If I gave you the numbers 2, 5, 11, ___ could you guess the next number? The human brain starts looking for patterns between the numbers. Most people find that the easiest pattern is the previous number times 2 plus 1.
(2 x 2) + 1 = 5
(5 x 2) + 1 = 11
(11 x 2) + 1 = 23
If you give the same problem to a computer, it finds the next number, but is a very unexpected efficient way. You can alternatively take the number and add a 5 and divide by 5.
25/5 = 5
55/5 = 11
115/5 = 23
As humans, we are restricted by the complexity of the math inside the pattern. We do not try every possible math equation, but instead rely on our amazing ability to spot patterns. Computers however, can calculate complex math fast and efficiently. My goal was to write a program that can find the pattern in any pattern. My first idea was simply to “brute force” the solution by simply trying every possibility. This technique requires no complex logic or equations. Instead of mimicking humans, it uses its own strengths to solve the problem. The following program was made in c# and the full source code is available for download at the bottom of this article.
Step 1: Understanding the problem
The code starts with a bank of numbers and math operations like below:
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1 |
const string stringbank = "123456789^+-/*"; |
The program will take the characters in the stringbank and try every single combination/permutation starting with one character, then two, then three etc. For example it would try calculating:
One Character: 1, 2, 3, 4, 5, 6, 7, 8, 9, ^, +, -, /, *
Two Characters: 11, 12, 13, 14, 15, 16, 17, 18, 19, 1^, 1+, 1-, 1/, 1*, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2^, 2+, 2-, 2/, 2*, 31, 32, 33, 34, 35, 36, 37, 38, 39, 3^, 3+, 3-, 3/, 3*, 41, 42, 43, 44, 45, 46, 47, 48, 49, 4^, 4+, 4-, 4/, 4*, 51, 52, 53, 54, 55, 56, 57, 58, 59, 5^, 5+, 5-, 5/, 5*, 61, 62, 63, 64, 65, 66, 67, 68, 69, 6^, 6+, 6-, 6/, 6*, 71, 72, 73, 74, 75, 76, 77, 78, 79, 7^, 7+, 7-, 7/, 7*, 81, 82, 83, 84, 85, 86, 87, 88, 89, 8^, 8+, 8-, 8/, 8*, 91, 92, 93, 94, 95, 96, 97, 98, 99, 9^, 9+, 9-, 9/, 9*, ^1, ^2, ^3, ^4, ^5, ^6, ^7, ^8, ^9, ^^, ^+, ^-, ^/, ^*, +1, +2, +3, +4, +5, +6, +7, +8, +9, +^, ++, +-, +/, +*, -1, -2, -3, -4, -5, -6, -7, -8, -9, -^, -+, –, -/, -*, /1, /2, /3, /4, /5, /6, /7, /8, /9, /^, /+, /-, //, /*, *1, *2, *3, *4, *5, *6, *7, *8, *9, *^, *+, *-, */, **
Three Characters: 111, 112, 113, 114, 115, 116, 117, 118, 119, 11^, 11+, 11-, 11/, 11*, 121, 122, 123, 124, 125, 126, 127, 128, 129, 12^, 12+, 12-, 12/, 12*, 131, 132, 133, 134, 135, 136, 137, 138, 139, 13^, 13+, 13-, 13/, 13*, 141, 142, 143, 144, 145, 146, 147, 148, 149, 14^, 14+, 14-, 14/, 14*, 151, 152, 153, 154, 155, 156, 157, 158, 159, 15^, 15+, 15-, 15/, 15*, 161, 162, 163, 164, 165, 166, 167, 168, 169, 16^, 16+, 16-, 16/, 16*, 171, 172, 173, 174, 175, 176, 177, 178, 179, 17^, 17+, 17-, 17/, 17*, 181, 182, 183, 184, 185, 186, 187, 188, 189, 18^, 18+, 18-, 18/, 18*, 191, 192, 193, 194, 195, 196, 197, 198, 199, 19^, 19+, 19-, 19/, 19*, 1^1, 1^2, 1^3, 1^4, 1^5, 1^6, 1^7, 1^8, 1^9, 1^^, 1^+, 1^-, 1^/, 1^*, 1+1, 1+2, 1+3, 1+4, 1+5, 1+6, 1+7, 1+8, 1+9, 1+^, 1++, 1+-, 1+/, 1+*, 1-1, 1-2, 1-3, 1-4, 1-5, 1-6, 1-7, 1-8, 1-9, 1-^, 1-+, 1–, 1-/, 1-*, 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/^, 1/+, 1/-, 1//, 1/*, 1*1, 1*2, 1*3, 1*4, 1*5, 1*6, 1*7, 1*8, 1*9, 1*^, 1*+, 1*-, 1*/, 1**, 211, 212, 213, 214, 215, 216, 217, 218, 219, 21^, 21+, 21-, 21/, 21*, 221, 222, 223, 224, 225, 226, 227, 228, 229, 22^, 22+, 22-, 22/, 22*, 231, 232, 233, 234, 235, 236, 237, 238, 239, 23^, 23+, 23-, 23/, 23*, 241, 242, 243, 244, 245, 246, 247, 248, 249, 24^, 24+, 24-, 24/, 24*, 251, 252, 253, 254, 255, 256, 257, 258, 259, 25^, 25+, 25-, 25/, 25*, 261, 262, 263, 264, 265, 266, 267, 268, 269, 26^, 26+, 26-, 26/, 26*, 271, 272, 273, 274, 275, 276, 277, 278, 279, 27^, 27+, 27-, 27/, 27*, 281, 282, 283, 284, 285, 286, 287, 288, 289, 28^, 28+, 28-, 28/, 28*, 291, 292, 293, 294, 295, 296, 297, 298, 299, 29^, 29+, 29-, 29/, 29*, 2^1, 2^2, 2^3, 2^4, 2^5, 2^6, 2^7, 2^8, 2^9, 2^^, 2^+, 2^-, 2^/, 2^*, 2+1, 2+2, 2+3, 2+4, 2+5, 2+6, 2+7, 2+8, 2+9, 2+^, 2++, 2+-, 2+/, 2+*, 2-1, 2-2, 2-3, 2-4, 2-5, 2-6, 2-7, 2-8, 2-9, 2-^, 2-+, 2–, 2-/, 2-*, 2/1, 2/2, 2/3, 2/4, 2/5, 2/6, 2/7, 2/8, 2/9, 2/^, 2/+, 2/-, 2//, 2/*, 2*1, 2*2, 2*3, 2*4, 2*5, 2*6, 2*7, 2*8, 2*9, 2*^, 2*+, 2*-, 2*/, 2**, 311, 312, 313, 314, 315, 316, 317, 318, 319, 31^, 31+, 31-, 31/, 31*, 321, 322, 323, 324, 325, 326, 327, 328, 329, 32^, 32+, 32-, 32/, 32*, 331, 332, 333, 334, 335, 336, 337, 338, 339, 33^, 33+, 33-, 33/, 33*, 341, 342, 343, 344, 345, 346, 347, 348, 349, 34^, 34+, 34-, 34/, 34*, 351, 352, 353, 354, 355, 356, 357, 358, 359, 35^, 35+, 35-, 35/, 35*, 361, 362, 363, 364, 365, 366, 367, 368, 369, 36^, 36+, 36-, 36/, 36*, 371, 372, 373, 374, 375, 376, 377, 378, 379, 37^, 37+, 37-, 37/, 37*, 381, 382, 383, 384, 385, 386, 387, 388, 389, 38^, 38+, 38-, 38/, 38*, 391, 392, 393, 394, 395, 396, 397, 398, 399, 39^, 39+, 39-, 39/, 39*, 3^1, 3^2, 3^3, 3^4, 3^5, 3^6, 3^7, 3^8, 3^9, 3^^, 3^+, 3^-, 3^/, 3^*, 3+1, 3+2, 3+3, 3+4, 3+5, 3+6, 3+7, 3+8, 3+9, 3+^, 3++, 3+-, 3+/, 3+*, 3-1, 3-2, 3-3, 3-4, 3-5, 3-6, 3-7, 3-8, 3-9, 3-^, 3-+, 3–, 3-/, 3-*, 3/1, 3/2, 3/3, 3/4, 3/5, 3/6, 3/7, 3/8, 3/9, 3/^, 3/+, 3/-, 3//, 3/*, 3*1, 3*2, 3*3, 3*4, 3*5, 3*6, 3*7, 3*8, 3*9, 3*^, 3*+, 3*-, 3*/, 3**, 411, 412, 413, 414, 415, 416, 417, 418, 419, 41^, 41+, 41-, 41/, 41*, 421, 422, 423, 424, 425, 426, 427, 428, 429, 42^, 42+, 42-, 42/, 42*, 431, 432, 433, 434, 435, 436, 437, 438, 439, 43^, 43+, 43-, 43/, 43*, 441, 442, 443, 444, 445, 446, 447, 448, 449, 44^, 44+, 44-, 44/, 44*, 451, 452, 453, 454, 455, 456, 457, 458, 459, 45^, 45+, 45-, 45/, 45*, 461, 462, 463, 464, 465, 466, 467, 468, 469, 46^, 46+, 46-, 46/, 46*, 471, 472, 473, 474, 475, 476, 477, 478, 479, 47^, 47+, 47-, 47/, 47*, 481, 482, 483, 484, 485, 486, 487, 488, 489, 48^, 48+, 48-, 48/, 48*, 491, 492, 493, 494, 495, 496, 497, 498, 499, 49^, 49+, 49-, 49/, 49*, 4^1, 4^2, 4^3, 4^4, 4^5, 4^6, 4^7, 4^8, 4^9, 4^^, 4^+, 4^-, 4^/, 4^*, 4+1, 4+2, 4+3, 4+4, 4+5, 4+6, 4+7, 4+8, 4+9, 4+^, 4++, 4+-, 4+/, 4+*, 4-1, 4-2, 4-3, 4-4, 4-5, 4-6, 4-7, 4-8, 4-9, 4-^, 4-+, 4–, 4-/, 4-*, 4/1, 4/2, 4/3, 4/4, 4/5, 4/6, 4/7, 4/8, 4/9, 4/^, 4/+, 4/-, 4//, 4/*, 4*1, 4*2, 4*3, 4*4, 4*5, 4*6, 4*7, 4*8, 4*9, 4*^, 4*+, 4*-, 4*/, 4**, 511, 512, 513, 514, 515, 516, 517, 518, 519, 51^, 51+, 51-, 51/, 51*, 521, 522, 523, 524, 525, 526, 527, 528, 529, 52^, 52+, 52-, 52/, 52*, 531, 532, 533, 534, 535, 536, 537, 538, 539, 53^, 53+, 53-, 53/, 53*, 541, 542, 543, 544, 545, 546, 547, 548, 549, 54^, 54+, 54-, 54/, 54*, 551, 552, 553, 554, 555, 556, 557, 558, 559, 55^, 55+, 55-, 55/, 55*, 561, 562, 563, 564, 565, 566, 567, 568, 569, 56^, 56+, 56-, 56/, 56*, 571, 572, 573, 574, 575, 576, 577, 578, 579, 57^, 57+, 57-, 57/, 57*, 581, 582, 583, 584, 585, 586, 587, 588, 589, 58^, 58+, 58-, 58/, 58*, 591, 592, 593, 594, 595, 596, 597, 598, 599, 59^, 59+, 59-, 59/, 59*, 5^1, 5^2, 5^3, 5^4, 5^5, 5^6, 5^7, 5^8, 5^9, 5^^, 5^+, 5^-, 5^/, 5^*, 5+1, 5+2, 5+3, 5+4, 5+5, 5+6, 5+7, 5+8, 5+9, 5+^, 5++, 5+-, 5+/, 5+*, 5-1, 5-2, 5-3, 5-4, 5-5, 5-6, 5-7, 5-8, 5-9, 5-^, 5-+, 5–, 5-/, 5-*, 5/1, 5/2, 5/3, 5/4, 5/5, 5/6, 5/7, 5/8, 5/9, 5/^, 5/+, 5/-, 5//, 5/*, 5*1, 5*2, 5*3, 5*4, 5*5, 5*6, 5*7, 5*8, 5*9, 5*^, 5*+, 5*-, 5*/, 5**, 611, 612, 613, 614, 615, 616, 617, 618, 619, 61^, 61+, 61-, 61/, 61*, 621, 622, 623, 624, 625, 626, 627, 628, 629, 62^, 62+, 62-, 62/, 62*, 631, 632, 633, 634, 635, 636, 637, 638, 639, 63^, 63+, 63-, 63/, 63*, 641, 642, 643, 644, 645, 646, 647, 648, 649, 64^, 64+, 64-, 64/, 64*, 651, 652, 653, 654, 655, 656, 657, 658, 659, 65^, 65+, 65-, 65/, 65*, 661, 662, 663, 664, 665, 666, 667, 668, 669, 66^, 66+, 66-, 66/, 66*, 671, 672, 673, 674, 675, 676, 677, 678, 679, 67^, 67+, 67-, 67/, 67*, 681, 682, 683, 684, 685, 686, 687, 688, 689, 68^, 68+, 68-, 68/, 68*, 691, 692, 693, 694, 695, 696, 697, 698, 699, 69^, 69+, 69-, 69/, 69*, 6^1, 6^2, 6^3, 6^4, 6^5, 6^6, 6^7, 6^8, 6^9, 6^^, 6^+, 6^-, 6^/, 6^*, 6+1, 6+2, 6+3, 6+4, 6+5, 6+6, 6+7, 6+8, 6+9, 6+^, 6++, 6+-, 6+/, 6+*, 6-1, 6-2, 6-3, 6-4, 6-5, 6-6, 6-7, 6-8, 6-9, 6-^, 6-+, 6–, 6-/, 6-*, 6/1, 6/2, 6/3, 6/4, 6/5, 6/6, 6/7, 6/8, 6/9, 6/^, 6/+, 6/-, 6//, 6/*, 6*1, 6*2, 6*3, 6*4, 6*5, 6*6, 6*7, 6*8, 6*9, 6*^, 6*+, 6*-, 6*/, 6**, 711, 712, 713, 714, 715, 716, 717, 718, 719, 71^, 71+, 71-, 71/, 71*, 721, 722, 723, 724, 725, 726, 727, 728, 729, 72^, 72+, 72-, 72/, 72*, 731, 732, 733, 734, 735, 736, 737, 738, 739, 73^, 73+, 73-, 73/, 73*, 741, 742, 743, 744, 745, 746, 747, 748, 749, 74^, 74+, 74-, 74/, 74*, 751, 752, 753, 754, 755, 756, 757, 758, 759, 75^, 75+, 75-, 75/, 75*, 761, 762, 763, 764, 765, 766, 767, 768, 769, 76^, 76+, 76-, 76/, 76*, 771, 772, 773, 774, 775, 776, 777, 778, 779, 77^, 77+, 77-, 77/, 77*, 781, 782, 783, 784, 785, 786, 787, 788, 789, 78^, 78+, 78-, 78/, 78*, 791, 792, 793, 794, 795, 796, 797, 798, 799, 79^, 79+, 79-, 79/, 79*, 7^1, 7^2, 7^3, 7^4, 7^5, 7^6, 7^7, 7^8, 7^9, 7^^, 7^+, 7^-, 7^/, 7^*, 7+1, 7+2, 7+3, 7+4, 7+5, 7+6, 7+7, 7+8, 7+9, 7+^, 7++, 7+-, 7+/, 7+*, 7-1, 7-2, 7-3, 7-4, 7-5, 7-6, 7-7, 7-8, 7-9, 7-^, 7-+, 7–, 7-/, 7-*, 7/1, 7/2, 7/3, 7/4, 7/5, 7/6, 7/7, 7/8, 7/9, 7/^, 7/+, 7/-, 7//, 7/*, 7*1, 7*2, 7*3, 7*4, 7*5, 7*6, 7*7, 7*8, 7*9, 7*^, 7*+, 7*-, 7*/, 7**, 811, 812, 813, 814, 815, 816, 817, 818, 819, 81^, 81+, 81-, 81/, 81*, 821, 822, 823, 824, 825, 826, 827, 828, 829, 82^, 82+, 82-, 82/, 82*, 831, 832, 833, 834, 835, 836, 837, 838, 839, 83^, 83+, 83-, 83/, 83*, 841, 842, 843, 844, 845, 846, 847, 848, 849, 84^, 84+, 84-, 84/, 84*, 851, 852, 853, 854, 855, 856, 857, 858, 859, 85^, 85+, 85-, 85/, 85*, 861, 862, 863, 864, 865, 866, 867, 868, 869, 86^, 86+, 86-, 86/, 86*, 871, 872, 873, 874, 875, 876, 877, 878, 879, 87^, 87+, 87-, 87/, 87*, 881, 882, 883, 884, 885, 886, 887, 888, 889, 88^, 88+, 88-, 88/, 88*, 891, 892, 893, 894, 895, 896, 897, 898, 899, 89^, 89+, 89-, 89/, 89*, 8^1, 8^2, 8^3, 8^4, 8^5, 8^6, 8^7, 8^8, 8^9, 8^^, 8^+, 8^-, 8^/, 8^*, 8+1, 8+2, 8+3, 8+4, 8+5, 8+6, 8+7, 8+8, 8+9, 8+^, 8++, 8+-, 8+/, 8+*, 8-1, 8-2, 8-3, 8-4, 8-5, 8-6, 8-7, 8-8, 8-9, 8-^, 8-+, 8–, 8-/, 8-*, 8/1, 8/2, 8/3, 8/4, 8/5, 8/6, 8/7, 8/8, 8/9, 8/^, 8/+, 8/-, 8//, 8/*, 8*1, 8*2, 8*3, 8*4, 8*5, 8*6, 8*7, 8*8, 8*9, 8*^, 8*+, 8*-, 8*/, 8**, 911, 912, 913, 914, 915, 916, 917, 918, 919, 91^, 91+, 91-, 91/, 91*, 921, 922, 923, 924, 925, 926, 927, 928, 929, 92^, 92+, 92-, 92/, 92*, 931, 932, 933, 934, 935, 936, 937, 938, 939, 93^, 93+, 93-, 93/, 93*, 941, 942, 943, 944, 945, 946, 947, 948, 949, 94^, 94+, 94-, 94/, 94*, 951, 952, 953, 954, 955, 956, 957, 958, 959, 95^, 95+, 95-, 95/, 95*, 961, 962, 963, 964, 965, 966, 967, 968, 969, 96^, 96+, 96-, 96/, 96*, 971, 972, 973, 974, 975, 976, 977, 978, 979, 97^, 97+, 97-, 97/, 97*, 981, 982, 983, 984, 985, 986, 987, 988, 989, 98^, 98+, 98-, 98/, 98*, 991, 992, 993, 994, 995, 996, 997, 998, 999, 99^, 99+, 99-, 99/, 99*, 9^1, 9^2, 9^3, 9^4, 9^5, 9^6, 9^7, 9^8, 9^9, 9^^, 9^+, 9^-, 9^/, 9^*, 9+1, 9+2, 9+3, 9+4, 9+5, 9+6, 9+7, 9+8, 9+9, 9+^, 9++, 9+-, 9+/, 9+*, 9-1, 9-2, 9-3, 9-4, 9-5, 9-6, 9-7, 9-8, 9-9, 9-^, 9-+, 9–, 9-/, 9-*, 9/1, 9/2, 9/3, 9/4, 9/5, 9/6, 9/7, 9/8, 9/9, 9/^, 9/+, 9/-, 9//, 9/*, 9*1, 9*2, 9*3, 9*4, 9*5, 9*6, 9*7, 9*8, 9*9, 9*^, 9*+, 9*-, 9*/, 9**, ^11, ^12, ^13, ^14, ^15, ^16, ^17, ^18, ^19, ^1^, ^1+, ^1-, ^1/, ^1*, ^21, ^22, ^23, ^24, ^25, ^26, ^27, ^28, ^29, ^2^, ^2+, ^2-, ^2/, ^2*, ^31, ^32, ^33, ^34, ^35, ^36, ^37, ^38, ^39, ^3^, ^3+, ^3-, ^3/, ^3*, ^41, ^42, ^43, ^44, ^45, ^46, ^47, ^48, ^49, ^4^, ^4+, ^4-, ^4/, ^4*, ^51, ^52, ^53, ^54, ^55, ^56, ^57, ^58, ^59, ^5^, ^5+, ^5-, ^5/, ^5*, ^61, ^62, ^63, ^64, ^65, ^66, ^67, ^68, ^69, ^6^, ^6+, ^6-, ^6/, ^6*, ^71, ^72, ^73, ^74, ^75, ^76, ^77, ^78, ^79, ^7^, ^7+, ^7-, ^7/, ^7*, ^81, ^82, ^83, ^84, ^85, ^86, ^87, ^88, ^89, ^8^, ^8+, ^8-, ^8/, ^8*, ^91, ^92, ^93, ^94, ^95, ^96, ^97, ^98, ^99, ^9^, ^9+, ^9-, ^9/, ^9*, ^^1, ^^2, ^^3, ^^4, ^^5, ^^6, ^^7, ^^8, ^^9, ^^^, ^^+, ^^-, ^^/, ^^*, ^+1, ^+2, ^+3, ^+4, ^+5, ^+6, ^+7, ^+8, ^+9, ^+^, ^++, ^+-, ^+/, ^+*, ^-1, ^-2, ^-3, ^-4, ^-5, ^-6, ^-7, ^-8, ^-9, ^-^, ^-+, ^–, ^-/, ^-*, ^/1, ^/2, ^/3, ^/4, ^/5, ^/6, ^/7, ^/8, ^/9, ^/^, ^/+, ^/-, ^//, ^/*, ^*1, ^*2, ^*3, ^*4, ^*5, ^*6, ^*7, ^*8, ^*9, ^*^, ^*+, ^*-, ^*/, ^**, +11, +12, +13, +14, +15, +16, +17, +18, +19, +1^, +1+, +1-, +1/, +1*, +21, +22, +23, +24, +25, +26, +27, +28, +29, +2^, +2+, +2-, +2/, +2*, +31, +32, +33, +34, +35, +36, +37, +38, +39, +3^, +3+, +3-, +3/, +3*, +41, +42, +43, +44, +45, +46, +47, +48, +49, +4^, +4+, +4-, +4/, +4*, +51, +52, +53, +54, +55, +56, +57, +58, +59, +5^, +5+, +5-, +5/, +5*, +61, +62, +63, +64, +65, +66, +67, +68, +69, +6^, +6+, +6-, +6/, +6*, +71, +72, +73, +74, +75, +76, +77, +78, +79, +7^, +7+, +7-, +7/, +7*, +81, +82, +83, +84, +85, +86, +87, +88, +89, +8^, +8+, +8-, +8/, +8*, +91, +92, +93, +94, +95, +96, +97, +98, +99, +9^, +9+, +9-, +9/, +9*, +^1, +^2, +^3, +^4, +^5, +^6, +^7, +^8, +^9, +^^, +^+, +^-, +^/, +^*, ++1, ++2, ++3, ++4, ++5, ++6, ++7, ++8, ++9, ++^, +++, ++-, ++/, ++*, +-1, +-2, +-3, +-4, +-5, +-6, +-7, +-8, +-9, +-^, +-+, +–, +-/, +-*, +/1, +/2, +/3, +/4, +/5, +/6, +/7, +/8, +/9, +/^, +/+, +/-, +//, +/*, +*1, +*2, +*3, +*4, +*5, +*6, +*7, +*8, +*9, +*^, +*+, +*-, +*/, +**, -11, -12, -13, -14, -15, -16, -17, -18, -19, -1^, -1+, -1-, -1/, -1*, -21, -22, -23, -24, -25, -26, -27, -28, -29, -2^, -2+, -2-, -2/, -2*, -31, -32, -33, -34, -35, -36, -37, -38, -39, -3^, -3+, -3-, -3/, -3*, -41, -42, -43, -44, -45, -46, -47, -48, -49, -4^, -4+, -4-, -4/, -4*, -51, -52, -53, -54, -55, -56, -57, -58, -59, -5^, -5+, -5-, -5/, -5*, -61, -62, -63, -64, -65, -66, -67, -68, -69, -6^, -6+, -6-, -6/, -6*, -71, -72, -73, -74, -75, -76, -77, -78, -79, -7^, -7+, -7-, -7/, -7*, -81, -82, -83, -84, -85, -86, -87, -88, -89, -8^, -8+, -8-, -8/, -8*, -91, -92, -93, -94, -95, -96, -97, -98, -99, -9^, -9+, -9-, -9/, -9*, -^1, -^2, -^3, -^4, -^5, -^6, -^7, -^8, -^9, -^^, -^+, -^-, -^/, -^*, -+1, -+2, -+3, -+4, -+5, -+6, -+7, -+8, -+9, -+^, -++, -+-, -+/, -+*, –1, –2, –3, –4, –5, –6, –7, –8, –9, –^, –+, —, –/, –*, -/1, -/2, -/3, -/4, -/5, -/6, -/7, -/8, -/9, -/^, -/+, -/-, -//, -/*, -*1, -*2, -*3, -*4, -*5, -*6, -*7, -*8, -*9, -*^, -*+, -*-
… ,***
Three thing can be observed by this.
- The number of possibilities increases exponentially with each character (ie 1 character = 14^1 possibilities, 2 characters = 14^2 possibilities….)
- The majority of the possibilities are invalid and must be dismissed. (+8^ or +8* are not valid equations)
- Even if it does calculate an answer there is no way to input any of the values into the math equation. In other words imagine the pattern was 2, 5, 11, __. The formula to find the next number is the (previous number x 2) + 1 —> (11 x 2) + 1 = 23 however our program currently does not have the ability to input any numbers (in this example the 11).
The solution is quite simple. We add a “variable” character to the string bank like below and replace the “variable” character with our input number right before the program calculates the solution.
|
1 |
const string stringbank = "a123456789^+-/*"; //Variable 'a' |
Now the program theoretically can get a*2+1 as a possible solution.
The general code logic
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 |
//a*2+1 is just an example of one of the many possible permutations. string trialEquation = "a*2+1"; //Insert function here that return every permutation. // Step 1 - Take the first, second and third number in the pattern and store it as input1, input2 and input3. string input1 = "2"; string input2 = "5"; string input3 = "11"; // Step 3 - Generate three test trials by replacing the 'a' with input1, input2 & input3. string trial1 = trialEquation.Replace("a", input1); //"a*2+1" --> "2*2+1" string trial2 = trialEquation.Replace("a", input2); //"a*2+1" --> "5*2+1" string trial3 = trialEquation.Replace("a", input3); //"a*2+1" --> "11*2+1" //Step 4. Try to calculate trial and if it passes both tests then we assume we found a functioning equation. try { if (*calculate trial1 answer here* == input2) if (*calculate trial2 answer here* == input3) { Console.WriteLine("Success! Solution found! {0}", trialEquation); Console.WriteLine("---------------------------"); } } catch { // ignored } |
Step 2: Coding the code
A function that returns all permutations of the stringbank
Here is the finished code.
Download finished project here.
Download compiled exe here.
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/* This brute force algorithm was originally written (by me) back in 1998, and has been collecting dust since then. However, for the purpose of testing Gist on GitHub I decided to rewrite the algorithm from VB6 to C#, make some improvements and release this fast, compact, non-recursive, brute force algorithm under the MIT license: http://opensource.org/licenses/MIT Notes: - Do a run with testLetters = "0123456789" and testLength = 3, to see what happens - Remember to keep the callback testCalback as fast as possible - Tweet some love to @fredrikdev :) Have fun! Fredrik https://github.com/johanssonrobotics */ using System; using System.Collections.Generic; using System.Linq; using System.Text; using System.Threading.Tasks; using System.Data;//import this namespace using MathParserTK; using System.Globalization; using System.Threading; using System.Diagnostics; using System.Threading.Tasks; namespace FJR { public static class Algorithm { // callback that is called for every combination to test public delegate bool BruteForceTest(ref char[] test, int testIndex, int testMax); // fast, compact, non-recursive, brute force algorithm public static bool BruteForce(string testLetters, int testLength, BruteForceTest testCallback) { // get the number of combinations we need to try (just for statistic purposes) int testIndex = 0, testMax = (int)Math.Pow(testLetters.Length, testLength); // initialize & perform first test var test = new char[testLength]; for (int x = 0; x < testLength; x++) test[x] = testLetters[0]; if (testCallback(ref test, ++testIndex, testMax)) return true; // start rotating chars from the back and forward for (int ti = testLength - 1, li = 0; ti > -1; ti--) { for (li = testLetters.IndexOf(test[ti]) + 1; li < testLetters.Length; li++) { // test test[ti] = testLetters[li]; if (testCallback(ref test, ++testIndex, testMax)) return true; // rotate to the right for (int z = ti + 1; z < testLength; z++) if (test[z] != testLetters[testLetters.Length - 1]) { ti = testLength; goto outerBreak; } } outerBreak: if (li == testLetters.Length) test[ti] = testLetters[0]; } // no match return false; } public static void Main(string[] args) { int i2 = 1; Console.WriteLine("Please enter var A:"); string varA = Console.ReadLine(); Console.WriteLine("Please enter var B:"); string varB = Console.ReadLine(); Console.WriteLine("Please enter var C "); string varC = Console.ReadLine(); //Console.WriteLine("Please enter var D "); //string varD = Console.ReadLine(); MathParser parser = new MathParser(); // example usage Console.WriteLine("Searching for combination..."); Stopwatch stopWatch = new Stopwatch(); stopWatch.Start(); // our 'test' routine that gets called for every combination to test BruteForceTest testCallback = delegate(ref char[] test, int testIndex, int testMax) { var str = new string(test); // Console.Title = Math.Round(100 * (testIndex / (double)testMax), 0) + "%" + " - " + testIndex + " - " + str; i2++; try { string newstr = str.Replace("a", varA); double answer = parser.Parse(newstr); //Console.WriteLine(answer); if (answer.ToString() == varB) { //Console.WriteLine("{0} = {1}", str, answer); string checkstr = str.Replace("a", varB); double check = parser.Parse(checkstr); if (check.ToString() == varC) { Console.WriteLine("--------------------------------------------------"); Console.WriteLine("Success! Solution found! ({0})=b", str); string nextstr = str.Replace("a", varC); double next = parser.Parse(nextstr); Console.WriteLine("next number is({0})", next); stopWatch.Stop(); TimeSpan ts = stopWatch.Elapsed; // Format and display the TimeSpan value. string elapsedTime = String.Format("{0:00}:{1:00}:{2:00}.{3:00}", ts.Hours, ts.Minutes, ts.Seconds, ts.Milliseconds / 10); Console.WriteLine("RunTime " + elapsedTime); Console.WriteLine("testIndex " + testIndex); stopWatch.Start(); } } } catch{} return(1==2); // Returns a false }; // test all combinations of the letters "abc" with the result length 3 // todo: you may want to add a for-loop here to test e.g. length 1 to 8 const string stringbank = "a123456789^+-/*"; for (int i = 0; i < 100; i++) { BruteForce(stringbank, i, testCallback); } Console.ReadLine(); } } } namespace MathParserTK { #region License agreement // This is light, fast and simple to understand mathematical parser // designed in one class, which receives as input a mathematical // expression (System.String) and returns the output value (System.Double). // For example, if you input string is "√(625)+25*(3/3)" then parser return double value — 50. // Copyright (C) 2012-2013 Yerzhan Kalzhani, kirnbas@gmail.com // This program is free software: you can redistribute it and/or modify // it under the terms of the GNU General Public License as published by // the Free Software Foundation, either version 3 of the License, or // (at your option) any later version. // This program is distributed in the hope that it will be useful, // but WITHOUT ANY WARRANTY; without even the implied warranty of // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the // GNU General Public License for more details. // You should have received a copy of the GNU General Public License // along with this program. If not, see <http://www.gnu.org/licenses/> #endregion #region Example usage // public static void Main() // { // MathParser parser = new MathParser(); // string s1 = "pi+5*5+5*3-5*5-5*3+1E1"; // string s2 = "sin(cos(tg(sh(ch(th(100))))))"; // bool isRadians = false; // double d1 = parser.Parse(s1, isRadians); // double d2 = parser.Parse(s2, isRadians); // // Console.WriteLine(d1); // 13.141592... // Console.WriteLine(d2); // 0.0174524023974442 // Console.ReadKey(true); // } #endregion #region Common tips to extend this math parser // If you want to add new operator, then add new item to supportedOperators // dictionary and check next methods: // (1) IsRightAssociated, (2) GetPriority, (3) SyntaxAnalysisRPN, (4) NumberOfArguments // // If you want to add new function, then add new item to supportedFunctions // dictionary and check next above methods: (2), (3), (4). // // if you want to add new constant, then add new item to supportedConstants #endregion public class MathParser { #region Fields #region Markers (each marker should have length equals to 1) private const string NumberMaker = "#"; private const string OperatorMarker = "$"; private const string FunctionMarker = "@"; #endregion #region Internal tokens private const string Plus = OperatorMarker + "+"; private const string UnPlus = OperatorMarker + "un+"; private const string Minus = OperatorMarker + "-"; private const string UnMinus = OperatorMarker + "un-"; private const string Multiply = OperatorMarker + "*"; private const string Divide = OperatorMarker + "/"; private const string Degree = OperatorMarker + "^"; private const string LeftParent = OperatorMarker + "("; private const string RightParent = OperatorMarker + ")"; private const string Sqrt = FunctionMarker + "sqrt"; private const string Sin = FunctionMarker + "sin"; private const string Cos = FunctionMarker + "cos"; private const string Tg = FunctionMarker + "tg"; private const string Ctg = FunctionMarker + "ctg"; private const string Sh = FunctionMarker + "sh"; private const string Ch = FunctionMarker + "ch"; private const string Th = FunctionMarker + "th"; private const string Log = FunctionMarker + "log"; private const string Ln = FunctionMarker + "ln"; private const string Exp = FunctionMarker + "exp"; private const string Abs = FunctionMarker + "abs"; #endregion #region Dictionaries (containts supported input tokens, exclude number) // Key -> supported input token, Value -> internal token or number /// /// Contains supported operators /// private readonly Dictionary<string, string> supportedOperators = new Dictionary<string, string> { { "+", Plus }, { "-", Minus }, { "*", Multiply }, { "/", Divide }, { "^", Degree }, { "(", LeftParent }, { ")", RightParent } }; /// /// Contains supported functions /// private readonly Dictionary<string, string> supportedFunctions = new Dictionary<string, string> { { "sqrt", Sqrt }, { "√", Sqrt }, { "sin", Sin }, { "cos", Cos }, { "tg", Tg }, { "ctg", Ctg }, { "sh", Sh }, { "ch", Ch }, { "th", Th }, { "log", Log }, { "exp", Exp }, { "abs", Abs } }; private readonly Dictionary<string, string> supportedConstants = new Dictionary<string, string> { {"pi", NumberMaker + Math.PI.ToString() }, {"e", NumberMaker + Math.E.ToString() } }; #endregion #endregion private readonly char decimalSeparator; private bool isRadians; #region Constructors /// /// Initialize new instance of MathParser /// (symbol of decimal separator is read /// from regional settings in system) /// public MathParser() { try { decimalSeparator = Char.Parse(System.Globalization.CultureInfo.CurrentCulture.NumberFormat.NumberDecimalSeparator); } catch (FormatException ex) { throw new FormatException("Error: can't read char decimal separator from system, check your regional settings.", ex); } } /// /// Initialize new instance of MathParser /// ///Set decimal separator public MathParser(char decimalSeparator) { this.decimalSeparator = decimalSeparator; } #endregion /// /// Produce result of the given math expression /// ///Math expression (infix/standard notation) /// Result public double Parse(string expression, bool isRadians = true) { this.isRadians = isRadians; try { return Calculate(ConvertToRPN(FormatString(expression))); } catch (DivideByZeroException e) { throw e; } catch (FormatException e) { throw e; } catch (InvalidOperationException e) { throw e; } catch (ArgumentOutOfRangeException e) { throw e; } catch (ArgumentException e) { throw e; } catch (Exception e) { throw e; } } /// /// Produce formatted string by the given string /// ///Unformatted math expression /// Formatted math expression private string FormatString(string expression) { if (string.IsNullOrEmpty(expression)) { throw new ArgumentNullException("Expression is null or empty"); } StringBuilder formattedString = new StringBuilder(); int balanceOfParenth = 0; // Check number of parenthesis // Format string in one iteration and check number of parenthesis // (this function do 2 tasks because performance priority) for (int i = 0; i < expression.Length; i++) { char ch = expression[i]; if (ch == '(') { balanceOfParenth++; } else if (ch == ')') { balanceOfParenth--; } if (Char.IsWhiteSpace(ch)) { continue; } else if (Char.IsUpper(ch)) { formattedString.Append(Char.ToLower(ch)); } else { formattedString.Append(ch); } } if (balanceOfParenth != 0) { throw new FormatException("Number of left and right parenthesis is not equal"); } return formattedString.ToString(); } #region Convert to Reverse-Polish Notation /// /// Produce math expression in reverse polish notation /// by the given string /// ///Math expression in infix notation /// Math expression in postfix notation (RPN) private string ConvertToRPN(string expression) { int pos = 0; // Current position of lexical analysis StringBuilder outputString = new StringBuilder(); Stack stack = new Stack(); // While there is unhandled char in expression while (pos < expression.Length) { string token = LexicalAnalysisInfixNotation(expression, ref pos); outputString = SyntaxAnalysisInfixNotation(token, outputString, stack); } // Pop all elements from stack to output string while (stack.Count > 0) { // There should be only operators if (stack.Peek()[0] == OperatorMarker[0]) { outputString.Append(stack.Pop()); } else { throw new FormatException("Format exception," + " there is function without parenthesis"); } } return outputString.ToString(); } /// /// Produce token by the given math expression /// ///Math expression in infix notation ///Current position in string for lexical analysis /// Token private string LexicalAnalysisInfixNotation(string expression, ref int pos) { // Receive first char StringBuilder token = new StringBuilder(); token.Append(expression[pos]); // If it is a operator if (supportedOperators.ContainsKey(token.ToString())) { // Determine it is unary or binary operator bool isUnary = pos == 0 || expression[pos - 1] == '('; pos++; switch (token.ToString()) { case "+": return isUnary ? UnPlus : Plus; case "-": return isUnary ? UnMinus : Minus; default: return supportedOperators[token.ToString()]; } } else if (Char.IsLetter(token[0]) || supportedFunctions.ContainsKey(token.ToString()) || supportedConstants.ContainsKey(token.ToString())) { // Read function or constant name while (++pos < expression.Length && Char.IsLetter(expression[pos])) { token.Append(expression[pos]); } if (supportedFunctions.ContainsKey(token.ToString())) { return supportedFunctions[token.ToString()]; } else if (supportedConstants.ContainsKey(token.ToString())) { return supportedConstants[token.ToString()]; } else { throw new ArgumentException("Unknown token"); } } else if (Char.IsDigit(token[0]) || token[0] == decimalSeparator) { // Read number // Read the whole part of number if (Char.IsDigit(token[0])) { while (++pos < expression.Length && Char.IsDigit(expression[pos])) { token.Append(expression[pos]); } } else { // Because system decimal separator // will be added below token.Clear(); } // Read the fractional part of number if (pos < expression.Length && expression[pos] == decimalSeparator) { // Add current system specific decimal separator token.Append(CultureInfo.CurrentCulture .NumberFormat.NumberDecimalSeparator); while (++pos < expression.Length && Char.IsDigit(expression[pos])) { token.Append(expression[pos]); } } // Read scientific notation (suffix) if (pos + 1 < expression.Length && expression[pos] == 'e' && (Char.IsDigit(expression[pos + 1]) || (pos + 2 < expression.Length && (expression[pos + 1] == '+' || expression[pos + 1] == '-') && Char.IsDigit(expression[pos + 2])))) { token.Append(expression[pos++]); // e if (expression[pos] == '+' || expression[pos] == '-') token.Append(expression[pos++]); // sign while (pos < expression.Length && Char.IsDigit(expression[pos])) { token.Append(expression[pos++]); // power } // Convert number from scientific notation to decimal notation return NumberMaker + Convert.ToDouble(token.ToString()); } return NumberMaker + token.ToString(); } else { throw new ArgumentException("Unknown token in expression"); } } /// /// Syntax analysis of infix notation /// ///Token ///Output string (math expression in RPN) ///Stack which contains operators (or functions) /// Output string (math expression in RPN) private StringBuilder SyntaxAnalysisInfixNotation(string token, StringBuilder outputString, Stack stack) { // If it's a number just put to string if (token[0] == NumberMaker[0]) { outputString.Append(token); } else if (token[0] == FunctionMarker[0]) { // if it's a function push to stack stack.Push(token); } else if (token == LeftParent) { // If its '(' push to stack stack.Push(token); } else if (token == RightParent) { // If its ')' pop elements from stack to output string // until find the ')' string elem; while ((elem = stack.Pop()) != LeftParent) { outputString.Append(elem); } // if after this a function is in the peek of stack then put it to string if (stack.Count > 0 && stack.Peek()[0] == FunctionMarker[0]) { outputString.Append(stack.Pop()); } } else { // While priority of elements at peek of stack >= (>) token's priority // put these elements to output string while (stack.Count > 0 && Priority(token, stack.Peek())) { outputString.Append(stack.Pop()); } stack.Push(token); } return outputString; } /// /// Is priority of token less (or equal) to priority of p /// private bool Priority(string token, string p) { return IsRightAssociated(token) ? GetPriority(token) < GetPriority(p) : GetPriority(token) <= GetPriority(p); } /// /// Is right associated operator /// private bool IsRightAssociated(string token) { return token == Degree; } /// /// Get priority of operator /// private int GetPriority(string token) { switch (token) { case LeftParent: return 0; case Plus: case Minus: return 2; case UnPlus: case UnMinus: return 6; case Multiply: case Divide: return 4; case Degree: case Sqrt: return 8; case Sin: case Cos: case Tg: case Ctg: case Sh: case Ch: case Th: case Log: case Ln: case Exp: case Abs: return 10; default: throw new ArgumentException("Unknown operator"); } } #endregion #region Calculate expression in RPN /// /// Calculate expression in reverse-polish notation /// ///Math expression in reverse-polish notation /// Result private double Calculate(string expression) { int pos = 0; // Current position of lexical analysis var stack = new Stack(); // Contains operands // Analyse entire expression while (pos < expression.Length) { string token = LexicalAnalysisRPN(expression, ref pos); stack = SyntaxAnalysisRPN(stack, token); } // At end of analysis in stack should be only one operand (result) if (stack.Count > 1) { throw new ArgumentException("Excess operand"); } return stack.Pop(); } /// /// Produce token by the given math expression /// ///Math expression in reverse-polish notation ///Current position of lexical analysis /// Token private string LexicalAnalysisRPN(string expression, ref int pos) { StringBuilder token = new StringBuilder(); // Read token from marker to next marker token.Append(expression[pos++]); while (pos < expression.Length && expression[pos] != NumberMaker[0] && expression[pos] != OperatorMarker[0] && expression[pos] != FunctionMarker[0]) { token.Append(expression[pos++]); } return token.ToString(); } /// /// Syntax analysis of reverse-polish notation /// ///Stack which contains operands ///Token /// Stack which contains operands private Stack SyntaxAnalysisRPN(Stack stack, string token) { // if it's operand then just push it to stack if (token[0] == NumberMaker[0]) { stack.Push(double.Parse(token.Remove(0, 1))); } // Otherwise apply operator or function to elements in stack else if (NumberOfArguments(token) == 1) { double arg = stack.Pop(); double rst; switch (token) { case UnPlus: rst = arg; break; case UnMinus: rst = -arg; break; case Sqrt: rst = Math.Sqrt(arg); break; case Sin: rst = ApplyTrigFunction(Math.Sin, arg); break; case Cos: rst = ApplyTrigFunction(Math.Cos, arg); break; case Tg: rst = ApplyTrigFunction(Math.Tan, arg); break; case Ctg: rst = 1 / ApplyTrigFunction(Math.Tan, arg); break; case Sh: rst = Math.Sinh(arg); break; case Ch: rst = rst = Math.Cosh(arg); break; case Th: rst = Math.Tanh(arg); break; case Ln: rst = Math.Log(arg); break; case Exp: rst = Math.Exp(arg); break; case Abs: rst = Math.Abs(arg); break; default: throw new ArgumentException("Unknown operator"); } stack.Push(rst); } else { // otherwise operator's number of arguments equals to 2 double arg2 = stack.Pop(); double arg1 = stack.Pop(); double rst; switch (token) { case Plus: rst = arg1 + arg2; break; case Minus: rst = arg1 - arg2; break; case Multiply: rst = arg1 * arg2; break; case Divide: if (arg2 == 0) { throw new DivideByZeroException("Second argument is zero"); } rst = arg1 / arg2; break; case Degree: rst = Math.Pow(arg1, arg2); break; case Log: rst = Math.Log(arg2, arg1); break; default: throw new ArgumentException("Unknown operator"); } stack.Push(rst); } return stack; } /// /// Apply trigonometric function /// ///Trigonometric function ///Argument /// Result of function private double ApplyTrigFunction(Func<double, double> func, double arg) { if (!isRadians) { arg = arg * Math.PI / 180; // Convert value to degree } return func(arg); } /// /// Produce number of arguments for the given operator /// private int NumberOfArguments(string token) { switch (token) { case UnPlus: case UnMinus: case Sqrt: case Tg: case Sh: case Ch: case Th: case Ln: case Ctg: case Sin: case Cos: case Exp: case Abs: return 1; case Plus: case Minus: case Multiply: case Divide: case Degree: case Log: return 2; default: throw new ArgumentException("Unknown operator"); } } #endregion } } |
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