Problems & Puzzles: Puzzles

 Puzzle 345. Magic square of cubic primes Here you are invited to discover: Q1. A magic square composed exclusively by distinct cubic prime numbers. This puzzle is just is the open problem #6 of the article “Some notes on the magic squares of squares problem”, by Christian Boyer.As a special gift for the readers of my pages Christian Boyer has written a special page in his site that will save to me a lot of words in order to explain all the theoretical elements involved in this puzzle (Thanks so much CB!...). The most important thing that I immediately rescue from his page is that a solution for this puzzle has been shown to be impossible for squares nxn of order n=3. So you should try for n>3.Boyer, who has became a very skilled expert on the subject, on the same page linked above adds: "I think that 4x4 are also impossible with distinct positive integers". A minor solution accepted is this one: Q2. A semi-magic square composed exclusively by distinct cubic prime numbers (not taking care of the diagonals)

Christian Boyer wrote (on December 26, 2005):

Thanks to have asked the puzzle 345.

Because it is a difficult puzzle, you may add the 2 less difficult questions authorizing negative integers (= -1×prime), as I proposed in my previous email:

Q3. Magic square using cubes of primes and cubes of (-1×primes), with authorization of a (p3, –p3) trick similar to CB10 and CB11 squares showed at http://www.multimagie.com/English/SquaresOfCubes.htm

Q4. Magic square using cubes of primes and cubes of (-1×primes), but above trick forbidden, all primes completely different

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