Ponder This Challenge - January 2019 - Charlie helps Bob defeat Alice

Alice and Bob are playing the following game: they start from a number N and each one of them in his or her turn (Alice starts) divides N by any divisor that is either a prime or a product of several distinct prime numbers. The winner is the one who gets to one - thus leaving the other player with no legal move.
To define the initial N, Alice chooses a number a from a set A, and a number b from a set B. The game is played with N=a+b. Charlie knows that Alice will start, and he wants to let Bob win. He does that by fixing the sets A and B.

He can do that, for example, by choosing A=[3,99] and B=[1,22]. (Why?)

Your challenge, this month, is to help Charlie find sets A and B with at least four different numbers each, that will allow Bob to win.

Bonus '*' for solutions with more than 4 elements in the set B.

Solution

• Click here to view the solution

  Thanks to Thomas Egense for suggesting the idea for this month's challenge.

  We obscured it by inventing a game and we postponed the credit to the solution page to not spoil it for you (see his page).

  Bob wins when a+b is a square of an integer. He wins by mirroring Alice's play.
  So Charlie should chose A and B such that all possible a+b will be squares.

  Many of you found the solution
  A=[9, 28224, 419904, 3968064]; B=[0, 47952, 259072, 2442960]
  but there are more, like
  A = [324, 54756, 119716, 264196]; B = [0, 79200, 227205, 1258560]

  Many of you found a solution with 4 and 5 elements:
  A = [46655, 1245455, 2742335, 22316175]; B = [1, 274834, 1225729, 3368449, 30159361].

  Harald Bögeholz found many more:
  A=[15875,729315,8398403,105637283]; B=[1,493921,9774766,19511686,46687681]
  A=[108899,864899,16080099,45239075]; B=[1,648001,5869126,17155126,109763425]
  A=[14399,1404224,26625599,247118399]; B=[1,1890001,25792001,85141585,237762001]
  A=[435599,3459599,64320399,180956303]; B=[1,2592001,23476501,68620501,439053697]
  A=[71288,2585663,16064063,24206399]; B=[1,2254337,9945937,29904337,779175937]
  A=[268323,2965283,13972643,139287203]; B=[1,3491398,13779181,21596653,34636141]
  A=[3154175,6110783,64256255,543076415]; B=[1,5232641,67557106,100763461,416931841]
  A=[63503,2917263,33593615,422549135]; B=[1,1975681,39099061,78046741,186750721]
  A=[129599,3920399,24206399,138862655]; B=[1,1346626,10368001,32223745,51136001]

  Uoti Urpala was the first to find a solution with 6 elements:
  A=[0, 16594560, 137675520, 1202947200]; B=[8928144, 18974736, 33489369, 124634896, 290566116, 573506704]

  Jess Thrysoee found a second 4,6 solution:
  A = [4847501376, 586414656, 35712576, 102090816]; B = [0, 1126551888, 40186368, 98244900, 462827008, 2258314240]

  Uoti Urpala found 76 different 4,6 solutions:
  A=[1899, 42156, 140884, 473004]; B=[1899, 17613, 59373, 69837, 80109, 1279251]
  A=[2988, 5052, 12108, 34812]; B=[2988, 4356, 5787, 11164, 17046, 23948]
  A=[3128, 273020, 1039720, 2292620]; B=[3128, 270453, 351645, 494872, 819672, 2893128]
  A=[3264, 44760, 171960, 303060]; B=[3264, 68736, 122739, 135264, 223840, 493239]
  A=[6587, 26712, 99288, 188712]; B=[6587, 22715, 56645, 133595, 199205, 272915]
  A=[6720, 37968, 64680, 139920]; B=[6720, 15855, 26880, 45440, 69510, 129290]
  A=[6765, 62304, 80160, 543840]; B=[6765, 58355, 168685, 225555, 233547, 494835]
  A=[11160, 476532, 1328040, 2354760]; B=[11160, 375720, 473401, 659835, 928440, 1752120]
  A=[13937, 124817, 187537, 1316623]; B=[13937, 24864, 139678, 588192, 938631, 4161696]
  A=[14280, 41580, 68376, 355320]; B=[14280, 26760, 31640, 67368, 100065, 136920]
  A=[14616, 73080, 127020, 370620]; B=[14616, 39384, 89784, 228375, 391384, 611784]
  A=[15400, 392140, 514360, 934640]; B=[15400, 231000, 299145, 548520, 835395, 951384]
  A=[22715, 34440, 101640, 189960]; B=[6587, 22715, 56645, 133595, 199205, 272915]
  A=[26112, 228480, 405888, 815388]; B=[26112, 160480, 261120, 294240, 606135, 1966560]
  A=[32067, 807933, 11897067, 31782933]; B=[32067, 661500, 870660, 4550875, 27703620, 41773500]
  A=[35468, 49357, 244468, 763532]; B=[35468, 91360, 119160, 176040, 450740, 794592]
  A=[39564, 211596, 237204, 1579116]; B=[39564, 374976, 600768, 1214208, 1800356, 7912107]
  A=[46800, 285168, 725400, 4396080]; B=[46800, 155376, 788560, 1496560, 1971450, 11439935]
  A=[47898, 77052, 144477, 198180]; B=[47898, 60390, 113498, 197190, 381402, 508410]
  A=[60375, 3912300, 5268564, 37142700]; B=[60375, 1895775, 9232545, 16298345, 41052585, 69466785]
  A=[63360, 98928, 235840, 628128]; B=[63360, 107205, 243180, 296604, 448896, 658560]
  A=[68145, 718305, 4010415, 8550465]; B=[68145, 218505, 813330, 1108030, 3899742, 4026582]
  A=[74640, 254640, 532692, 2753640]; B=[74640, 193200, 414925, 459760, 865200, 2092944]
  A=[86240, 137360, 266560, 590920]; B=[86240, 116585, 283338, 436590, 548640, 949536]
  A=[93808, 926288, 4771052, 17975672]; B=[93808, 1626768, 2642211, 5974137, 11469744, 18265104]
  A=[106656, 668096, 1290432, 2972112]; B=[106656, 352044, 1158399, 1677984, 1751904, 3267936]
  A=[107205, 131397, 251195, 634053]; B=[63360, 107205, 243180, 296604, 448896, 658560]
  A=[126720, 281760, 691680, 1199808]; B=[126720, 381780, 435456, 785455, 1932120, 5818120]
  A=[128060, 2436940, 4355560, 10180808]; B=[128060, 668325, 1252860, 3836100, 4223100, 5021700]
  A=[130308, 172308, 623308, 857892]; B=[78579, 130308, 179940, 212790, 326790, 625410]
  A=[134640, 871080, 6026160, 13563144]; B=[134640, 1177335, 2744208, 4738410, 5567760, 11449058]
  A=[148656, 1418844, 4566156, 12070320]; B=[148656, 3752880, 5268000, 8405695, 15846000, 19380000]
  A=[151326, 195174, 447426, 706426]; B=[151326, 189630, 273870, 299670, 474453, 817530]
  A=[152768, 2974853, 9777152, 31382912]; B=[152768, 1947792, 4102168, 7867056, 11199804, 20190016]
  A=[159180, 612045, 1510068, 2327220]; B=[159180, 408660, 678360, 1055040, 1882776, 2205560]
  A=[193200, 310800, 561708, 2759400]; B=[74640, 193200, 414925, 459760, 865200, 2092944]
  A=[215880, 1333920, 9515940, 25499040]; B=[215880, 1450680, 2438205, 4266680, 4996761, 18729480]
  A=[229383, 595287, 1161468, 4677948]; B=[229383, 452529, 1718073, 2387847, 2860209, 11519431]
  A=[231000, 454860, 563640, 962640]; B=[15400, 231000, 299145, 548520, 835395, 951384]
  A=[231840, 3004260, 4539924, 20807640]; B=[231840, 699545, 1969632, 3199185, 9472320, 66306240]
  A=[244720, 417781, 2077712, 4724720]; B=[244720, 590140, 907200, 1217160, 3935680, 21160160]
  A=[254100, 651420, 3672900, 5205420]; B=[254100, 381675, 881034, 4150300, 8332940, 19464060]
  A=[323323, 13995267, 56034363, 182130333]; B=[323323, 20624681, 35913724, 47146372, 80905363, 221948804]
  A=[324240, 592305, 1189440, 5478480]; B=[88212, 324240, 335860, 1248240, 2078916, 4468940]
  A=[399000, 536484, 4313400, 11722620]; B=[399000, 518035, 1576200, 2837688, 7266840, 14364840]
  A=[400320, 577080, 947520, 2582720]; B=[400320, 1025952, 1293120, 1436640, 1743165, 15994080]
  A=[400365, 1707240, 2583000, 4911144]; B=[400365, 974925, 1548365, 3717675, 4264533, 15763475]
  A=[411895, 1010240, 3092320, 7385840]; B=[411895, 860145, 1143857, 1560945, 3115255, 8599305]
  A=[555520, 1413104, 7747520, 17295520]; B=[555520, 1127850, 3443790, 4166400, 4761600, 23994775]
  A=[561085, 1979480, 6968360, 34372555]; B=[561085, 2239461, 6241635, 6945435, 26359515, 46981285]
  A=[607376, 907120, 2494580, 11298980]; B=[607376, 2092640, 3565376, 4488649, 5073376, 13845601]
  A=[664020, 2165460, 3121965, 5733420]; B=[664020, 1643152, 2232720, 4292568, 6982920, 13623120]
  A=[664076, 1295476, 2107924, 9966476]; B=[664076, 704060, 1290240, 5523840, 8536960, 29745765]
  A=[742560, 4405856, 13302744, 27101256]; B=[742560, 1617408, 6211842, 9185967, 10509408, 23613408]
  A=[919921, 15637756, 21728516, 113781884]; B=[919921, 5298447, 12436503, 46948407, 55338519, 130195401]
  A=[969272, 5338340, 18330760, 32563160]; B=[969272, 3002472, 7267095, 14630472, 30718728, 79284600]
  A=[1014860, 1347885, 2223540, 5252940]; B=[1014860, 1427888, 1602920, 3101560, 6195980, 12005840]
  A=[1055700, 2416176, 6104700, 13851600]; B=[1055700, 1643220, 3548340, 9452340, 13013415, 15806260]
  A=[1177335, 1458345, 6138615, 13613481]; B=[134640, 1177335, 2744208, 4738410, 5567760, 11449058]
  A=[1624536, 3719952, 21364872, 38753424]; B=[1624536, 4820634, 8490664, 31979619, 37993176, 237817944]
  A=[1754808, 20436108, 39416608, 107222808]; B=[1754808, 6134535, 12319560, 15782136, 186050040, 297618915]
  A=[1929240, 4320504, 5914680, 20237355]; B=[1929240, 2422728, 3854340, 9517860, 15478540, 50502760]
  A=[2528310, 11504130, 12535050, 24640550]; B=[2528310, 8811810, 18636435, 24229815, 30766806, 49656690]
  A=[2650604, 19864229, 30754796, 134295700]; B=[2650604, 9039360, 18573060, 28061440, 37485096, 652477440]
  A=[2884104, 3245355, 4416520, 5505480]; B=[9696, 2884104, 3647196, 5467104, 7164996, 14409696]
  A=[6017895, 29670375, 42957369, 112966119]; B=[6017895, 9608058, 24695658, 42304208, 52369200, 70979792]
  A=[6328080, 117702288, 284368095, 467905680]; B=[6328080, 44622620, 54067860, 177186240, 246584184, 1166476080]
  A=[8558550, 40347450, 87297210, 162507114]; B=[8558550, 17763200, 23151975, 54774720, 156499200, 916563648]
  A=[14842800, 37931600, 63018288, 99261225]; B=[5147520, 14842800, 32866200, 69761160, 129627120, 229568640]
  A=[16869545, 40791520, 178529120, 387890272]; B=[16869545, 50939735, 86963415, 229181271, 353828055, 816442935]
  A=[17555300, 141727040, 368704700, 1044447040]; B=[17555300, 92298780, 176746500, 273078225, 1009193220, 2015297028]
  A=[18198675, 22792275, 59135725, 165669075]; B=[18198675, 28936050, 45841950, 51351300, 156456300, 495399996]
  A=[33558000, 122151120, 501068880, 749266245]; B=[33558000, 466256112, 525070840, 636930840, 930563340, 1496399160]
  A=[50250144, 116166939, 268033809, 521032224]; B=[50250144, 144323088, 191540748, 213709188, 1227407068, 1783421016]
  A=[67041780, 118174485, 150158580, 845395980]; B=[67041780, 159942120, 258931920, 582481380, 1097199480, 1268345552]
  A=[194285520, 411016905, 495199120, 1171600560]; B=[194285520, 234123864, 277690920, 1121557320, 1238324880, 13340292648]

  And still, the existence of a 5,5 solution (a solution with 5 elements both in A and B) is still open.

Solvers

• *David Greer (31/12/18 02:06 AM IDT)
• **Andreas Stiller (31/12/18 02:15 AM IDT)
• *Alper Halbutogullari (31/12/18 02:26 AM IDT)
• Alexander Daryin (31/12/18 05:19 AM IDT)
• Justin Rogers (31/12/18 07:59 AM IDT)
• *Anders Kaseorg (31/12/18 08:45 AM IDT)
• Bert Dobbelaere (31/12/18 01:20 PM IDT)
• Thomas Huet (31/12/18 02:18 PM IDT)
• *Florian Fischer (31/12/18 04:43 PM IDT)
• Eden Saig (31/12/18 06:12 PM IDT)
• JJ Rabeyrin (31/12/18 06:28 PM IDT)
• **Deron Stewart (01/01/19 12:31 AM IDT)
• *Haye Chan (02/01/19 08:53 AM IDT)
• Graham Hemsley (02/01/19 10:42 AM IDT)
• Hugo Pfoertner (02/01/19 12:52 PM IDT)
• *Adam Daire (02/01/19 21:13 PM IDT)
• Noam Lasry (03/01/19 01:00 AM IDT)
• Reiner Martin (03/01/19 01:52 AM IDT)
• Rashid Naimi (03/01/19 05:49 AM IDT)
• Adrienne Stanley (04/01/19 12:56 AM IDT)
• *Paolo Farinelli (04/01/19 02:02 AM IDT)
• *Emma Kee (04/01/19 11:47 AM IDT)
• **Jess Thrysoee (04/01/19 01:16 AM IDT)
• **Dieter Beckerle (05/01/19 10:44 AM IDT)
• *Harald Bögeholz (05/01/19 02:48 PM IDT)
• *Antonio García Astudillo (06/01/19 01:52 AM IDT)
• *Walter Schmidt (06/01/19 09:29 AM IDT)
• *David F.H. Dunkley (06/01/19 12:00 PM IDT)
• *Daniel Bitin (06/01/19 07:31 PM IDT)
• Daniel Chong Jyh Tar (07/01/19 10:19 AM IDT)
• Charles Joscelyne (08/01/19 01:51 AM IDT)
• Victor Chang (08/01/19 08:30 AM IDT)
• **Uoti Urpala (08/01/19 08:34 AM IDT)
• **Vladimir Volevich (08/01/19 08:52 AM IDT)
• Louis Faucon (08/01/19 10:21 AM IDT)
• *David Stevens (08/01/19 11:45 AM IDT)
• *Todd Will (09/01/19 08:17 PM IDT)
• *Loukas Sidiropoulos (09/01/19 04:07 PM IDT)
• *Kang Jin Cho (11/01/19 08:23 AM IDT)
• *Dieter Dobbelaere (11/01/19 04:16 PM IDT)
• Armin Krauss (11/01/19 09:29 PM IDT)
• Kelvin Kim (12/01/19 06:04 PM IDT)
• Young Joon Kim (12/01/19 06:21 PM IDT)
• Jeewoo Lee (12/01/19 07:30 PM IDT)
• Joonsoo Lee (12/01/19 08:16 PM IDT)
• Almog Yair (13/01/19 01:13 PM IDT)
• *Dominik Reichl (13/01/19 02:06 PM IDT)
• Joehyun Kim (13/01/19 03:12 PM IDT)
• Jiho Lee (13/01/19 08:08 PM IDT)
• Cynthia Beauchemin (13/01/19 09:37 PM IDT)
• Rajko Nenadov (13/01/19 10:00 PM IDT)
• Rustam Salikhov (14/01/19 09:36 AM IDT)
• Mykola Zubach (16/01/19 11:25 AM IDT)
• *Shouky Dan & Tamir Ganor (17/01/19 08:49 AM IDT)
• Alex Fleischer (17/01/19 08:58 AM IDT)
• Joseph DeVincentis (17/01/19 06:23 PM IDT)
• Francis Golding (18/01/19 03:01 PM IDT)
• Danny Antonetti (19/01/19 05:05 AM IDT)
• Marco Bellocchi (20/01/19 12:50 AM IDT)
• Taeyang David Park (20/01/19 06:54 PM IDT)
• *Boris Silantyev (21/01/19 11:52 AM IDT)
• Yi Won Kim (22/01/19 04:22 AM IDT)
• Denys Kopiychenko (22/01/19 05:56 AM IDT)
• Phil Muhm (22/01/19 09:45 PM IDT)
• *Atul Gupta (23/01/19 01:49 AM IDT)
• Tyler Mullen (23/01/19 03:00 AM IDT)
• Michael Rosola (23/01/19 06:36 PM IDT)
• **Tom Sirgedas (24/01/19 12:24 AM IDT)
• Eric Edward Pauley (24/01/19 10:02 AM IDT)
• *Pieter Vandercammen (25/01/19 12:28 AM IDT)
• DukeBG (25/01/19 02:45 PM IDT)
• Klaus Hoffmann (25/01/19 07:44 PM IDT)
• Deane Stewart (25/01/19 09:20 PM IDT)
• Motty Porat (27/01/19 12:32 AM IDT)
• Andrew Lee (27/01/19 03:33 PM IDT)
• Andreas Knüpfer (27/01/19 06:22 PM IDT)
• *Toke Eskildsen (27/01/19 08:40 PM IDT)
• Henry Lee (27/01/19 08:51 PM IDT)
• Jingran Lin (28/01/19 02:19 AM IDT)
• Fabio Filatrella (28/01/19 02:30 PM IDT)
• Luke Pebody (29/01/19 04:32 AM IDT)
• *Chris Shannon (29/01/19 04:36 AM IDT)
• Dan Dima (29/01/19 03:21 PM IDT)
• *Aviv Nisgav (30/01/19 09:35 AM IDT)
• Robert Lang (31/01/19 06:56 PM IDT)
• Erik Wünstel (31/01/19 06:57 PM IDT)
• Radu-Alexandru Todor (01/02/19 02:20 AM IDT)
• **Oscar Volpatti (01/02/19 05:36 AM IDT)
• Noam Zaks & Ayal Zaks (02/02/19 06:54 PM IDT)