Heronian triangle has integer sides and integer area. Find a Heronian triangle where all 3 sides are squares. Only one solution is known.
Extra credit for anyone who finds a second solution.
clearvars,clc
for a=1:1000
a2=a^2;
for b=a:1000
b2=b^2;
for c=b:1000
c2=c^2;
if c2<a2+b2
s=(a2+b2+c2)/2;
A=sqrt(s*(s-a2)*(s-b2)*(s-c2));
if A==round(A)
fprintf('%8d %8d %8d %4d %4d %4d %15d\n',a2, b2, c2, a, b, c, A)
if round(sqrt(A))^2==A
disp('-----------------------')
return
end
end
end
end
end
end
finds plenty of such triangles:
sides square roots Area
of sides
7396 16129 20164 86 127 142 55059062
7396 313600 314721 86 560 561 1148263825
8281 26244 33856 91 162 184 48525302
8464 124609 126736 92 353 356 514486772
9604 61009 64516 98 247 254 279750812
10609 38416 43264 103 196 208 191061302
11025 176400 180625 105 420 425 908491456
11236 77841 81796 106 279 286 418681568
12544 29929 42436 112 173 206 17153672
12996 21316 24964 114 146 158 138511367
13924 128164 132496 118 358 364 861095423
16900 36864 52441 130 192 229 143702606
17689 156816 162409 133 396 403 1337211657
19044 44100 54289 138 210 233 388262856
19044 152100 158404 138 390 398 1392340296
19321 57600 67081 139 240 259 518407199
19600 78400 87025 140 280 295 722753906
20449 279841 285156 143 529 534 2787308402
21904 391876 396900 148 626 630 4202565095
21904 609961 624100 148 781 790 5160471811
22201 48400 60516 149 220 246 495917271
23716 451584 463761 154 672 681 4655517566
23716 898704 902500 154 948 950 10540737820
24025 56644 65536 155 238 256 668420829
24025 108241 118336 155 329 344 1227949799
24336 364816 370881 156 604 609 4332378108
25921 77284 90000 161 278 300 933228445
28224 774400 779689 168 880 883 10769617724
29584 64516 80656 172 254 284 880944992
29929 132496 145161 173 364 381 1871353208
30625 171396 183184 175 414 428 2496224695
30976 35344 55696 176 188 236 499777532
30976 103041 117649 176 321 343 1492127713
30976 321489 330625 176 567 575 4819883015
32400 303601 313600 180 551 560 4748745710
32761 40401 61504 181 201 248 604515673
33124 104976 135424 182 324 368 776404832
33856 498436 506944 184 706 712 8231788352
34969 146689 160000 187 383 400 2463141799
35344 60025 67081 188 245 259 1057098245
36100 112896 134689 190 336 367 1762322135
36481 42436 74529 191 206 273 481931842
38416 244036 258064 196 494 508 4476012992
39601 277729 291600 199 527 540 5266635545
40000 846400 848241 200 920 921 16923735271
41616 327184 341056 204 572 584 6542032892
41616 813604 822649 204 902 907 16611252410
42436 153664 173056 206 392 416 3056980832
44100 64009 91204 210 253 302 1291598056
44100 435600 448900 210 660 670 9285999900
44100 705600 722500 210 840 850 14535863296
44944 235225 253009 212 485 503 5016671801
44944 311364 327184 212 558 572 6698905088
49284 148996 173056 222 386 416 3422232572
49284 450241 452929 222 671 673 11094838718
50176 119716 169744 224 346 412 274458752
51984 85264 99856 228 292 316 2216181872
51984 120409 148225 228 347 385 2893577425
52900 123904 152100 230 352 390 3031179260
53361 848241 861184 231 921 928 22112387568
54289 492804 509796 233 702 714 12904878912
54756 209764 234256 234 458 484 5394761167
55225 120409 150544 235 347 388 3069008801
55696 210681 248004 236 459 498 4705490863
55696 512656 529984 236 716 728 13777526768
57121 195364 231361 239 442 481 4688855152
61009 90000 127449 247 300 357 2513083449
61009 214369 251001 247 463 501 5627041977
61009 577600 625681 247 760 791 11282524060
65025 302500 329476 255 550 574 9298167590
66564 145161 181476 258 381 426 4459784022
67081 189225 222784 259 435 472 5902867865
67600 147456 209764 260 384 458 2299241696
67600 585225 606841 260 765 779 19057486625
67600 837225 855625 260 915 925 27507031225
68121 765625 784996 261 875 886 25292843650
69696 568516 568516 264 754 754 19774391869
70225 887364 906304 265 942 952 30299910551
70756 627264 649636 266 792 806 21395386512
71289 84100 96721 267 290 311 2914828827
71289 254016 308025 267 504 555 6485299191
72361 559504 600625 269 748 775 17235300241
73984 544644 570025 272 738 755 19323040241
74529 236196 304704 273 486 552 3930549462
75076 574564 599076 274 758 774 20778239367
75625 366025 396900 275 605 630 13102343650
76176 176400 217156 276 420 466 6212205696
76176 407044 436921 276 638 661 14724326107
76176 608400 633616 276 780 796 22277444736
76729 374544 405769 277 612 637 13606398788
77284 230400 268324 278 480 518 8294515184
78400 313600 348100 280 560 590 11564062496
80656 164836 209764 284 406 458 6125924732
81796 110889 126736 286 333 356 4475798717
81796 125316 189225 286 354 435 3876309795
82944 693889 720801 288 833 849 27700238463
84100 324900 362404 290 570 602 12836933960
84100 470596 476100 290 686 690 19783084755
85264 724201 751689 292 851 867 29730616632
86436 549081 580644 294 741 762 22659815772
87025 157609 207936 295 397 456 6301584655
87616 142129 194481 296 377 441 5708384992
88804 193600 242064 298 440 492 7934676336
90000 268324 312481 300 518 559 11249567545
91204 162409 246016 302 403 496 3627022373
91204 431649 469225 302 657 685 18620311966
93025 887364 915849 305 942 957 39868403934
93636 335241 378225 306 579 615 14709433806
95481 345744 389376 309 588 624 15475965462
95481 436921 529984 309 661 728 5136064192
96100 226576 262144 310 476 512 10694733264
96100 432964 473344 310 658 688 19647196784
97969 399424 442225 313 632 665 18416512784
99225 217156 252004 315 466 502 10650256190
101124 700569 736164 318 837 858 33913207008
103041 224676 280900 321 474 530 10685032206
103041 304704 378225 321 552 615 12184896672
103041 481636 555025 321 694 745 18652366914
103684 309136 360000 322 556 600 14931655120
106929 828100 864900 327 910 930 42408218916
108241 308025 326041 329 555 571 16670306322
108900 224676 285156 330 474 534 11276390696
110889 611524 654481 333 782 809 32231711502
111556 802816 833569 334 896 913 43766708950
112225 173056 241081 335 416 491 8895517775
112225 260100 285156 335 510 534 14592421143
112896 156025 226576 336 395 476 8054923086
112896 269361 381924 336 519 618 1389447432
112896 499849 547600 336 707 740 26632528370
112896 717409 758641 336 847 871 38668589687
114921 274576 319225 339 524 565 15422617764
114921 589824 646416 339 768 804 30778605344
116281 409600 436921 341 640 661 23692939264
116964 129600 207025 342 360 455 6918252795
116964 182329 253009 342 427 503 9769670062
116964 191844 224676 342 438 474 11219420727
117649 641601 717409 343 801 847 30452462544
118336 258064 322624 344 508 568 14095119872
118336 259081 292681 344 509 541 15287306512
118336 624100 670761 344 790 819 35056117575
119716 529984 580644 346 728 762 29941651328
119716 910116 950625 346 954 975 52296442104
121104 616225 717409 348 785 847 22094354754
121801 850084 861184 349 922 928 51760095546
122500 685584 732736 350 828 856 39939595120
123201 309136 327184 351 556 572 19020523883
123201 720801 767376 351 849 876 42289004088
123904 141376 222784 352 376 472 7996440512
123904 412164 470596 352 642 686 23874043408
124609 597529 619369 353 773 787 37126117161
125316 150544 187489 354 388 433 9398470492
125316 454276 516961 354 674 719 26127296767
126025 813604 859329 355 902 927 48976631150
126736 230400 303601 356 480 551 13417090706
127449 484416 535824 357 696 732 29512357345
128164 518400 574564 358 720 758 31260876775
128164 690561 715716 358 831 846 43998240210
128881 260100 343396 359 510 586 14495667509
130321 284089 355216 361 533 596 17086466657
131044 161604 246016 362 402 496 9672250768
131044 284089 330625 362 533 575 18393292706
132496 310249 375769 364 557 613 19378950487
132496 419904 541696 364 648 736 12422477312
133225 697225 749956 365 835 866 44075942450
133225 727609 741321 365 853 861 48464161994
133225 751689 802816 365 867 896 47634491295
135424 293764 350464 368 542 592 19364667846
136900 528529 644809 370 727 803 21049694449
136900 784996 894916 370 886 946 34157871624
137641 217156 300304 371 466 548 13678597109
138384 238144 270400 372 488 520 16463347204
138384 579121 608400 372 761 780 39879826281
138384 739600 793881 372 860 891 48601533360
139876 586756 640000 374 766 800 39410268784
139876 700569 756900 374 837 870 46435218330
141376 240100 268324 376 490 518 16913571920
141376 788544 808201 376 888 899 55667690219
142884 547600 611524 378 740 782 36747873671
144400 451584 538756 380 672 734 28197154160
145924 169744 298116 382 412 546 7710909472
147456 321489 401956 384 567 634 21878992617
149769 763876 894916 387 874 946 29951413663
151321 662596 680625 389 814 825 50131260353
152100 331776 471969 390 576 687 11639911086
152100 881721 940900 390 939 970 63621507373
152881 649636 688900 391 806 830 49119611650
153664 826281 919681 392 909 959 53054221650
155236 378225 398161 394 615 631 29277862299
159201 753424 819025 399 868 905 56730585625
160000 693889 762129 400 833 873 52358983199
160801 171396 209764 401 414 458 13491167672
163216 213444 316969 404 462 563 15920346173
165649 361201 451584 407 601 672 27615321207
167281 758641 810000 409 871 900 62076570075
168100 603729 622521 410 777 789 50726678009
168100 693889 758641 410 833 871 55953620188
168921 739600 799236 411 860 894 60433451428
169744 614656 692224 412 784 832 48911693312
170569 614656 762129 413 784 873 29272378729
171396 396900 488601 414 630 699 31449291336
172225 908209 976144 415 953 988 74242465796
173056 212521 372100 416 461 610 9347207079
173056 228484 272484 416 478 522 19671222103
173889 518400 603729 417 720 777 41990983119
176400 256036 364816 420 506 604 20665568896
176400 705600 783225 420 840 885 58543066386
177241 605284 667489 421 778 817 52294897907
178929 252004 399424 423 502 632 15879299292
178929 422500 458329 423 650 677 37798579064
179776 367236 476100 424 606 690 29469978722
179776 770884 811801 424 878 901 68817012180
180625 413449 435600 425 643 660 37179469217
180625 576081 662596 425 759 814 48575610754
180625 813604 850084 425 902 922 73142841227
181476 319225 425104 426 565 652 26598608795
181476 518400 678976 426 720 824 25017477948
184041 271441 384400 429 521 620 22865787225
184041 781456 887364 429 884 942 62412329770
184900 240100 357604 430 490 598 20285878280
184900 403225 504100 430 635 710 34411913750
186624 543169 602176 432 737 776 50018313145
188356 258064 322624 434 508 568 24298749695
192721 574564 669124 439 758 818 51581806292
193600 864900 883600 440 930 940 83713776908
194481 413449 440896 441 643 664 40043076710
195364 715716 804609 442 846 897 65574062502
197136 595984 692224 444 772 832 54755721152
197136 883600 889249 444 940 943 86795575781
198025 367236 458329 445 606 677 35230202777
198916 446224 600625 446 668 775 32223617664
198916 480249 585225 446 693 765 44214453810
199809 435600 544644 447 660 738 40169298951
200704 478864 678976 448 692 824 4391340032
206116 247009 413449 454 497 643 19071901741
207025 656100 846400 455 810 920 30328313750
207936 341056 399424 456 584 632 35458909952
207936 481636 592900 456 694 770 46297238800
211600 259081 395641 460 509 629 25036705796
211600 495616 608400 460 704 780 48498868160
212521 394384 480249 461 628 693 41233801664
212521 633616 737881 461 796 859 62728696697
213444 413449 534361 462 643 731 40607891122
. . .
At first I tried to find those where the area is also a perfect square, to limit the quantity of solutions, but the program found none.
But then, looking into Wikipedia, I found the problem: these are not primitive Heronian triangles (GCD of sides is 1 in a primitive one).
What should have been asked for are primitive Heronian triangles. Wikipedia actually gives two such, one having been discovered more recently than the puzzle maker's source:
As of February 2021, only two primitive Heronian triangles with perfect square sides are known:
(1853², 4380², 4427², Area=32918611718880), published in 2013.
(11789², 68104² , 68595², Area=284239560530875680), published in 2018.
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Posted by Charlie
on 2024-01-02 13:26:33 |
computer attempt; researched answer (spoiler)