From a 3x3x3 grid, select 8 points such that no 4 lie in a plane.  The solution is unique. (answer below)

The no-3-in-line, in 2-space, is a well known problem (by Dudeney, 1917). http://www.research.att.com/~njas/sequences/A000769 I did find "No-three-in-line-in-3D" by  Attila Por, David R. Wood http://citeseer.ist.psu.edu/por04nothreelined.html I haven't found any references to the no-4-in-plane problem, though. I was looking for small grid embeddings of complete graphs with non-crossing edges, and the no-4-in-plane seemed like the simplest criteria for selecting the points. The unique solution for 8 points in the 3x3x3 grid is 000  011  012  101  110  120  201  222. For 2x2x2, 5 points.  I haven't yet solved the 4x4x4 case. --Ed Pegg Jr

I haven't yet solved the 4x4x4 case. For the problem of placing points in a 4x4x4 grid such that no four points lie in a plane, i've found 197 distinct solutions with 10 points, after eliminating rotations and reflections. No idea if this is all, or if 11 points is possible. A point cannot be added to any of these these solutions. To decode these solutions, convert to ascii code, subtract 49, and convert to base-4. From 000 49 1 to 333 112 p 123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnop 126LMT_eln 127LMX]cfo 128DINX_gj 128KNU]bgp 12:LMX]cfp 12:LNX]ekp 12<HIP^bek 12<IOT_fjm 12?HIO]dgl 136ELNT_ek 136HIOR]hl 136HIOX]bj 136IPX]bhj 137DIN`eln 137INR`eln 138IOTU^fn 138IOU^dln 138IOX\]bk 138OTU^fln 13:ELO]dfl 13:LMX`cin 13:LNX]ckp 13:LNX]ekp 13:LPS^hkm 13<EIOX_df 13<EOX_dfi 13<FPX]cgj 13<HIOX]bj 13<HIX]bgj 13=HOU`bhk 13>ELP_chi 13>HIP_dgi 146ELM_bhk 146IOR]hlo 14:HIPVbgi 14:HIP^bgi 14>EPX_cek 14>HIPV_gi 167CPT^eln 167DIPRYhn 167DLM\cen 167IPTXYbo 167LMT\cen 167MT\_cel 167MT\_eln 167PTZ^cel 168BHO\]ck 168CPU]cln 168NTY_bko 168NT\_bio 168NX\]bio 168OS\]cep 16<CLMUhno 16<CMNX]bk 16<CMN\bko 16<CMPXben 16<CMU\hko 16<DEN_cjk 16<DISUgno 16<DIS`egn 16<DLMZcen 16<DNOUbko 16<DNSUgno 16<HMNSU\b 16<HMNU\bk 16<HMNUbko 16<HMOSU\k 16<HMOSUbk 16<HMOSUbp 16<HMOSUcp 16<HMOS]bk 16<HMOU\ck 16<HMOUbko 16<HMSUckn 16<HMU\bcj 16<HMU\bck 16<HNOSU\j 16<HNOSUbo 16<HNOUbko 16<HOSU]bk 16<HOSU]bp 16<HOSU]jk 16<HOS]hjk 16<MNTU\bo 16<MNU\bko 16<NOTUbko 16<OPSU]cj 16DGO]_bhi 16DHO\]_ce 16DHO]_cel 16DNORU`el 16HLNR]ekl 16HOR]_cel 16HPS]^cel 178CLM\fin 178CNU^blo 178MTY^cfn 178MTY^cln 178NRU`cln 178NR\`cen 178NTU^blo 178NTY^clo 178ORUYcln 17:CNPSXei 17:DNS^`ei 17:HPY^`ce 17;CNPT^el 17;CNPTeln 17;CPT^eln 17;DN^`cel 17;LMT_eln 17;MNT^cel 17;MPT^cel 17;NPT^cel 17<DNY`cen 17<DNY`cjn 17<HIP^bce 17>CLM`cfh 17>DIR`fhi 17>DMR\^hi 17>DS\^fhi 17>DZ\^chi 17>HMR\hij 17>IPST\jo 17>IR\`fhi 17>LMTYfko 17>LMT_fik 17>LNSTYfo 17>LPTY^ce 17>MR\`hij 17>NPTVYcl 17>PTY^cel 17?CLM\^fi 17?DIS`hin 17?DIS`iln 17?DS\^fhi 17?DS\^hin 17?HT^_bel 17?LMSTeln 17DHNY^`ce 17DNX\]cen 18?OSTYfln 18HOSY]fkl 236CLMU_hj 239LNSX]ek 239LNTU^gk 239LPTU^ek 23:IPSU^gl 23:IPSX^ek 23:LMSU^hk 256PSX]bio 257DOUY`cl 258ALNTYgo 25;CMNQXgl 25;DHOR`gi 25;DHOY`gi 25;DHR]`gi 25;DH]`bgi 25;DNPQXgi 25;HP]^bgi 25<CDNXYfo 25<DNOQgio 25<DNOQhio 25<DNQ^ghi 25<DNQ^gik 25<DNQ^gio 25<DNQ^iko 25<DOU^`ck 25<HIMS^fl 25<HIOQ^cl 25<HMOQ^cl 25<MNXYbko 25<MOXYcjn 25BHOY^`cg 25DHOY^`ce 25DHOY^`gi 25DNOQXgio 25DOPQW^gi 268PSZ]cel 269BOPS]ek 26<CLMQ_hj 26<DNOQ_hk 26<DOQ^_hk 26<HMOSU^c 26<LMOSU^h 26<MOSU^hk 26<MQX_hjk 26?DMQ`ghi 26?ELOQShi 26?HIS^ekl 26?HMQ\hik 26?LMQX_ij 27:CNP]cel 27;DIOU`hj 27;DIO^fhi 27;DIO`fhi 27;DMOR`hi