Comments on: Perplex City and Magic Squares

By: Rick

Rick — Wed, 09 May 2012 20:59:50 +0000

There was a third solution you didnt identify… that I did for 4×4.. .each equaling 34

By: Chris Cowan

Chris Cowan — Mon, 23 Apr 2012 12:26:42 +0000

i found a one that you have listed as unsovibaule 3 17 16
14 9 11
10 12 7
5 4 18

By: Tim Dunn

Tim Dunn — Fri, 20 Apr 2012 22:08:53 +0000

Hi there,
Interesting stuff. I wrote an Excel spreadsheet that finds a solution to 4×4 (not always the same solution)
It uses an evolutionary approach. Start with random arrangement. Generate 10 offspring by swapping two random locations. Calculate the ‘goodness’ of each offspring (sum of differences in all row col diag sums). Use the best offspring as the parent for the next generation. I find that it get a solution in less than 100 generations if it ever gets one. Sometimes it gets stuck in local minima from which there is no escape. To get round this I simply set a limit (150) on generations before reverting to the original arrangement and re-starting.
One interesting thing is that it was not told that the total had to be 34, only that all the totals had to match.
Obviously it would be saner to code it in C or something rather than Excel but at the time I had no access to a compiler.

By: Boris

Boris — Wed, 08 Feb 2012 17:33:24 +0000

I am interesting only in those magic squares that to addition to sums in rows, columns and diagonales also have the same amount in all 4 quarters. One of possible solutions for that is the following one:

14 3 5 12
8 9 15 2
11 6 4 13
1 16 10 7

By the way, it was missing in your 924 solutions (giving that they all are shown in order of the first two numbers).

By: Leo

Leo — Thu, 08 Sep 2011 02:07:19 +0000

I found a new magic square in my homework.
1 14 8 11
15 4 10 5
12 7 13 2
6 9 3 16

By: gm

gm — Sat, 14 May 2011 10:06:54 +0000

Hey guys
Have you been able to solve any EVEN number magic squares?
eg: 6×6, 8×8, etc
There are more than one solutions as for 4×4. Please have a go.
gm

By: SÃ¸ren Blaabjerg

SÃ¸ren Blaabjerg — Thu, 13 Aug 2009 19:22:44 +0000

Dear Danny

I have studied your list a bit further and found quite a number of omissions and duplicates
as well. In the end I decided to write a small computer program myself in order to find the complete list, and I have just succeeded doing so. As expected the conclusion was, that there are indeed all in all 880 unique solutions or 880 x 8 solutions if you count all rotations and reflexions. I made the program in such a way, that you can choose either and step through all the solutions displayed one by one.

Most of the solutions are not particularly interesting though. For instance I find those, where both the 2 x 2 subsquares at the corners and the subsquare at the center add up to the same sum as the rows, columns and two diagonals particularly pleasing. There are quite a number of those, around 350 unique ones I believe.

By: Lawid

Lawid — Mon, 10 Aug 2009 11:02:54 +0000

Hi,

I have set of 100 integers organized in 10*10 table. Only the sums of two rows are different by +1 and -1 compared to the sum required for the magic square. Can you help me in checking if we can get with these numbers a magic square table?

Thanks

By: SÃ¸ren Blaabjerg

SÃ¸ren Blaabjerg — Sat, 01 Aug 2009 10:49:18 +0000

Dear Danny

I have just studied your list further and can confirm, that your list is far from comprehensive and indeed also contain quite a number of duplications (i.e. solutions that can be derived through reflexion and/or rotation of other solutions on the list). I you are interested, I might myself have a look on your code :-)

By: Danny Dawson

Danny Dawson — Wed, 29 Jul 2009 17:05:45 +0000

The 924 solutions I listed are indeed unique, but not rotationally or reflexively so.

Also, as several commenters have pointed out, it seems that my list is not comprehensive. I haven’t taken the time yet to look back at my old code and determine why this is so, but at some point, I certainly should do so.

Thanks to all of you for your comments, and especially to SÃ¸ren for continuing to work on this puzzle. Even though I haven’t done any work on this myself in a long time, it’s still quite fun to think about. :)