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/////////////////////////////////////////////////////////////////////////////// // // Benjamin Franklin's 8x8 Magic Square // // This magic square was constructed by Benjamin Franklin and first mentioned // in a letter to Peter Collinson of England about 1752. It was subsequently // published in Franklin's Experiments and Observations in Electricity // (London, 1569) // // 52 61 4 13 20 29 36 45 // 14 3 62 51 46 35 30 19 // 53 60 5 12 21 28 37 44 // 11 6 59 54 43 38 27 22 // 55 58 7 10 23 26 39 42 // 9 8 57 56 41 40 25 24 // 50 63 2 15 18 31 34 47 // 16 1 64 49 48 33 32 17 // // This square has Franklin's trademark, the bent diagonals. However, because // the main diagonals do not sum correctly, it is not a true magic square. // // The bent diagonals can be visualized as follows: // // # - - - - - - - # - - - - - - # // - # - - - - - - - # - - - - # - // - - # - - - - - - - # - - # - - // - - - # - - - - - - - # # - - - // - - - # - - - - - - - - - - - - // - - # - - - - - - - - - - - - - // - # - - - - - - - - - - - - - - // # - - - - - - - - - - - - - - - // // - - - - - - - - - - - - - - - # // - - - - - - - - - - - - - - # - // - - - - - - - - - - - - - # - - // - - - - - - - - - - - - # - - - // - - - # # - - - - - - - # - - - // - - # - - # - - - - - - - # - - // - # - - - - # - - - - - - - # - // # - - - - - - # - - - - - - - # // // // There are eight bent diagonals in each of the four different directions, // note the diagonals can wrap around. In the following table each letter // (a,b,c,d,e,f,g,h) represents one bent diagonal. // // +---+---+---+---+---+---+---+---+ // | a | b | c | d | e | f | g | h | // |---+---+---+---+---+---+---+---| // | h | a | b | c | d | e | f | g | // |---+---+---+---+---+---+---+---| // | g | h | a | b | c | d | e | f | // |---+---+---+---+---+---+---+---| // | f | g | h | a | b | c | d | e | // |---+---+---+---+---+---+---+---| // | f | g | h | a | b | c | d | e | // |---+---+---+---+---+---+---+---| // | g | h | a | b | c | d | e | f | // |---+---+---+---+---+---+---+---| // | h | a | b | c | d | e | f | g | // |---+---+---+---+---+---+---+---| // | a | b | c | d | e | f | g | h | // +---+---+---+---+---+---+---+---+ // // The other three bent diagonal directions can be obtained by rotating the // table. // // As any web search can show, there are various interpretations of the magic // square rules. We summarize them as follows: // // 1. All half rows and half columns sum to 130. (32 constraints) // 2. The four entries in all 2x2 subsquares sum to 130. This includes wrap // around cases. (64 constraints) // 3. All bent diagonals sum to 260. (32 constraints) // // All in all there are 128 distinct constraints on 64 variables. // // In 2006 in the article published online by the Proceedings of the Royal // Society "Enumerating the bent diagonal squares of Dr Benjamin Franklin FRS" // by Daniel Schindel, Matthew Rempel and Peter Loly // the authors showed there are 1,105,920 variations of his magic square. // (http://www.physics.umanitoba.ca/news/loly_paper.pdf) // // The following F1 program indeed found exactly 1,105,920 solutions. // /////////////////////////////////////////////////////////////////////////////// // // To execute the program, run the query // // all Franklin8x8(a) // /////////////////////////////////////////////////////////////////////////////// // // NOTES: // Although the total number of solutions and individual solutions remain the // same, the execution time and order of various individual solutions // depends significantly on the order of the constraints. In other words, if // you rearrange the same constraints in different order, you will get results // in different order (and in different time). Note that the results are not // generated evenly in time, they come in "bunches". So if you get 300 results // in the very first minute, don't expect 30000 in 100 minutes. To generate // all solutions, the program can run a long time. It took about four weeks // to find all solutions, albeit as an idle low priority background task on // a fairly slow machine. // // You can find the actual results (25MB zipped text file) at // // www.f1compiler.com/samples/FRANKLIN8X8.zip // // /////////////////////////////////////////////////////////////////////////////// Square8x8 = [0..63]->>L[1..64] Sum :< I = 260 Sum2 :< I = Sum/2 pred Franklin8x8(a::Square8x8) iff // An array of 64 variables of an 8x8 square a = [ a1, a2, a3, a4, a5, a6, a7, a8, b1, b2, b3, b4, b5, b6, b7, b8, c1, c2, c3, c4, c5, c6, c7, c8, d1, d2, d3, d4, d5, d6, d7, d8, e1, e2, e3, e4, e5, e6, e7, e8, f1, f2, f3, f4, f5, f6, f7, f8, g1, g2, g3, g4, g5, g6, g7, g8, h1, h2, h3, h4, h5, h6, h7, h8] & // Constraints to to remove symmetries. These have nothing to do // with the Benjamin Franklin's constraints. a1 > a8 & a1 > h1 & a1 > h8 & a8 > h1 & // The actual Benjamin Franklin's constraints for 8x8 Magic Square // Half rows and half columns (32 constraints) a1 + a2 + a3 + a4 = Sum2 & b1 + b2 + b3 + b4 = Sum2 & c1 + c2 + c3 + c4 = Sum2 & d1 + d2 + d3 + d4 = Sum2 & a1 + b1 + c1 + d1 = Sum2 & a2 + b2 + c2 + d2 = Sum2 & a3 + b3 + c3 + d3 = Sum2 & a4 + b4 + c4 + d4 = Sum2 & a5 + a6 + a7 + a8 = Sum2 & b5 + b6 + b7 + b8 = Sum2 & c5 + c6 + c7 + c8 = Sum2 & d5 + d6 + d7 + d8 = Sum2 & a5 + b5 + c5 + d5 = Sum2 & a6 + b6 + c6 + d6 = Sum2 & a7 + b7 + c7 + d7 = Sum2 & a8 + b8 + c8 + d8 = Sum2 & e1 + f1 + g1 + h1 = Sum2 & e2 + f2 + g2 + h2 = Sum2 & e3 + f3 + g3 + h3 = Sum2 & e4 + f4 + g4 + h4 = Sum2 & e1 + e2 + e3 + e4 = Sum2 & f1 + f2 + f3 + f4 = Sum2 & g1 + g2 + g3 + g4 = Sum2 & h1 + h2 + h3 + h4 = Sum2 & e5 + e6 + e7 + e8 = Sum2 & f5 + f6 + f7 + f8 = Sum2 & g5 + g6 + g7 + g8 = Sum2 & h5 + h6 + h7 + h8 = Sum2 & e5 + f5 + g5 + h5 = Sum2 & e6 + f6 + g6 + h6 = Sum2 & e7 + f7 + g7 + h7 = Sum2 & e8 + f8 + g8 + h8 = Sum2 & // Bent Diagonals (8 constraints for each directions) // 8 7 6 5 4 3 2 1 // 7 6 5 4 3 2 1 8 // 6 5 4 3 2 1 8 7 // 5 4 3 2 1 8 7 6 // 5 4 3 2 1 8 7 6 // 6 5 4 3 2 1 8 7 // 7 6 5 4 3 2 1 8 // 1 7 6 5 4 3 2 1 // < a8 + b7 + c6 + d5 + e5 + f6 + g7 + h8 = Sum & a7 + b6 + c5 + d4 + e4 + f5 + g6 + h7 = Sum & a6 + b5 + c4 + d3 + e3 + f4 + g5 + h6 = Sum & a5 + b4 + c3 + d2 + e2 + f3 + g4 + h5 = Sum & a4 + b3 + c2 + d1 + e1 + f2 + g3 + h4 = Sum & a3 + b2 + c1 + d8 + e8 + f1 + g2 + h3 = Sum & a2 + b1 + c8 + d7 + e7 + f8 + g1 + h2 = Sum & a1 + b8 + c7 + d6 + e6 + f7 + g8 + h1 = Sum & // The four entries in every 2x2 subsquare sum to 130. // 8x8 = 64 constraints a1 + a2 + b1 + b2 = Sum2 & a2 + a3 + b2 + b3 = Sum2 & a3 + a4 + b3 + b4 = Sum2 & a4 + a5 + b4 + b5 = Sum2 & a5 + a6 + b5 + b6 = Sum2 & a6 + a7 + b6 + b7 = Sum2 & a7 + a8 + b7 + b8 = Sum2 & a8 + a1 + b8 + b1 = Sum2 & // wrap b1 + b2 + c1 + c2 = Sum2 & b2 + b3 + c2 + c3 = Sum2 & b3 + b4 + c3 + c4 = Sum2 & b4 + b5 + c4 + c5 = Sum2 & b5 + b6 + c5 + c6 = Sum2 & b6 + b7 + c6 + c7 = Sum2 & b7 + b8 + c7 + c8 = Sum2 & b8 + b1 + c8 + c1 = Sum2 & // wrap c1 + c2 + d1 + d2 = Sum2 & c2 + c3 + d2 + d3 = Sum2 & c3 + c4 + d3 + d4 = Sum2 & c4 + c5 + d4 + d5 = Sum2 & c5 + c6 + d5 + d6 = Sum2 & c6 + c7 + d6 + d7 = Sum2 & c7 + c8 + d7 + d8 = Sum2 & c8 + c1 + d8 + d1 = Sum2 & // wrap d1 + d2 + e1 + e2 = Sum2 & d2 + d3 + e2 + e3 = Sum2 & d3 + d4 + e3 + e4 = Sum2 & d4 + d5 + e4 + e5 = Sum2 & d5 + d6 + e5 + e6 = Sum2 & d6 + d7 + e6 + e7 = Sum2 & d7 + d8 + e7 + e8 = Sum2 & d8 + d1 + e8 + e1 = Sum2 & // wrap e1 + e2 + f1 + f2 = Sum2 & e2 + e3 + f2 + f3 = Sum2 & e3 + e4 + f3 + f4 = Sum2 & e4 + e5 + f4 + f5 = Sum2 & e5 + e6 + f5 + f6 = Sum2 & e6 + e7 + f6 + f7 = Sum2 & e7 + e8 + f7 + f8 = Sum2 & e8 + e1 + f8 + f1 = Sum2 & // wrap f1 + f2 + g1 + g2 = Sum2 & f2 + f3 + g2 + g3 = Sum2 & f3 + f4 + g3 + g4 = Sum2 & f4 + f5 + g4 + g5 = Sum2 & f5 + f6 + g5 + g6 = Sum2 & f6 + f7 + g6 + g7 = Sum2 & f7 + f8 + g7 + g8 = Sum2 & f8 + f1 + g8 + g1 = Sum2 & // wrap g1 + g2 + h1 + h2 = Sum2 & g2 + g3 + h2 + h3 = Sum2 & g3 + g4 + h3 + h4 = Sum2 & g4 + g5 + h4 + h5 = Sum2 & g5 + g6 + h5 + h6 = Sum2 & g6 + g7 + h6 + h7 = Sum2 & g7 + g8 + h7 + h8 = Sum2 & g8 + g1 + h8 + h1 = Sum2 & // wrap h1 + h2 + a1 + a2 = Sum2 & h2 + h3 + a2 + a3 = Sum2 & h3 + h4 + a3 + a4 = Sum2 & h4 + h5 + a4 + a5 = Sum2 & h5 + h6 + a5 + a6 = Sum2 & h6 + h7 + a6 + a7 = Sum2 & h7 + h8 + a7 + a8 = Sum2 & h8 + h1 + a8 + a1 = Sum2 & // wrap // Bent Diagonals (8 constraints for each directions) // 1 8 7 6 6 7 8 1 // 2 1 8 7 7 8 1 2 // 3 2 1 8 8 1 2 3 // 4 3 2 1 1 2 3 4 // 5 4 3 2 2 3 4 5 // 6 5 4 3 3 4 5 6 // 7 6 5 4 4 5 6 7 // 8 7 6 5 5 6 7 8 // \/ a1 + b2 + c3 + d4 + d5 + c6 + b7 + a8 = Sum & b1 + c2 + d3 + e4 + e5 + d6 + c7 + b8 = Sum & c1 + d2 + e3 + f4 + f5 + e6 + d7 + c8 = Sum & d1 + e2 + f3 + g4 + g5 + f6 + e7 + d8 = Sum & e1 + f2 + g3 + h4 + h5 + g6 + f7 + e8 = Sum & f1 + g2 + h3 + a4 + a5 + h6 + g7 + f8 = Sum & g1 + h2 + a3 + b4 + b5 + a6 + h7 + g8 = Sum & h1 + a2 + b3 + c4 + c5 + b6 + a7 + h8 = Sum & // Bent Diagonals (8 constraints for each directions) // 8 7 6 5 5 6 7 8 // 7 6 5 4 4 5 6 7 // 6 5 4 3 3 4 5 6 // 5 4 3 2 2 3 4 5 // 4 3 2 1 1 2 3 4 // 3 2 1 8 8 1 2 3 // 2 1 8 7 7 8 1 2 // 1 8 7 6 6 7 8 1 // /\ h1 + g2 + f3 + e4 + e5 + f6 + g7 + h8 = Sum & g1 + f2 + e3 + d4 + d5 + e6 + f7 + g8 = Sum & f1 + e2 + d3 + c4 + c5 + d6 + e7 + f8 = Sum & e1 + d2 + c3 + b4 + b5 + c6 + d7 + e8 = Sum & d1 + c2 + b3 + a4 + a5 + b6 + c7 + d8 = Sum & c1 + b2 + a3 + h4 + h5 + a6 + b7 + c8 = Sum & b1 + a2 + h3 + g4 + g5 + h6 + a7 + b8 = Sum & a1 + h2 + g3 + f4 + f5 + g6 + h7 + a8 = Sum & // Bent Diagonals (8 constraints for each directions) // 1 2 3 4 5 6 7 8 // 8 1 2 3 4 5 6 7 // 7 8 1 2 3 4 5 6 // 6 7 8 1 2 3 4 5 // 6 7 8 1 2 3 4 5 // 7 8 1 2 3 4 5 6 // 8 1 2 3 4 5 6 7 // 1 2 3 4 5 6 7 8 // > a1 + b2 + c3 + d4 + e4 + f3 + g2 + h1 = Sum & a2 + b3 + c4 + d5 + e5 + f4 + g3 + h2 = Sum & a3 + b4 + c5 + d6 + e6 + f5 + g4 + h3 = Sum & a4 + b5 + c6 + d7 + e7 + f6 + g5 + h4 = Sum & a5 + b6 + c7 + d8 + e8 + f7 + g6 + h5 = Sum & a6 + b7 + c8 + d1 + e1 + f8 + g7 + h6 = Sum & a7 + b8 + c1 + d2 + e2 + f1 + g8 + h7 = Sum & a8 + b1 + c2 + d3 + e3 + f2 + g1 + h8 = Sum
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