Contributions came from Christian Boyer and Giovanni Resta. J. C. Rosa
has sent a contribution to the Puzzle 79,
close related to this puzzle, so it would be a good idea to go there.
***
Boyer has sent the minimal 4x4 magic square of squares. As a
matter of fact this square was found by other person in 2001 and Boyer has
verified that it certainly is the minimal one. But I will not publish it in
order to encourage the readers of these pages to re-discover it.
Boyer has confirmed also that his prime magic square of squares
shown above, is in fact the minimal one and has sent another one that he
thinks is the second smallest.
23² 353² 761² 179²
641² 331² 263² 383²
409² 691² 283² 107²
397² 157² 89² 739²
Magic sum = 736300.
He has also sent the smallest solution to the case 5x5, for the question
2. Again I will not publish it yet until other puzzler send one more
solution.
Boyer announces that he "will let to other people the pleasure
to find examples for k>=6."
***
This 'other people' seems to be - by the moment Giovanni Resta,
who has confirmed the two 4x4 prime magic squares of squares already sent by
Boyer are in fact the smallest and the second smallest (which means that GR
sent the same second smallest than CB) and sent also the third smallest:
701 571 79 193
269 317 311 769
67 659 479 439
541 11 727 199
Magical sum = 860932
Giovanni says" I'm in the process to find the minimal 5 x 5,
but I've not the result yet".
***
Christian Boyer wrote (July 2005):
Good news. My article "Some notes on the magic squares of squares
problem"
is now published in the new issue of The Mathematical Intelligencer.
I was happy to quote your name, pages 58 and 64, with a reference to your
puzzles 79, 287 and 288. And the names of Luke Pebody and Jean-Claude Rosa
are also quoted for their proofs of the impossible 4x4 bimagic square
(puzzle 287).
I have created a new page "Magic squares of squares" in my web site
www.multimagie.com/indexengl.htm And there is an unpublished
supplement to my article, available from this new page: because you will
see that the square CB18 is the 5x5 solution of your puzzle 288, I think
that you can now update your puzzle with this solution (I do not know if
Giovanni has finished his "process to find the minimal 5 x 5").
See
also this
page by Boyer
|
11² |
23² |
53² |
139² |
107² |
|
13² |
103² |
149² |
31² |
17² |
|
71² |
137² |
47² |
67² |
61² |
|
113² |
59² |
41² |
97² |
83² |
|
127² |
29² |
73² |
7² |
109² |
CB18. The smallest 5×5 magic square of squares of prime numbers, S2 =
34229.
Latest news below: I have constructed 10 days ago the first known 6x6 and
7x7 magic squares of squares.
4x4 and 5x5 were already known (since 1770 and 2004 resp.). And the
well-known bimagic squares of order 8 and + (since 1890) are magic squares
of squares.
It means that the main remaining problem is the VERY difficult problem of
3x3 magic squares of squares!
Best regards.
Christian.
-------------------6x6
If I am right, 6x6 magic squares of squares using squared consecutive
integers (0² to 35², or 1² to 36²) are impossible.
My 6x6 magic square of squares is NOT using squared consecutive
integers...
but it is interesting to see the used numbers:
0² to 36² ONLY EXCLUDING 30².
It is impossible to construct a 6x6 magic square of squares with a smaller
magic sum. But it is possible to construct other samples with the same
magic sum S2 = 2551.
2² 1² 36² 5² 0² 35²
6² 33² 20² 29² 4² 13²
25² 7² 14² 24² 31² 12²
21² 32² 11² 15² 22² 16²
34² 18² 23² 10² 19² 9²
17² 8² 3² 28² 27² 26²
I have added a funny supplemental feature in this sample: the 3 smallest
used integers (0, 1, 2) and the 2 biggest (35, 36) are used together in
the first row.
-------------------7x7
If I am right, the smallest order allowing magic squares of squares using
squared CONSECUTIVE integers seems to be the order 7.
An indirect consequence: the impossibility of 7x7 bimagic squares was not
coming from a problem with squared numbers!
Here is a sample using integers 0² to 48², magic sum S2=5432.
25² 45² 15² 14² 44² 5² 20²
16² 10² 22² 6² 46² 26² 42²
48² 9² 18² 41² 27² 13² 12²
34² 37² 31² 33² 0² 29² 4²
19² 7² 35² 30² 1² 36² 40²
21² 32² 2² 39² 23² 43² 8²
17² 28² 47² 3² 11² 24² 38²
I have added a funny supplemental feature in this sample: the 7 rows are
magic (S1=168) when the integers are not squared.
***
