## Some solutions to Project Euler 62: Cubic permutations

Project Euler 62
https://projecteuler.net/index.php?section=problems&id=62

The cube,   $41063625 \; = \; 345^3$,   can be permuted to produce two other cubes:

$56623104 \; = \; 384^3$    and
$66430125 \; = \; 405^3$

In fact,   41063625   is the smallest cube which has exactly three permutations of its digits which are also cube.

Find the smallest cube for which exactly five permutations of its digits are cube.

****************************************

Here are some solutions:

The 12-digit cube   127035954683   has four other permutations that are cubes:

$127035954683 \; = \; 5027^3$
$352045367981 \; = \; 7061^3$
$373559126408 \; = \; 7202^3$
$569310543872 \; = \; 8288^3$
$589323567104 \; = \; 8384^3$

The next set is :

$1402837695364 \; = \; 5196^3$
$5361789306244 \; = \; 8124^3$
$6132584079364 \; = \; 8496^3$
$9132376564084 \; = \; 9702^3$
$9363024516874 \; = \; 9783^3$

The 13-digit cube   1961574655832   has four other permutations that are cubes:

$1961574655832 \; = \; 12518^3$
$2981631556457 \; = \; 14393^3$
$5657831164259 \; = \; 17819^3$
$8631525145697 \; = \; 20513^3$
$9625537115648 \; = \; 21272^3$

The next set is :

$2097643558401 \; = \; 12801^3$
$2501609484375 \; = \; 13575^3$
$2595641083407 \; = \; 13743^3$
$6140550947328 \; = \; 18312^3$
$9083540714625 \; = \; 20865^3$

Then,

$2546097986375 \; = \; 13655^3$
$2567976059384 \; = \; 13694^3$
$5429503678976 \; = \; 17576^3 \; = \; 26^9$
$6587579024936 \; = \; 18746^3$
$7736546098952 \; = \; 19778^3$

The 13-digit cube   1000600120008   has five other permutations that are cubes:

$1000600120008 \; = \; 10002^3$
$1006012008000 \; = \; 10020^3$
$1061208000000 \; = \; 10200^3$
$8001200060001 \; = \; 20001^3$
$8012006001000 \; = \; 20010^3$
$8120601000000 \; = \; 20100^3$

The next set is :

$1426487591593 \; = \; 11257^3$
$1432197595648 \; = \; 11272^3$
$3496581419752 \; = \; 15178^3$
$4275981654391 \; = \; 16231^3$
$4813967954125 \; = \; 16885^3$
$7591941538264 \; = \; 19654^3$

math grad - Interest: Number theory
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### 4 Responses to Some solutions to Project Euler 62: Cubic permutations

1. paul says:

!4 digit cubes starting with 6 permutations, then 7, 8 and 9.

6 Perms

11064224539187 = 22283^3
14679130284125 = 24485^3
16480142715392 = 25448^3
19322185461704 = 26834^3
32464811925701 = 31901^3
46512291801347 = 35963^3

11228890332457 = 22393^3
27590148228313 = 30217^3
28302913421875 = 30475^3
32425138819072 = 31888^3
32820251748913 = 32017^3
87534298213201 = 44401^3

11824302795768 = 22782^3
14637802579128 = 24462^3
21563878129704 = 27834^3
21780751246983 = 27927^3
22771068931584 = 28344^3
90488612217375 = 44895^3

12297598507368 = 23082^3
29857358209167 = 31023^3
55227991087368 = 38082^3
69517788295032 = 41118^3
75735288109629 = 42309^3
82876153250799 = 43599^3

12435568197632 = 23168^3
23654732519816 = 28706^3
28557213361496 = 30566^3
32685125964317 = 31973^3
48639116235527 = 36503^3
63832146957125 = 39965^3

12852368769024 = 23424^3
21738662850429 = 27909^3
23472268051968 = 28632^3
27048629165832 = 30018^3
43822698057216 = 35256^3
52242319660887 = 37383^3

14290527842163 = 24267^3
14827105492263 = 24567^3
15482932761024 = 24924^3
22742159410368 = 28332^3
42286042531719 = 34839^3
48022145376192 = 36348^3

14861533742056 = 24586^3
15845403276613 = 25117^3
41013654783256 = 34486^3
45331267580416 = 35656^3
60814367453125 = 39325^3
75162133546408 = 42202^3

15282532970496 = 24816^3
20453927568192 = 27348^3
36192428790552 = 33078^3
38157029452296 = 33666^3
38906527429125 = 33885^3
86345027212599 = 44199^3

17859401723619 = 26139^3
21591780491736 = 27846^3
24979031175168 = 29232^3
34917687019125 = 32685^3
60879341715219 = 39339^3
97196410781352 = 45978^3

20429256361753 = 27337^3
32516740952632 = 31918^3
35365069172224 = 32824^3
37209526341625 = 33385^3
53259231706624 = 37624^3
92063755312264 = 45154^3

20645238791737 = 27433^3
22831375767409 = 28369^3
30234279618577 = 31153^3
57146703927832 = 38518^3
83327476901752 = 43678^3
90367732421875 = 44875^3

24469825120768 = 29032^3
42082459126687 = 34783^3
42640229817856 = 34936^3
42816226875904 = 34984^3
52827241960648 = 37522^3
88174602564229 = 44509^3

25979045828113 = 29617^3
31590457812928 = 31612^3
32415988079125 = 31885^3
52839914078125 = 37525^3
58451728309129 = 38809^3
87109155423289 = 44329^3

26615737485793 = 29857^3
38672765749513 = 33817^3
49176737526853 = 36637^3
65816927734375 = 40375^3
68297557341376 = 40876^3
79257646133875 = 42955^3

38187639421875 = 33675^3
38197846381752 = 33678^3
38813597864712 = 33858^3
69837818532741 = 41181^3
71623315478889 = 41529^3
86239578381741 = 44181^3

7 perms

10569784298536 = 21946^3
18083756564992 = 26248^3
18965807549632 = 26668^3
50628361995784 = 36994^3
75966439518208 = 42352^3
97831984066552 = 46078^3
99852138660457 = 46393^3

10849635532072 = 22138^3
21559230074368 = 27832^3
24538160052379 = 29059^3
25523849601307 = 29443^3
40725336980125 = 34405^3
41260329503875 = 34555^3
81503765420293 = 43357^3

12356831794159 = 23119^3
12938153594176 = 23476^3
31491628517539 = 31579^3
35219817465139 = 32779^3
47198521353619 = 36139^3
81469933172551 = 43351^3
96557281941133 = 45877^3

12804692354875 = 23395^3
13097526845248 = 23572^3
15490388426752 = 24928^3
16443257028589 = 25429^3
17205482538496 = 25816^3
38405782562419 = 33739^3
42607284155983 = 34927^3

15354698826375 = 24855^3
15696358354872 = 25038^3
35197258563648 = 32772^3
55659832683741 = 38181^3
56689485123375 = 38415^3
58596436857321 = 38841^3
97825614658533 = 46077^3

16577473805992 = 25498^3
34776859950721 = 32641^3
50714629978375 = 37015^3
70578124593976 = 41326^3
75998730546712 = 42358^3
79856973251047 = 43063^3
79923751046875 = 43075^3

16708497237125 = 25565^3
25981677417032 = 29618^3
29432076171875 = 30875^3
47638707921251 = 36251^3
78501712967432 = 42818^3
86427102371597 = 44213^3
91283041277576 = 45026^3

8 perms

10314675896832 = 21768^3
12417863650893 = 23157^3
13974881626503 = 24087^3
19317865402368 = 26832^3
71096865321483 = 41427^3
78342316815069 = 42789^3
89260685431371 = 44691^3
98534308116672 = 46188^3

10340284735656 = 21786^3
18437405506632 = 26418^3
26055434078136 = 29646^3
35245610867403 = 32787^3
36015465037824 = 33024^3
36310726085544 = 33114^3
43410085673625 = 35145^3
65724053438016 = 40356^3

10418796526321 = 21841^3
10591472326681 = 21961^3
20194756231168 = 27232^3
27328019461561 = 30121^3
43191827216056 = 35086^3
61051282149376 = 39376^3
74826025131619 = 42139^3
90567241861312 = 44908^3

12620043581768 = 23282^3
16602842507831 = 25511^3
16702615804328 = 25562^3
18618053272064 = 26504^3
23102857608641 = 28481^3
25207880643161 = 29321^3
63287460812051 = 39851^3
68724510023681 = 40961^3

15302864497283 = 24827^3
22380469853741 = 28181^3
25084283369417 = 29273^3
42064313878952 = 34778^3
42930148628375 = 35015^3
72490848321536 = 41696^3
82562904733184 = 43544^3
85340912247683 = 44027^3

9 Perms

13465983902671 = 23791^3
18102364796593 = 26257^3
23667095189431 = 28711^3
23719065439168 = 28732^3
38160429751963 = 33667^3
68312596071439 = 40879^3
68523370149961 = 40921^3
80997264315163 = 43267^3
96803741135296 = 45916^3

Paul.

2. paul says:

15 Digits cubes have quite a lot, so here are those with 12 permutations

106739258417856 = 47436^3
109785265614387 = 47883^3
114876307556928 = 48612^3
131856684750792 = 50898^3
169048165238757 = 55293^3
183066597827541 = 56781^3
186453210757689 = 57129^3
214190675578368 = 59832^3
669531482107587 = 87483^3
674018635189752 = 87678^3
765458231098176 = 91476^3
885731701294656 = 96036^3

112706583998464 = 48304^3
151436482979608 = 53302^3
164368470918952 = 54778^3
198124698564073 = 58297^3
204117986685439 = 58879^3
364928067945181 = 71461^3
396401268841579 = 73459^3
416548203169987 = 74683^3
572696830149184 = 83044^3
841686945209713 = 94417^3
864597134281096 = 95266^3
950267986183144 = 98314^3

and 11

101729840326568 = 46682^3
104680231527689 = 47129^3
136781926005824 = 51524^3
210528068347169 = 59489^3
304712526961088 = 67292^3
319200124657688 = 68342^3
502668472109831 = 79511^3
598672128360041 = 84281^3
625860113940872 = 85538^3
828147650069312 = 93908^3
840751262310968 = 94382^3

104573652871897 = 47113^3
128948357570761 = 50521^3
147055817932687 = 52783^3
158472163970875 = 54115^3
204867717859351 = 58951^3
250487817791653 = 63037^3
341810677278559 = 69919^3
397470681578125 = 73525^3
679731084578125 = 87925^3
739867054871125 = 90445^3
788914637557201 = 92401^3

125631058992704 = 50084^3
212060594589137 = 59633^3
239540911807625 = 62105^3
316851909572024 = 68174^3
405865029721319 = 74039^3
516100498223597 = 80213^3
524250018996137 = 80633^3
692003421597851 = 88451^3
904932561078125 = 96725^3
930428791015625 = 97625^3
973802164502159 = 99119^3

126120833457949 = 50149^3
127369820359144 = 50314^3
129523494108736 = 50596^3
196538434097221 = 58141^3
223947119460583 = 60727^3
269811549437032 = 64618^3
297061398351424 = 66724^3
304698942523171 = 67291^3
340229194658731 = 69811^3
341019324789625 = 69865^3
803449196732125 = 92965^3

129308576850432 = 50568^3
186042283957503 = 57087^3
239552483087016 = 62106^3
259633020184875 = 63795^3
267513238098504 = 64434^3
401898256032573 = 73797^3
500698587232143 = 79407^3
512826043839507 = 80043^3
545679003228813 = 81717^3
800859237314625 = 92865^3
901260738352584 = 96594^3

Paul.