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///////////////////////////////////////////////////////////////////////////////
//
// 1-----10-----19
// | 2     11    |20
// |   3-----12-----21
// 4   | 13     22   |
// | 5 |   14    |23 |
// |   6     15  |  24
// 7---|-16-----25   |
//   8 |   17     26 |
//     9-----18-----27
//
///////////////////////////////////////////////////////////////////////////////
//
// To generate solutions, run the query 
//
//      all MagicCube3x3x3()
//
///////////////////////////////////////////////////////////////////////////////
//
// Solutions:
//     
//  6 26 10       16  3 23       20 13  9
// 17  1 24       21 14  7        4 27 11
// 19 15  8        5 25 12       18  2 22   
//
// ___ Solution: 1 ___ [00:00:00] __ [Backtracks: 263] ____
//
//  4 26 12       17  3 22       21 13  8
// 18  1 23       19 14  9        5 27 10
// 20 15  7        6 25 11       16  2 24
//
// ___ Solution: 2 ___ [00:00:00] __ [Backtracks: 435] ____
// 
//  2 24 16       15  7 20       25 11  6
// 18  1 23       19 14  9        5 27 10
// 22 17  3        8 21 13       12  4 26
//
// ___ Solution: 3 ___ [00:00:00] __ [Backtracks: 635] ____
//
//  1 23 18       15  7 20       26 12  4
// 17  3 22       19 14  9        6 25 11
// 24 16  2        8 21 13       10  5 27
//
// ___ Solution: 4 ___ [00:00:00] __ [Backtracks: 758] ____
//
// Number of solutions: 4   Number of backtracks: 812
// Elapsed time: 00:00:00
//                         


local C :< L = 42

pred MagicCube3x3x3() iff

    // Express the cube as an array 
    mc ::[0..26]->>L[1..27] &

    // Decompose the array into 27 elements for easier coding
    mc = [n01,n02,n03,
          n04,n05,n06,
          n07,n08,n09,
          n10,n11,n12,
          n13,n14,n15,
          n16,n17,n18,
          n19,n20,n21,
          n22,n23,n24,
          n25,n26,n27] &

    // Remove symmetries...
    n01 < n03 & n01 < n07 & n01 < n09 & n01 < n19 &  n01 < n21 & n01 < n25 & n01 < n27 &
    n02 > n04 & n04 > n10 &

    // Setup constraints between elements
    n01 + n02 + n03 = C & n01 + n10 + n19 = C & n01 + n04 + n07 = C &
    n04 + n05 + n06 = C & n02 + n11 + n20 = C & n02 + n05 + n08 = C &
    n07 + n08 + n09 = C & n03 + n12 + n21 = C & n03 + n06 + n09 = C &
    n10 + n11 + n12 = C & n04 + n13 + n22 = C & n10 + n13 + n16 = C &
    n13 + n14 + n15 = C & n05 + n14 + n23 = C & n11 + n14 + n17 = C &
    n16 + n17 + n18 = C & n06 + n15 + n24 = C & n12 + n15 + n18 = C &
    n19 + n20 + n21 = C & n07 + n16 + n25 = C & n19 + n22 + n25 = C &
    n22 + n23 + n24 = C & n08 + n17 + n26 = C & n20 + n23 + n26 = C &
    n25 + n26 + n27 = C & n09 + n18 + n27 = C & n21 + n24 + n27 = C &
    n01 + n14 + n27 = C & n03 + n14 + n25 = C &
    n07 + n14 + n21 = C & n09 + n14 + n19 = C &

    // Printout the magic cube as three 3x3 matrices
    PrintMC3(mc,0)

///////////////////////////////////////////////////////////////////////////////
// n1 n2 n3    n10 n11 n12    n19 n20 n21
// n4 n5 n6    n13 n14 n15    n22 n23 n24
// n7 n8 n9    n16 n17 n18    n25 n26 n27
local proc PrintMC3(mc :< [0..26]->L, r:<I) iff 
    if r < 3 then 
        Print('\n') &
        PrintDigit2(mc(3*r+0))  & PrintDigit2(mc(3*r+1))  & PrintDigit2(mc(3*r+2)) &
        Print('      ') &
        PrintDigit2(mc(3*r+9))  & PrintDigit2(mc(3*r+10)) & PrintDigit2(mc(3*r+11)) &
        Print('      ') &
        PrintDigit2(mc(3*r+18)) & PrintDigit2(mc(3*r+19)) & PrintDigit2(mc(3*r+20)) &
        PrintMC3(mc,r+1)
    else
        Print('\n') 
    end


local proc PrintDigit2(d:<L) iff
    if d < 10 then
        Print(' ',d,' ')
    else
        Print(d,' ')
    end

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