From: John Conway Date: Wed, 2 Aug 95 10:25:27 EDT Subject: Re: Magic Cubes If 1 is subtracted from the numbers in Hellenius's cubes (I only checked this for the first one, but it'll be true for all four), we get a "Greco-Latino-Sanskrit" square - in other words, each row of numbers (when written in base 3) has each of the three digits 0,1,2 occurring once in each of the three possible places (ones, threes, nines). Richard Guy and I have just checked that there is just one such square, up to permutation of coordinates, slices, and ternary digits, namely 000 211 122 112 020 201 221 102 010 121 002 210 200 111 022 012 220 101 212 120 001 021 202 110 100 011 222 If each coordinate takes the values 0,1,2, the entry in the xyz place is x.211 + y.121 + z.112, (the coordinates being added modulo 3 without carry), so that the whole thing is linear. If care is taken to keep the middle number (111) in the middle place, then the four body-diagonals will also be magic (which is the definition I've always heard for "magic cube"). All four of the Hellenius cubes can be found from this by suitably interpreting the digits, which can be done independently in each place - for instance one can swap 0 and 2 in the units place only if one likes. John Conway --------------------------------------------------------------------- Date: 02 Aug 95 04:38:47 EDT From: "Fred W. Helenius" <75240.125@compuserve.com> Subject: Re: Magic Cubes Yes, there are 3x3x3 orthogonal-magic-only cubes. Here are four: 1 14 27 17 21 4 24 7 11 15 25 2 19 5 18 8 12 22 26 3 13 6 16 20 10 23 9 1 14 27 18 19 5 23 9 10 15 25 2 20 6 16 7 11 24 26 3 13 4 17 21 12 22 8 1 14 27 18 19 5 23 9 10 17 21 4 22 8 12 3 13 26 24 7 11 2 15 25 16 20 6 1 15 26 18 20 4 23 7 12 17 19 6 22 9 11 3 14 25 24 8 10 2 13 27 16 21 5 Unless I made some programming or copying error, these are all of the solutions, up to shuffling the slices and renaming the axes. That makes 4*1296 = 5184 solutions in all. There are interesting symmetries among the four solutions. Every number appears in combination with only four other pairs of values (taking 12 as an example, (4, 26), (5,25), (7,23), (8, 22)), and each solution contains a different choice of three of the pairs. Notice also that the values 3n+1, 3n+2, 3n+3 are always coplanar.