perplexus.info :: Just Math : Linear and Quadratic Sum Puzzle

Linear and Quadratic Sum Puzzle (Posted on 2015-07-25)

Given that each of A,B,C and D is a distinct positive integer.

Find the four smallest values of A+B+C+D such that each of A+B+C+D and A²+ B²+ C²+ D² is a perfect square.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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re: computer-aided solution

Comment 3 of 3 |

(In reply to computer-aided solution by Charlie)

An example of each total from 5^2 to 105^2:

1 4 8 12 25 225 5 15

1 5 13 17 36 484 6 22

1 2 14 32 49 1225 7 35

3 5 19 37 64 1764 8 42

1 10 30 40 81 2601 9 51

2 10 14 74 100 5776 10 76

1 2 30 88 121 8649 11 93

1 7 25 111 144 12996 12 114

1 6 16 146 169 21609 13 147

1 5 73 117 196 19044 14 138

1 2 20 202 225 41209 15 203

1 3 65 187 256 39204 16 198

1 2 80 206 289 48841 17 221

1 5 125 193 324 52900 18 230

1 8 78 274 361 81225 19 285

1 3 27 369 400 136900 20 370

1 8 142 290 441 104329 21 323

1 17 121 345 484 133956 22 366

1 10 52 466 529 219961 23 469

1 7 273 295 576 161604 24 402

1 2 200 422 625 218089 25 467

1 5 77 593 676 357604 26 598

1 2 106 620 729 395641 27 629

1 7 65 711 784 509796 28 714

1 2 68 770 841 597529 29 773

1 5 41 853 900 729316 30 854

1 8 328 624 961 497025 31 705

1 11 187 825 1024 715716 32 846

1 10 256 822 1089 741321 33 861

1 13 45 1097 1156 1205604 34 1098

1 4 124 1096 1225 1216609 35 1103

1 3 355 937 1296 1004004 36 1002

1 2 376 990 1369 1121481 37 1059

1 5 377 1061 1444 1267876 38 1126

1 8 92 1420 1521 2024929 39 1423

1 3 177 1419 1600 2044900 40 1430

1 2 422 1256 1681 1755625 41 1325

1 25 557 1181 1764 1705636 42 1306

1 18 400 1430 1849 2205225 43 1485

1 17 103 1815 1936 3305124 44 1818

1 2 394 1628 2025 2805625 45 1675

1 5 577 1533 2116 2683044 46 1638

1 4 112 2092 2209 4389025 47 2095

1 7 359 1937 2304 3880900 48 1970

1 2 150 2248 2401 5076009 49 2253

1 5 153 2341 2500 5503716 50 2346

1 8 404 2188 2601 4950625 51 2225

1 15 343 2345 2704 5616900 52 2370

1 6 522 2280 2809 5470921 53 2339

1 13 165 2737 2916 7518564 54 2742

1 2 628 2394 3025 6125625 55 2475

1 15 1207 1913 3136 5116644 56 2262

1 12 256 2980 3249 8946081 57 2991

1 17 1349 1997 3364 5808100 58 2410

1 2 422 3056 3481 9517225 59 3085

1 3 385 3211 3600 10458756 60 3234

1 2 250 3468 3721 12089529 61 3477

1 5 345 3493 3844 12320100 62 3510

1 10 866 3092 3969 10310521 63 3211

1 7 553 3535 4096 12802084 64 3578

1 2 452 3770 4225 14417209 65 3797

1 13 2017 2325 4356 9474084 66 3078

1 10 352 4126 4489 17147881 67 4141

1 7 1055 3561 4624 13793796 68 3714

1 2 166 4592 4761 21114025 69 4595

1 5 1761 3133 4900 12916836 70 3594

1 10 292 4738 5041 22534009 71 4747

1 11 425 4747 5184 22714756 72 4766

1 6 102 5220 5329 27258841 73 5221

1 5 305 5165 5476 26770276 74 5174

1 4 974 4646 5625 22534009 75 4747

1 15 345 5415 5776 29441476 76 5426

1 2 1410 4516 5929 22382361 77 4731

1 21 565 5497 6084 30536676 78 5526

1 8 1184 5048 6241 26884225 79 5185

1 3 3075 3321 6400 20484676 80 4526

1 2 1892 4666 6561 25351225 81 5035

1 5 1297 5421 6724 31069476 82 5574

1 2 440 6446 6889 41744521 83 6461

1 3 1363 5689 7056 34222500 84 5850

1 2 1940 5282 7225 31663129 85 5627

1 5 773 6617 7396 44382244 86 6662

1 2 122 7444 7569 55428025 87 7445

1 11 2081 5651 7744 36264484 88 6022

1 2 628 7290 7921 53538489 89 7317

1 5 1141 6953 8100 49646116 90 7046

1 2 1600 6678 8281 47155689 91 6867

1 7 3001 5455 8464 38763076 92 6226

1 2 4232 4414 8649 37393225 93 6115

1 5 3205 5625 8836 41912676 94 6474

1 6 718 8300 9025 69405561 95 8331

1 3 2467 6745 9216 51581124 96 7182

1 6 136 9266 9409 85877289 97 9267

1 9 2297 7297 9604 58522500 98 7650

1 4 1828 7968 9801 66830625 99 8175

1 3 4051 5945 10000 51753636 100 7194

1 4 3078 7118 10201 60140025 101 7755

1 17 3281 7105 10404 61246276 102 7826

1 2 2840 7766 10609 68376361 103 8269

1 3 859 9953 10816 99800100 104 9990

1 8 254 10762 11025 115885225 105 10765

Posted by Charlie on 2015-07-25 13:30:24

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