------
-- Matrix Sum
-- Problem 345
--
-- We define the Matrix Sum of a matrix as the maximum sum of matrix elements with
-- each element being the only one in his row and column. For example, the Matrix Sum
-- of the matrix below equals 3315 ( = 863 + 383 + 343 + 959 + 767):
--
--   7  53 183 439 863
-- 497 383 563  79 973
-- 287  63 343 169 583
-- 627 343 773 959 943
-- 767 473 103 699 303
--
-- Find the Matrix Sum of:
--
--   7  53 183 439 863 497 383 563  79 973 287  63 343 169 583
-- 627 343 773 959 943 767 473 103 699 303 957 703 583 639 913
-- 447 283 463  29  23 487 463 993 119 883 327 493 423 159 743
-- 217 623   3 399 853 407 103 983  89 463 290 516 212 462 350
-- 960 376 682 962 300 780 486 502 912 800 250 346 172 812 350
-- 870 456 192 162 593 473 915  45 989 873 823 965 425 329 803
-- 973 965 905 919 133 673 665 235 509 613 673 815 165 992 326
-- 322 148 972 962 286 255 941 541 265 323 925 281 601  95 973
-- 445 721  11 525 473  65 511 164 138 672  18 428 154 448 848
-- 414 456 310 312 798 104 566 520 302 248 694 976 430 392 198
-- 184 829 373 181 631 101 969 613 840 740 778 458 284 760 390
-- 821 461 843 513  17 901 711 993 293 157 274  94 192 156 574
--  34 124   4 878 450 476 712 914 838 669 875 299 823 329 699
-- 815 559 813 459 522 788 168 586 966 232 308 833 251 631 107
-- 813 883 451 509 615  77 281 613 459 205 380 274 302  35 805
------


main = do
    print (testMatrix)
    print (backtrack testMatrix 0 0 0 0)

backtrack :: [[Int]] -> Int -> Int -> Int -> Int -> Int
backtrack mat col row maxsum prevSum
    | col == 4 = backtrack mat col row (max currSum maxSum) 0
    | row == 4 = backtrack mat (succ col) 0 maxSum currSum
    | otherwise = backtrack mat col (succ row) maxSum currSum
    where currSum = prevSum + mat !! row !! col

testMatrix = [ [  7,  53, 183, 439, 863]
             , [497, 383, 563,  79, 973]
             , [287,  63, 343, 169, 583]
             , [627, 343, 773, 959, 943]
             , [767, 473, 103, 699, 303]
             ]

matrix = [ [  7,  53, 183, 439, 863, 497, 383, 563,  79, 973, 287,  63, 343, 169, 583]
         , [627, 343, 773, 959, 943, 767, 473, 103, 699, 303, 957, 703, 583, 639, 913]
         , [447, 283, 463,  29,  23, 487, 463, 993, 119, 883, 327, 493, 423, 159, 743]
         , [217, 623,   3, 399, 853, 407, 103, 983,  89, 463, 290, 516, 212, 462, 350]
         , [960, 376, 682, 962, 300, 780, 486, 502, 912, 800, 250, 346, 172, 812, 350]
         , [870, 456, 192, 162, 593, 473, 915,  45, 989, 873, 823, 965, 425, 329, 803]
         , [973, 965, 905, 919, 133, 673, 665, 235, 509, 613, 673, 815, 165, 992, 326]
         , [322, 148, 972, 962, 286, 255, 941, 541, 265, 323, 925, 281, 601,  95, 973]
         , [445, 721,  11, 525, 473,  65, 511, 164, 138, 672,  18, 428, 154, 448, 848]
         , [414, 456, 310, 312, 798, 104, 566, 520, 302, 248, 694, 976, 430, 392, 198]
         , [184, 829, 373, 181, 631, 101, 969, 613, 840, 740, 778, 458, 284, 760, 390]
         , [821, 461, 843, 513,  17, 901, 711, 993, 293, 157, 274,  94, 192, 156, 574]
         , [ 34, 124,   4, 878, 450, 476, 712, 914, 838, 669, 875, 299, 823, 329, 699]
         , [815, 559, 813, 459, 522, 788, 168, 586, 966, 232, 308, 833, 251, 631, 107]
         , [813, 883, 451, 509, 615,  77, 281, 613, 459, 205, 380, 274, 302,  35, 805]
         ]