## Thursday, August 24, 2017

### Cube wonders

On Monday, Futility Closet posted — without explanation — five base-ten 12-digit "cube wonders" (my expression). After finding a smaller example (512) I wondered if I might be able to discover larger ones. Alas, my brute-force Mathematica code was taking its time rediscovering even those five examples and I eventually gave up.

A web search soon led me to OEIS A006887 where those five numbers are the cubes of terms #9 - #13. Jack Brennen kindly supplied a C-program that calculated the sequence's additional terms. All I had to do was cull from the newly generated list those terms whose cubes could be split into three equal digit-length parts, wherein (for the sake of aesthetics) the second and third parts did not start with a zero. Here are the first 33 solutions:

512 = (5+1+2)^3
121213882349 = (1212+1388+2349)^3
128711132649 = (1287+1113+2649)^3
162324571375 = (1623+2457+1375)^3
171323771464 = (1713+2377+1464)^3
368910352448 = (3689+1035+2448)^3
171471879319616 = (17147+18793+19616)^3
220721185826504 = (22072+11858+26504)^3
470511577514952 = (47051+15775+14952)^3
109294197946170875 = (109294+197946+170875)^3
116367227503144344 = (116367+227503+144344)^3
120706126590246912 = (120706+126590+246912)^3
160769107975275008 = (160769+107975+275008)^3
175471156639227736 = (175471+156639+227736)^3
214714120150263943 = (214714+120150+263943)^3
293736149984221021 = (293736+149984+221021)^3
435235164725157875 = (435235+164725+157875)^3
495862183018112625 = (495862+183018+112625)^3
526727149679131192 = (526727+149679+131192)^3
119673015472102184000 = (1196730+1547210+2184000)^3
171467712620032578875 = (1714677+1262003+2578875)^3
236392428062461013000 = (2363924+2806246+1013000)^3
104126812463311912000000 = (10412681+24633119+12000000)^3
129994501742459020233216 = (12999450+17424590+20233216)^3
170717891744681120955851 = (17071789+17446811+20955851)^3
176905291183926726606912 = (17690529+11839267+26606912)^3
201817431964353718831681 = (20181743+19643537+18831681)^3
257700981126972226596544 = (25770098+11269722+26596544)^3
339601941949863016309248 = (33960194+19498630+16309248)^3
420352701580617017068241 = (42035270+15806170+17068241)^3
469218322003481810750024 = (46921832+20034818+10750024)^3
526085181538162612736833 = (52608518+15381626+12736833)^3
529262771767141910291487 = (52926277+17671419+10291487)^3

I have placed a larger 236-solutions set here.