Problems & Puzzles: Puzzles

Puzzle 412. Semimagic square of cubes

This will be the third in a row puzzle related to magic squares. After this we will switch of subject.

This is the turn of a nice puzzle sent by Christian Boyer, even if this is not particularly related to primes:

A 3x3 magic square of cubes is proved impossible. (www.multimagie.com/English/SquaresOfCubes.htm), but it is still unknown if a 3x3 semi-magic square of cubes is possible or impossible.

Reminder:

-a magic square has its rows, columns and 2 diagonals having the same sum
-a semi-magic square has its rows and columns having the same sum

Here is my sample of a 3x3 semi-magic square using “only” 8 cubes out of 9:

 Magic sum = 241,801,435 513 6193 1653 6183 1623 1153 1783 723 235,788,435

In October 2006, Frank Rubin searched a 3x3 semi-magic square of cubes, and concluded –if his computing is right- that there is no solution using 9 numbers all smaller than 300,000^3.

Q1. Construct a 3x3 semi-magic square of 9 distinct cubes, or prove that it is impossible. (Note: please send your best 'almost solutions' to Q1)

Christian Boyer wrote on Nov. 07:

The November 2007 update of www.multimagie.com/indexengl.htm is now online, with a lot of news. You will see that your Puzzle 412 is mentioned. And you will also see the first known solution of the Open problem #6 (using prime numbers) recently found by J. Wroblewski & H. Pfoertner.

If you want to link directly to this nice result on prime numbers: www.multimagie.com/English/WroblewskiPfoertner42&44.htm.

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