The number 10000 can be split into 100 + 0 + 0 = 100, which is sqrt(10000).

That's rather banal, but there are two other such 5-digit numbers.

One is 55225:

5 + 5 + 225 = 235 = sqrt(55225)

What is the other one?

Each digit must be used exactly once, in proper order.

Bonus:

How many such 6-digit squares can you find?

For fun here are solutions I found for 7-, 8-, 9- and 10-digit squares. 

sqrt(1000000) = 1000 = 1000 + 0 + 0 + 0

sqrt(1679616) = 1296 = 1 + 679 + 616

sqrt(2896804) = 1702 = 2 + 896 + 804

sqrt(3175524) = 1782 = 3 + 1755 + 24

sqrt(4941729) = 2223 = 494 + 1729

sqrt(7441984) = 2728 = 744 + 1984

sqrt(11329956) = 3366 = 11 + 3299 + 56

sqrt(13293316) = 3646 = 1 + 329 + 3316

sqrt(13557124) = 3682 = 1 + 3557 + 124

sqrt(17073424) = 4132 = 1 + 707 + 3424

sqrt(23804641) = 4879 = 238 + 0 + 4641

sqrt(24068836) = 4906 = 2 + 4068 + 836

sqrt(24502500) = 4950 = 2450 + 2500

sqrt(25502500) = 5050 = 2550 + 2500

sqrt(25502500) = 5050 = 25 + 5025 + 0 + 0

sqrt(26512201) = 5149 = 26 + 5122 + 0 + 1

sqrt(28005264) = 5292 = 28 + 0 + 0 + 5264

sqrt(46676224) = 6832 = 46 + 6762 + 24

sqrt(51710481) = 7191 = 5 + 1 + 7104 + 81

sqrt(52881984) = 7272 = 5288 + 1984

sqrt(54597321) = 7389 = 5 + 4 + 59 + 7321

sqrt(54597321) = 7389 = 54 + 5 + 9 + 7321

sqrt(56746089) = 7533 = 56 + 7460 + 8 + 9

sqrt(56896849) = 7543 = 5 + 689 + 6849

sqrt(57562569) = 7587 = 5 + 7562 + 5 + 6 + 9

sqrt(60481729) = 7777 = 6048 + 1729

sqrt(63297936) = 7956 = 6 + 3 + 2 + 9 + 7936

sqrt(70829056) = 8416 = 70 + 8290 + 56

sqrt(71284249) = 8443 = 7 + 1 + 2 + 8424 + 9

sqrt(76860289) = 8767 = 76 + 8602 + 89

sqrt(78428736) = 8856 = 78 + 42 + 8736

sqrt(79388100) = 8910 = 7 + 93 + 8810 + 0

sqrt(79887844) = 8938 = 7 + 9 + 8878 + 44

sqrt(84787264) = 9208 = 8478 + 726 + 4

sqrt(86769225) = 9315 = 8 + 6 + 76 + 9225

sqrt(86955625) = 9325 = 8695 + 5 + 625

sqrt(91891396) = 9586 = 9189 + 1 + 396

sqrt(92563641) = 9621 = 9256 + 364 + 1

sqrt(92929600) = 9640 = 9 + 29 + 2 + 9600

sqrt(95355225) = 9765 = 9535 + 5 + 225

sqrt(98029801) = 9901 = 98 + 0 + 2 + 9801

sqrt(98029801) = 9901 = 9802 + 98 + 0 + 1

sqrt(98188281) = 9909 = 9818 + 8 + 2 + 81

sqrt(98188281) = 9909 = 9818 + 82 + 8 + 1

sqrt(98366724) = 9918 = 9836 + 6 + 72 + 4

sqrt(98903025) = 9945 = 9890 + 30 + 25

sqrt(99102025) = 9955 = 9910 + 20 + 25

sqrt(99281296) = 9964 = 9928 + 1 + 29 + 6

sqrt(99800100) = 9990 = 9980 + 0 + 10 + 0

sqrt(99820081) = 9991 = 9982 + 0 + 0 + 8 + 1

sqrt(99980001) = 9999 = 9998 + 0 + 0 + 0 + 1

sqrt(100000000) = 10000 = 10000 + 0 + 0 + 0 + 0

sqrt(110502144) = 10512 = 1 + 10502 + 1 + 4 + 4

sqrt(149377284) = 12222 = 1 + 4937 + 7284

sqrt(161976529) = 12727 = 1 + 6197 + 6529

sqrt(298287441) = 17271 = 2 + 9828 + 7441

sqrt(300814336) = 17344 = 3008 + 14336

sqrt(493817284) = 22222 = 4938 + 17284

sqrt(494217361) = 22231 = 494 + 21736 + 1

sqrt(642825316) = 25354 = 6 + 4 + 28 + 25316

sqrt(751527396) = 27414 = 7 + 5 + 1 + 5 + 27396

sqrt(1133601561) = 33669 = 11 + 33601 + 56 + 1

sqrt(1133601561) = 33669 = 1 + 1 + 33601 + 5 + 61

sqrt(1178342929) = 34327 = 1 + 17 + 8 + 34292 + 9

sqrt(1178342929) = 34327 = 11 + 7 + 8 + 34292 + 9

sqrt(1256135364) = 35442 = 12 + 5 + 61 + 35364

sqrt(1336487364) = 36558 = 1 + 3 + 36487 + 3 + 64

sqrt(1358291025) = 36855 = 1 + 35829 + 1025

sqrt(1518037444) = 38962 = 1518 + 0 + 37444

sqrt(1553936400) = 39420 = 1 + 55 + 39364 + 0 + 0

sqrt(1693240201) = 41149 = 16 + 932 + 40201

sqrt(1748410596) = 41814 = 1 + 748 + 41059 + 6

sqrt(1818425449) = 42643 = 1 + 81 + 8 + 42544 + 9

sqrt(2084561649) = 45657 = 20 + 8 + 45616 + 4 + 9

sqrt(2141745841) = 46279 = 21 + 417 + 45841

sqrt(2380464100) = 48790 = 2380 + 46410 + 0

sqrt(2384857225) = 48835 = 238 + 48572 + 25

sqrt(2449458064) = 49492 = 24 + 49458 + 0 + 6 + 4

sqrt(2456490969) = 49563 = 2 + 456 + 49096 + 9

sqrt(2490608836) = 49906 = 2 + 49060 + 8 + 836

sqrt(2505002500) = 50050 = 25 + 0 + 50025 + 0 + 0

sqrt(2647514116) = 51454 = 26 + 4 + 7 + 51411 + 6

sqrt(2800526400) = 52920 = 280 + 0 + 52640 + 0

sqrt(2805291225) = 52965 = 28 + 0 + 52912 + 25

sqrt(2853162225) = 53415 = 28 + 53162 + 225

sqrt(3293956449) = 57393 = 3 + 2 + 939 + 56449

sqrt(3311657209) = 57547 = 331 + 1 + 6 + 57209

sqrt(3514592656) = 59284 = 3 + 5 + 1 + 4 + 59265 + 6

sqrt(3862125316) = 62146 = 3 + 8 + 62125 + 3 + 1 + 6

sqrt(4139764281) = 64341 = 41 + 3 + 9 + 7 + 64281

sqrt(4256518564) = 65242 = 42 + 5 + 65185 + 6 + 4

sqrt(4506705424) = 67132 = 4 + 50 + 67054 + 24

sqrt(4682391184) = 68428 = 4 + 68239 + 1 + 184

sqrt(4769007364) = 69058 = 4 + 7 + 69007 + 36 + 4

sqrt(4782690649) = 69157 = 4 + 78 + 2 + 69064 + 9

sqrt(4853769561) = 69669 = 48 + 53 + 7 + 69561

sqrt(5272502544) = 72612 = 52 + 72502 + 54 + 4

sqrt(5572174609) = 74647 = 5 + 5 + 7 + 21 + 74609

sqrt(5674759561) = 75331 = 5 + 6 + 74759 + 561

sqrt(5906076201) = 76851 = 590 + 60 + 76201

sqrt(5947648641) = 77121 = 594 + 76486 + 41

sqrt(6039776656) = 77716 = 6 + 0 + 39 + 77665 + 6

sqrt(6049417284) = 77778 = 60494 + 17284

sqrt(6067786816) = 77896 = 6 + 0 + 6 + 77868 + 16

sqrt(6322794256) = 79516 = 63 + 22 + 79425 + 6

sqrt(6580129924) = 81118 = 65 + 80129 + 924

sqrt(6731382025) = 82045 = 6 + 7 + 3 + 1 + 3 + 82025

sqrt(6826064400) = 82620 = 6 + 82606 + 4 + 4 + 0 + 0

sqrt(6832014336) = 82656 = 68320 + 14336

sqrt(7261084944) = 85212 = 7 + 261 + 0 + 84944

sqrt(7385855481) = 85941 = 7 + 385 + 85548 + 1

sqrt(7385855481) = 85941 = 73 + 85855 + 4 + 8 + 1

sqrt(7668680041) = 87571 = 766 + 86800 + 4 + 1

sqrt(7879757824) = 88768 = 7 + 87975 + 782 + 4

sqrt(7887571344) = 88812 = 7 + 88757 + 1 + 3 + 44

sqrt(7887748969) = 88813 = 7 + 88774 + 8 + 9 + 6 + 9

sqrt(8033895424) = 89632 = 80 + 3 + 3 + 89542 + 4

sqrt(8248090761) = 90819 = 8 + 2 + 48 + 0 + 90761

sqrt(8419163536) = 91756 = 84 + 1 + 91635 + 36

sqrt(8723933604) = 93402 = 8 + 7 + 23 + 93360 + 4

sqrt(8756093476) = 93574 = 87 + 5 + 6 + 0 + 93476

sqrt(8862527881) = 94141 = 8 + 86252 + 7881

sqrt(8945565561) = 94581 = 8 + 94556 + 5 + 5 + 6 + 1

sqrt(9048004641) = 95121 = 90480 + 0 + 4641

sqrt(9435596769) = 97137 = 9 + 4 + 355 + 96769

sqrt(9811893025) = 99055 = 98118 + 930 + 2 + 5

sqrt(9845799076) = 99226 = 9 + 84 + 57 + 99076

sqrt(9845799076) = 99226 = 98 + 45 + 7 + 99076

sqrt(9849371536) = 99244 = 98493 + 715 + 36

sqrt(9869031649) = 99343 = 98690 + 3 + 1 + 649

sqrt(9911994481) = 99559 = 99 + 11 + 99448 + 1

sqrt(9926136900) = 99630 = 99261 + 369 + 0 + 0

sqrt(9926336161) = 99631 = 99263 + 361 + 6 + 1

sqrt(9940688209) = 99703 = 99406 + 88 + 209

sqrt(9953055225) = 99765 = 99530 + 5 + 5 + 225

sqrt(9980010000) = 99900 = 99800 + 100 + 0 + 0

sqrt(9980209801) = 99901 = 99802 + 0 + 98 + 0 + 1

sqrt(9981808281) = 99909 = 99818 + 0 + 8 + 2 + 81

sqrt(9981808281) = 99909 = 99818 + 0 + 82 + 8 + 1

sqrt(9983606724) = 99918 = 99836 + 0 + 6 + 72 + 4

sqrt(9989003025) = 99945 = 99890 + 0 + 30 + 25

sqrt(9991002025) = 99955 = 99910 + 0 + 20 + 25

sqrt(9992801296) = 99964 = 99928 + 0 + 1 + 29 + 6

sqrt(9998000100) = 99990 = 99980 + 0 + 0 + 10 + 0

sqrt(9998200081) = 99991 = 99982 + 0 + 0 + 0 + 8 + 1

Posted by tomarken on 2022-04-06 08:58:40