limited properties: not all the 2x2 sub-squares add to the magic number.

As a related aside (and one that might be of more interest to the magical community), I will suggest that when presenting a magic square to an audience, one should never refer to it as such. Point out its many amazing properties, to be sure, but don't give it a label. It makes it too easy for someone to spend a few minutes with Google and discover that you're not as clever as they originally thought you to be!

... Doug
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stanalger
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Quote:
On 2005-05-02 14:29, ddyment wrote:
Technically, in fact, in order for a "15 puzzle" square to be a true "order 4" magic square, the blank should be treated as "16", not as zero. ... Doug


Doug,

Are the numbers 0-15 not sequential? Are you saying there is only ONE possible
target number for a 4x4 magic square, namely 34?

Stan Alger
stanalger
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Quote:
On 2005-05-02 14:29, ddyment wrote:
Stan is correct in his comments. I was referring to the more formal (i.e., mathematical) definition of magic squares, which holds that the numbers must be not only unique, but also sequential. I assumed that this was implied by the nature of the topic, but I should have been more specific.

Technically, in fact, in order for a "15 puzzle" square to be a true "order 4" magic square, the blank should be treated as "16", not as zero. These are all technical subtleties that are of little consequence to the magician, though.

... Doug


Someone should tell Berlekamp, Conway, and Guy that a 4x4 magic square should
contain the numbers 1-16. They use 0-15 in their recreational mathematics
classic, WINNING WAYS FOR YOUR MATHEMATICAL PLAYS.

"I assumed this was implied by the nature of the topic"????
Really? In a magic forum? How many magicians who do the "magic square" use
sequential numbers? I think most use four key squares, EVEN when the target
number is congruent to 2 mod 4!!!

Very few realize that it's possible to construct a square that totals the
target number in 52 different ways FOR ANY EVEN TARGET. If the even target
is not divisible by four, you can use sequential numbers. Otherwise, you
need only skip one number. A square containing the numbers 16-23 and 25-32
(i.e. 16-32 without the 24) LOOKS much nicer than a square containing
1-12 and 75-78. Those numbers in the 70s stick out like a sore thumb.

And with 52 cards in a deck (and 52 weeks in a year)....


Stan Alger
ddyment
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Stan wrote:
Quote:
Are the numbers 0-15 not sequential? Are you saying there is only ONE possible target number for a 4x4 magic square, namely 34?


No, I said what I said, which is that a true "order 4" magic square (as distinct from a generic 4x4 magic square), must contain the numbers 1 through 16. There are countless 4x4 magic squares, but (ignoring reflections and rotations) there are only 880 order 4 magic squares. By way of contrast, there is only one order 3 magic square, and there are 275,305,224 order 5 magic squares. I did note that these are technical subtleties that are of little consequence to the magician.


Quote:
Someone should tell Berlekamp, Conway, and Guy that a 4x4 magic square should contain the numbers 1-16...

Why? This appears nonsensical to me.


Quote:
"I assumed this was implied by the nature of the topic"????
Really? In a magic forum? How many magicians who do the "magic square" use sequential numbers?

The title of this topic is "Magic Square on Fifteen Puzzle?"; the tiles in said puzzle are sequential numbers from one through fifteen.


Quote:
Very few realize that it's possible to construct a square that totals the target number in 52 different ways FOR ANY EVEN TARGET.

I suspect that most people who know much about magic squares are aware of this; it's just that many of them find it difficult to construct such squares on the fly, and the more common 26 ways of reaching the magic total not only work for all targets, but are sufficient for the majority of entertainment goals (actually, most just point out 24 of those combinations, given that the remaining pair does not exhibit the same symmetry as the others).
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stanalger
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Quote:
On 2005-05-05 14:37, ddyment wrote:
Stan wrote:
Quote:
Are the numbers 0-15 not sequential? Are you saying there is only ONE possible target number for a 4x4 magic square, namely 34?


No, I said what I said, which is that a true "order 4" magic square (as distinct from a generic 4x4 magic square), must contain the numbers 1 through 16. There are countless 4x4 magic squares, but (ignoring reflections and rotations) there are only 880 order 4 magic squares. By way of contrast, there is only one order 3 magic square, and there are 275,305,224 order 5 magic squares. I did note that these are technical subtleties that are of little consequence to the magician.


I don't understand the relevance of these figures. You get 880
order 4 squares regardless of whether you are using 0-15 or 1-16.

Quote:
Quote:
Someone should tell Berlekamp, Conway, and Guy that a 4x4 magic square should contain the numbers 1-16...

Why? This appears nonsensical to me.


Agreed.

Quote:
Quote:
"I assumed this was implied by the nature of the topic"????
Really? In a magic forum? How many magicians who do the "magic square" use sequential numbers?

The title of this topic is "Magic Square on Fifteen Puzzle?"; the tiles in said puzzle are sequential numbers from one through fifteen.


And you can extend the sequence in either direction, can't you?
There's no sixteen square in the Fifteen Puzzle. Just seems more
natural to let the blank represent zero rather than sixteen.

Quote:
Quote:
Very few realize that it's possible to construct a square that totals the target number in 52 different ways FOR ANY EVEN TARGET.

I suspect that most people who know much about magic squares are aware of this; it's just that many of them find it difficult to construct such squares on the fly, and the more common 26 ways of reaching the magic total not only work for all targets, but are sufficient for the majority of entertainment goals (actually, most just point out 24 of those combinations, given that the remaining pair does not exhibit the same symmetry as the others).

Difficult to construct such squares? Not at all. The technique
discussed in the Sam Dalal book is very easy to use. And same
technique does work for all targets, with the added bonus
that you automatically get a "perfect" square 50% of the time!
(I'm using "perfect" here in the sense that I thought you were using it in message #2 of this thread, i.e."it sums to the magic number in 52 different symmetrical ways.")

With just a bit of cleverness, you can ensure that you will get an
even target number from the spectator. Reread message #2 in this thread.
"Much better", indeed!

Ken Weber urges us to "Never settle for good enough. Sweat the details. Raise your level."

Stan Alger
ddyment
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Stan wrote:
Quote:
I don't understand the relevance of these figures.

This is clear; Stan's confusion seems to arise from a conceptual misunderstanding of an order 4 square, which he consistently (and unsurprisingly) interprets as referring to any 4x4 square. Mathematicians have very specific meanings for such terms, for good reasons, but (as I have already said twice, and will repeat here for the last time, as I am done with this particular topic) these are technical subtleties that are of little consequence to the magician).

He has also chosen to ignore the fact that this particular topic is about squares that can be constructed with the "Fifteen Puzzle", not squares in general.

Anyone interested in this subject (the mathematical one, not the entertainment one) would probably enjoy Clifford Pickover's The Zen of Magic Squares, Circles, and Stars, Princeton University Press (2003), 432 pages. Subtitled "An Exhibition of Surprising Structures across Dimensions", this is a contemporary and comprehensive book on current magic square topics and ideas, by an accomplished (and popular) mathematical author.
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drkptrs1975
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Just use the one with 16, but saubract 1 from each square.
stanalger
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Hey, check out the "order-4" magic square on page 278 of the Pickover book.
What were you saying, Doug?

Stan
Scott Cram
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Stanalger provided the arrangement I currently use for the magic square on the 15 puzzle.

If you want to see me perform the 15 puzzle magic square, it's now available on YouTube.
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