For example, 625 = 25^2, and 256 = 16^2
The only digit that can work is 6. We can see it easily by listing all the 3-digit square numbers:
10^2 = 100 21^2 = 441
11^2 = 121 22^2 = 484
12^2 = 144 23^2 = 529
13^2 = 169 24^2 = 576
14^2 = 196 25^2 = 625
15^2 = 225 26^2 = 676
16^2 = 256 27^2 = 729
17^2 = 289 28^2 = 784
18^2 = 324 29^2 = 841
19^2 = 361 30^2 = 900
20^2 = 400 31^2 = 961
Can we find 2 digits, x, y, such as xyAB is a square and ABxy is also a square?
32^2 = 1024 49^2 = 2401 66^2 = 4356 83^2 = 6889
33^2 = 1089 50^2 = 2500 67^2 = 4489 84^2 = 7056
34^2 = 1156 51^2 = 2601 68^2 = 4624 85^2 = 7225
35^2 = 1225 52^2 = 2704 69^2 = 4761 86^2 = 7396
36^2 = 1296 53^2 = 2809 70^2 = 4900 87^2 = 7569
37^2 = 1369 54^2 = 2916 71^2 = 5041 88^2 = 7744
38^2 = 1444 55^2 = 3025 72^2 = 5184 89^2 = 7921
39^2 = 1521 56^2 = 3136 73^2 = 5329 90^2 = 8100
40^2 = 1600 57^2 = 3249 74^2 = 5476 91^2 = 8281
41^2 = 1681 58^2 = 3364 75^2 = 5625 92^2 = 8464
42^2 = 1764 59^2 = 3481 76^2 = 5776 93^2 = 8649
43^2 = 1849 60^2 = 3600 77^2 = 5929 94^2 = 8836
44^2 = 1936 61^2 = 3721 78^2 = 6084 95^2 = 9025
45^2 = 2025 62^2 = 3844 79^2 = 6241 96^2 = 9216
46^2 = 2116 63^2 = 3969 80^2 = 6400 97^2 = 9409
47^2 = 2209 64^2 = 4096 81^2 = 6561 98^2 = 9604
48^2 = 2304 65^2 = 4225 82^2 = 6724 99^2 = 9801
Let's look at 5-digit square numbers
---------------------------------------------------------------
Here I look at numbers of the form xyABC and ABCxy
100^2 = 10000 154^2 = 23716 208^2 = 43264 262^2 = 68644
101^2 = 10201 155^2 = 24025 209^2 = 43681 263^2 = 69169
102^2 = 10404 156^2 = 24336 210^2 = 44100 264^2 = 69696
103^2 = 10609 157^2 = 24649 211^2 = 44521 265^2 = 70225
104^2 = 10816 158^2 = 24964 212^2 = 44944 266^2 = 70756
105^2 = 11025 159^2 = 25281 213^2 = 45369 267^2 = 71289
106^2 = 11236 160^2 = 25600 214^2 = 45796 268^2 = 71824
107^2 = 11449 161^2 = 25921 215^2 = 46225 269^2 = 72361
108^2 = 11664 162^2 = 26244 216^2 = 46656 270^2 = 72900
109^2 = 11881 163^2 = 26569 217^2 = 47089 271^2 = 73441
110^2 = 12100 164^2 = 26896 218^2 = 47524 272^2 = 73984
111^2 = 12321 165^2 = 27225 219^2 = 47961 273^2 = 74529
112^2 = 12544 166^2 = 27556 220^2 = 48400 274^2 = 75076
113^2 = 12769 167^2 = 27889 221^2 = 48841 275^2 = 75625
114^2 = 12996 168^2 = 28224 222^2 = 49284 276^2 = 76176
115^2 = 13225 169^2 = 28561 223^2 = 49729 277^2 = 76729
116^2 = 13456 170^2 = 28900 224^2 = 50176 278^2 = 77284
117^2 = 13689 171^2 = 29241 225^2 = 50625 279^2 = 77841
118^2 = 13924 172^2 = 29584 226^2 = 51076 280^2 = 78400
119^2 = 14161 173^2 = 29929 227^2 = 51529 281^2 = 78961
120^2 = 14400 174^2 = 30276 228^2 = 51984 282^2 = 79524
121^2 = 14641 175^2 = 30625 229^2 = 52441 283^2 = 80089
122^2 = 14884 176^2 = 30976 230^2 = 52900 284^2 = 80656
123^2 = 15129 177^2 = 31329 231^2 = 53361 285^2 = 81225
124^2 = 15376 178^2 = 31684 232^2 = 53824 286^2 = 81796
125^2 = 15625 179^2 = 32041 233^2 = 54289 287^2 = 82369
126^2 = 15876 180^2 = 32400 234^2 = 54756 288^2 = 82944
127^2 = 16129 181^2 = 32761 235^2 = 55225 289^2 = 83521
128^2 = 16384 182^2 = 33124 236^2 = 55696 290^2 = 84100
129^2 = 16641 183^2 = 33489 237^2 = 56169 291^2 = 84681
130^2 = 16900 184^2 = 33856 238^2 = 56644 292^2 = 85264
131^2 = 17161 185^2 = 34225 239^2 = 57121 293^2 = 85849
132^2 = 17424 186^2 = 34596 240^2 = 57600 294^2 = 86436
133^2 = 17689 187^2 = 34969 241^2 = 58081 295^2 = 87025
134^2 = 17956 188^2 = 35344 242^2 = 58564 296^2 = 87616
135^2 = 18225 189^2 = 35721 243^2 = 59049 297^2 = 88209
136^2 = 18496 190^2 = 36100 244^2 = 59536 298^2 = 88804
137^2 = 18769 191^2 = 36481 245^2 = 60025 299^2 = 89401
138^2 = 19044 192^2 = 36864 246^2 = 60516 300^2 = 90000
139^2 = 19321 193^2 = 37249 247^2 = 61009 301^2 = 90601
140^2 = 19600 194^2 = 37636 248^2 = 61504 302^2 = 91204
141^2 = 19881 195^2 = 38025 249^2 = 62001 303^2 = 91809
142^2 = 20164 196^2 = 38416 250^2 = 62500 304^2 = 92416
143^2 = 20449 197^2 = 38809 251^2 = 63001 305^2 = 93025
144^2 = 20736 198^2 = 39204 252^2 = 63504 306^2 = 93636
145^2 = 21025 199^2 = 39601 253^2 = 64009 307^2 = 94249
146^2 = 21316 200^2 = 40000 254^2 = 64516 308^2 = 94864
147^2 = 21609 201^2 = 40401 255^2 = 65025 309^2 = 95481
148^2 = 21904 202^2 = 40804 256^2 = 65536 310^2 = 96100
149^2 = 22201 203^2 = 41209 257^2 = 66049 311^2 = 96721
150^2 = 22500 204^2 = 41616 258^2 = 66564 312^2 = 97344
151^2 = 22801 205^2 = 42025 259^2 = 67081 313^2 = 97969
152^2 = 23104 206^2 = 42436 260^2 = 67600 314^2 = 98596
153^2 = 23409 207^2 = 42849 261^2 = 68121 315^2 = 99225
316^2 = 99856
6-digit square numbers
-----------------------------------------
Here I look at numbers of the forms (xyABCD, ABCDxy) and (xyzABC, ABCxyz)
317^2 = 100489 488^2 = 238144 658^2 = 432964 828^2 = 685584
318^2 = 101124 489^2 = 239121 659^2 = 434281 829^2 = 687241
319^2 = 101761 490^2 = 240100 660^2 = 435600 830^2 = 688900
320^2 = 102400 491^2 = 241081 661^2 = 436921 831^2 = 690561
321^2 = 103041 492^2 = 242064 662^2 = 438244 832^2 = 692224
322^2 = 103684 493^2 = 243049 663^2 = 439569 833^2 = 693889
323^2 = 104329 494^2 = 244036 664^2 = 440896 834^2 = 695556
324^2 = 104976 495^2 = 245025 665^2 = 442225 835^2 = 697225
325^2 = 105625 496^2 = 246016 666^2 = 443556 836^2 = 698896
326^2 = 106276 497^2 = 247009 667^2 = 444889 837^2 = 700569
327^2 = 106929 498^2 = 248004 668^2 = 446224 838^2 = 702244
328^2 = 107584 499^2 = 249001 669^2 = 447561 839^2 = 703921
329^2 = 108241 500^2 = 250000 670^2 = 448900 840^2 = 705600
330^2 = 108900 501^2 = 251001 671^2 = 450241 841^2 = 707281
331^2 = 109561 502^2 = 252004 672^2 = 451584 842^2 = 708964
332^2 = 110224 503^2 = 253009 673^2 = 452929 843^2 = 710649
333^2 = 110889 504^2 = 254016 674^2 = 454276 844^2 = 712336
334^2 = 111556 505^2 = 255025 675^2 = 455625 845^2 = 714025
335^2 = 112225 506^2 = 256036 676^2 = 456976 846^2 = 715716
336^2 = 112896 507^2 = 257049 677^2 = 458329 847^2 = 717409
337^2 = 113569 508^2 = 258064 678^2 = 459684 848^2 = 719104
338^2 = 114244 509^2 = 259081 679^2 = 461041 849^2 = 720801
339^2 = 114921 510^2 = 260100 680^2 = 462400 850^2 = 722500
340^2 = 115600 511^2 = 261121 681^2 = 463761 851^2 = 724201
341^2 = 116281 512^2 = 262144 682^2 = 465124 852^2 = 725904
342^2 = 116964 513^2 = 263169 683^2 = 466489 853^2 = 727609
343^2 = 117649 514^2 = 264196 684^2 = 467856 854^2 = 729316
344^2 = 118336 515^2 = 265225 685^2 = 469225 855^2 = 731025
345^2 = 119025 516^2 = 266256 686^2 = 470596 856^2 = 732736
346^2 = 119716 517^2 = 267289 687^2 = 471969 857^2 = 734449
347^2 = 120409 518^2 = 268324 688^2 = 473344 858^2 = 736164
348^2 = 121104 519^2 = 269361 689^2 = 474721 859^2 = 737881
349^2 = 121801 520^2 = 270400 690^2 = 476100 860^2 = 739600
350^2 = 122500 521^2 = 271441 691^2 = 477481 861^2 = 741321
351^2 = 123201 522^2 = 272484 692^2 = 478864 862^2 = 743044
352^2 = 123904 523^2 = 273529 693^2 = 480249 863^2 = 744769
353^2 = 124609 524^2 = 274576 694^2 = 481636 864^2 = 746496
354^2 = 125316 525^2 = 275625 695^2 = 483025 865^2 = 748225
355^2 = 126025 526^2 = 276676 696^2 = 484416 866^2 = 749956
356^2 = 126736 527^2 = 277729 697^2 = 485809 867^2 = 751689
357^2 = 127449 528^2 = 278784 698^2 = 487204 868^2 = 753424
358^2 = 128164 529^2 = 279841 699^2 = 488601 869^2 = 755161
359^2 = 128881 530^2 = 280900 700^2 = 490000 870^2 = 756900
360^2 = 129600 531^2 = 281961 701^2 = 491401 871^2 = 758641
361^2 = 130321 532^2 = 283024 702^2 = 492804 872^2 = 760384
362^2 = 131044 533^2 = 284089 703^2 = 494209 873^2 = 762129
363^2 = 131769 534^2 = 285156 704^2 = 495616 874^2 = 763876
364^2 = 132496 535^2 = 286225 705^2 = 497025 875^2 = 765625
365^2 = 133225 536^2 = 287296 706^2 = 498436 876^2 = 767376
366^2 = 133956 537^2 = 288369 707^2 = 499849 877^2 = 769129
367^2 = 134689 538^2 = 289444 708^2 = 501264 878^2 = 770884
368^2 = 135424 539^2 = 290521 709^2 = 502681 879^2 = 772641
369^2 = 136161 540^2 = 291600 710^2 = 504100 880^2 = 774400
370^2 = 136900 541^2 = 292681 711^2 = 505521 881^2 = 776161
371^2 = 137641 542^2 = 293764 712^2 = 506944 882^2 = 777924
372^2 = 138384 543^2 = 294849 713^2 = 508369 883^2 = 779689
373^2 = 139129 544^2 = 295936 714^2 = 509796 884^2 = 781456
374^2 = 139876 545^2 = 297025 715^2 = 511225 885^2 = 783225
375^2 = 140625 546^2 = 298116 716^2 = 512656 886^2 = 784996
376^2 = 141376 547^2 = 299209 717^2 = 514089 887^2 = 786769
377^2 = 142129 548^2 = 300304 718^2 = 515524 888^2 = 788544
378^2 = 142884 549^2 = 301401 719^2 = 516961 889^2 = 790321
379^2 = 143641 550^2 = 302500 720^2 = 518400 890^2 = 792100
380^2 = 144400 551^2 = 303601 721^2 = 519841 891^2 = 793881
381^2 = 145161 552^2 = 304704 722^2 = 521284 892^2 = 795664
382^2 = 145924 553^2 = 305809 723^2 = 522729 893^2 = 797449
383^2 = 146689 554^2 = 306916 724^2 = 524176 894^2 = 799236
384^2 = 147456 555^2 = 308025 725^2 = 525625 895^2 = 801025
385^2 = 148225 556^2 = 309136 726^2 = 527076 896^2 = 802816
386^2 = 148996 557^2 = 310249 727^2 = 528529 897^2 = 804609
387^2 = 149769 558^2 = 311364 728^2 = 529984 898^2 = 806404
388^2 = 150544 559^2 = 312481 729^2 = 531441 899^2 = 808201
389^2 = 151321 560^2 = 313600 730^2 = 532900 900^2 = 810000
390^2 = 152100 561^2 = 314721 731^2 = 534361 901^2 = 811801
391^2 = 152881 562^2 = 315844 732^2 = 535824 902^2 = 813604
392^2 = 153664 563^2 = 316969 733^2 = 537289 903^2 = 815409
393^2 = 154449 564^2 = 318096 734^2 = 538756 904^2 = 817216
394^2 = 155236 565^2 = 319225 735^2 = 540225 905^2 = 819025
395^2 = 156025 566^2 = 320356 736^2 = 541696 906^2 = 820836
396^2 = 156816 567^2 = 321489 737^2 = 543169 907^2 = 822649
397^2 = 157609 568^2 = 322624 738^2 = 544644 908^2 = 824464
398^2 = 158404 569^2 = 323761 739^2 = 546121 909^2 = 826281
399^2 = 159201 570^2 = 324900 740^2 = 547600 910^2 = 828100
401^2 = 160801 571^2 = 326041 741^2 = 549081 911^2 = 829921
402^2 = 161604 572^2 = 327184 742^2 = 550564 912^2 = 831744
403^2 = 162409 573^2 = 328329 743^2 = 552049 913^2 = 833569
404^2 = 163216 574^2 = 329476 744^2 = 553536 914^2 = 835396
405^2 = 164025 575^2 = 330625 745^2 = 555025 915^2 = 837225
406^2 = 164836 576^2 = 331776 746^2 = 556516 916^2 = 839056
407^2 = 165649 577^2 = 332929 747^2 = 558009 917^2 = 840889
408^2 = 166464 578^2 = 334084 748^2 = 559504 918^2 = 842724
409^2 = 167281 579^2 = 335241 749^2 = 561001 919^2 = 844561
410^2 = 168100 580^2 = 336400 750^2 = 562500 920^2 = 846400
411^2 = 168921 581^2 = 337561 751^2 = 564001 921^2 = 848241
412^2 = 169744 582^2 = 338724 752^2 = 565504 922^2 = 850084
413^2 = 170569 583^2 = 339889 753^2 = 567009 923^2 = 851929
414^2 = 171396 584^2 = 341056 754^2 = 568516 924^2 = 853776
415^2 = 172225 585^2 = 342225 755^2 = 570025 925^2 = 855625
416^2 = 173056 586^2 = 343396 756^2 = 571536 926^2 = 857476
417^2 = 173889 587^2 = 344569 757^2 = 573049 927^2 = 859329
418^2 = 174724 588^2 = 345744 758^2 = 574564 928^2 = 861184
419^2 = 175561 589^2 = 346921 759^2 = 576081 929^2 = 863041
420^2 = 176400 590^2 = 348100 760^2 = 577600 930^2 = 864900
421^2 = 177241 591^2 = 349281 761^2 = 579121 931^2 = 866761
422^2 = 178084 592^2 = 350464 762^2 = 580644 932^2 = 868624
423^2 = 178929 593^2 = 351649 763^2 = 582169 933^2 = 870489
424^2 = 179776 594^2 = 352836 764^2 = 583696 934^2 = 872356
425^2 = 180625 595^2 = 354025 765^2 = 585225 935^2 = 874225
426^2 = 181476 596^2 = 355216 766^2 = 586756 936^2 = 876096
427^2 = 182329 597^2 = 356409 767^2 = 588289 937^2 = 877969
428^2 = 183184 598^2 = 357604 768^2 = 589824 938^2 = 879844
429^2 = 184041 599^2 = 358801 769^2 = 591361 939^2 = 881721
430^2 = 184900 600^2 = 360000 770^2 = 592900 940^2 = 883600
431^2 = 185761 601^2 = 361201 771^2 = 594441 941^2 = 885481
432^2 = 186624 602^2 = 362404 772^2 = 595984 942^2 = 887364
433^2 = 187489 603^2 = 363609 773^2 = 597529 943^2 = 889249
434^2 = 188356 604^2 = 364816 774^2 = 599076 944^2 = 891136
435^2 = 189225 605^2 = 366025 775^2 = 600625 945^2 = 893025
436^2 = 190096 606^2 = 367236 776^2 = 602176 946^2 = 894916
437^2 = 190969 607^2 = 368449 777^2 = 603729 947^2 = 896809
438^2 = 191844 608^2 = 369664 778^2 = 605284 948^2 = 898704
439^2 = 192721 609^2 = 370881 779^2 = 606841 949^2 = 900601
440^2 = 193600 610^2 = 372100 780^2 = 608400 950^2 = 902500
441^2 = 194481 611^2 = 373321 781^2 = 609961 951^2 = 904401
442^2 = 195364 612^2 = 374544 782^2 = 611524 952^2 = 906304
443^2 = 196249 613^2 = 375769 783^2 = 613089 953^2 = 908209
444^2 = 197136 614^2 = 376996 784^2 = 614656 954^2 = 910116
445^2 = 198025 615^2 = 378225 785^2 = 616225 955^2 = 912025
446^2 = 198916 616^2 = 379456 786^2 = 617796 956^2 = 913936
447^2 = 199809 617^2 = 380689 787^2 = 619369 957^2 = 915849
448^2 = 200704 618^2 = 381924 788^2 = 620944 958^2 = 917764
449^2 = 201601 619^2 = 383161 789^2 = 622521 959^2 = 919681
450^2 = 202500 620^2 = 384400 790^2 = 624100 960^2 = 921600
451^2 = 203401 621^2 = 385641 791^2 = 625681 961^2 = 923521
452^2 = 204304 622^2 = 386884 792^2 = 627264 962^2 = 925444
453^2 = 205209 623^2 = 388129 793^2 = 628849 963^2 = 927369
454^2 = 206116 624^2 = 389376 794^2 = 630436 964^2 = 929296
455^2 = 207025 625^2 = 390625 795^2 = 632025 965^2 = 931225
456^2 = 207936 626^2 = 391876 796^2 = 633616 966^2 = 933156
457^2 = 208849 627^2 = 393129 797^2 = 635209 967^2 = 935089
458^2 = 209764 628^2 = 394384 798^2 = 636804 968^2 = 937024
459^2 = 210681 629^2 = 395641 799^2 = 638401 969^2 = 938961
460^2 = 211600 630^2 = 396900 800^2 = 640000 970^2 = 940900
461^2 = 212521 631^2 = 398161 801^2 = 641601 971^2 = 942841
462^2 = 213444 632^2 = 399424 802^2 = 643204 972^2 = 944784
463^2 = 214369 633^2 = 400689 803^2 = 644809 973^2 = 946729
464^2 = 215296 634^2 = 401956 804^2 = 646416 974^2 = 948676
465^2 = 216225 635^2 = 403225 805^2 = 648025 975^2 = 950625
466^2 = 217156 636^2 = 404496 806^2 = 649636 976^2 = 952576
467^2 = 218089 637^2 = 405769 807^2 = 651249 977^2 = 954529
468^2 = 219024 638^2 = 407044 808^2 = 652864 978^2 = 956484
469^2 = 219961 639^2 = 408321 809^2 = 654481 979^2 = 958441
470^2 = 220900 640^2 = 409600 810^2 = 656100 980^2 = 960400
471^2 = 221841 641^2 = 410881 811^2 = 657721 981^2 = 962361
472^2 = 222784 642^2 = 412164 812^2 = 659344 982^2 = 964324
473^2 = 223729 643^2 = 413449 813^2 = 660969 983^2 = 966289
474^2 = 224676 644^2 = 414736 814^2 = 662596 984^2 = 968256
475^2 = 225625 645^2 = 416025 815^2 = 664225 985^2 = 970225
476^2 = 226576 646^2 = 417316 816^2 = 665856 986^2 = 972196
477^2 = 227529 647^2 = 418609 817^2 = 667489 987^2 = 974169
478^2 = 228484 648^2 = 419904 818^2 = 669124 988^2 = 976144
479^2 = 229441 649^2 = 421201 819^2 = 670761 989^2 = 978121
480^2 = 230400 650^2 = 422500 820^2 = 672400 990^2 = 980100
481^2 = 231361 651^2 = 423801 821^2 = 674041 991^2 = 982081
482^2 = 232324 652^2 = 425104 822^2 = 675684 992^2 = 984064
483^2 = 233289 653^2 = 426409 823^2 = 677329 993^2 = 986049
484^2 = 234256 654^2 = 427716 824^2 = 678976 994^2 = 988036
485^2 = 235225 655^2 = 429025 825^2 = 680625 995^2 = 990025
486^2 = 236196 656^2 = 430336 826^2 = 682276 996^2 = 992016
487^2 = 237169 657^2 = 431649 827^2 = 683929 997^2 = 994009
998^2 = 996004
999^2 = 998001
No comments:
Post a Comment