Some words about the method used, by Walter and me, for our perfect magic cube of order 5. We have first computed a large amount of auxiliary cubes of order 3. These auxiliary cubes are central "symmetrical", meaning that all their 13 alignments of 3 numbers going through the center have x+63+y = 189. These auxiliary cubes have some other partial magic characteristics: 29/49 alignments (including 14/18 diagonals and 4/4 triagonals) have already the same sum = 189. You can check these characteristics, looking at the central cube of order 3 included in our magic cube of order 5. And after that, using these auxiliary cubes, we have tried to fill always by computer the 5^3 - 3^3 = 98 missing numbers, mainly using complementary numbers x+189+y = 315. That's why you can seen a lot of various symmetries in our cube. Running several computers during several weeks before to find the first solution. With this method, it is very easy to find cubes of order 5 with 26 magic diagonals. And more than 1500 cubes with 28 magic diagonals were found before to find our first cube with 30 magic diagonals. We used more than 80,000 different auxiliary cubes of order 3 before to find the first solution. The method used previously by Walter for his cube of order 6 is similar, using a central auxiliary cube of order 4. Christian Boyer. www.multimagie.com/indexengl.htm

-----Message d'origine----- De: Richard Schroeppel [mailto:rcs@CS.Arizona.EDU] Date: lundi 17 novembre 2003 21:24 À: cboyer@club-internet.fr Objet: magic cubes

Feel free to send more details about the construction/search methods for the 5^3 and 6^3 to the math-fun list.

Rich Schroeppel rcs@cs.arizona.edu