Message #2924

From: mananself@gmail.com
Subject: RE: 120Z solved!!!
Date: Thu, 30 Jan 2014 15:13:23 -0800

Hi Andrey,



Thanks for the advice. We’ll see if I can endure the 3C situation…



About the matrices, let me clarify my definitions. As you said, there are 72 orbits for 2C pieces. I define Matrix2C to have 72 rows, 120 columns. The component at the i-th row and j-th column is 1 if flipping cell j changes the parity of orbit i, and is 0 otherwise. Since there are 60 orbits for 3C, Matrix3C has 60 rows 120 columns. The definition is similar.



Under this definition:



The rank of the 2C matrix is: 36
The rank of the 3C matrix is: 44




You can play with the Sage code here http://aleph.sagemath.org/?z=eJzNXWtP29gW_V6p_8HSlW7DHYbGZKaVRkJXEwqd0umTITPTqkIGQkghj3EMtPz6cQiPxGc_1nk5SBYk9jn7sfbey_vYTvKf5KQoxr88fZqddccna5Os1x1kxcnaKO89_f_VRnfn6upFJ7vsHLff_r45fJ3_2jvf3fr2Lv2r3WpevdhNt7cu2_2dt3_98_z71zffDzuv_vwwKnb2Xqc_nbd3X1y2d0_fp_3i9dXWxc7VT-n7q73h2z8Om3nzaHNr92U2eH389Z9PHw9–_Rj1hl_zDqfNg_Oxi_How8Xf7Y_bO98zAav1nuTi987nc29QafbH737OR9_-Pnrj99fbfX-_rvX7563r3a_vWt3Trf3nr3cHp9-am-me0e_tn979v7jm43_nmXD3sbUq8ePHj_qD8ajvEjybHg0Gkx3vNld30w2kjdZkfe_7Y6zw27j5XZjfWU1eb6-mqTrzZVyzPXB2bhyeOPz53Q1afKbfDTUlhovYtjAyUkjeFqV-WU1-azaEtAERJQ5JrWUUI-d8sTqiyrS8zMctKlTQJn1wOmpiMwINilYpHEl5utINRCbQ_yFSBw4Rfoh8LS_33VysIwuzX6B2QM0KpJM-bTqaadnPqY00qFMVu2yVWRlT2wvEKv43gMXXCer1NMD2gKpWkWdEYP0DGoEgiSu6nfNG-qURU5H6vtC4bSUBlPFxI89SKPqXFDE4BBnX1IwB5fP00uhXkER6Z0PZRlrxHhlHru5w0cile7W3ugTl9PluQl3xsAKaVy-3ZkhZO8RFeZ4otRmCWQSYcDcGvHhXBkKG1Jbs0EcXGg7epfnjyWec1FXtL7u13fdA88DGdra1tigSaglge8E-C8QHbyPnRG4Uxp7BDwj22Lj304ERxqUb82By1wjxqbVIOj6-5Uu9B7LdQuMgJsun7bK6tADXrnUsFmslufequ670GDE1bhzMgU5b_gXW-Cze8QrTDUzd_CTK3giRPUu4Y6tbSrHOBWoFgZfXsU8I0Yl-6jAOxStXgCBn_dwgydsoxyqv0OkWfi-hOseoTCw3WmlAu86UJm15nRzLlwBexjPWNkqchRS08qlzor2kRnQkerIutkjBkjqMCQLQ7GQltOxkzKsZOczqNyLOR-Ftrqfn8bRqrPx8O83bFbj5tT6uaI2bkmNv_FAMJAOft3DBzm1lkOZF3V5xSAtKHSm0tjmOzSbzv2pLzhxz4jx8PaRieRxwLbUWLn4s0fqTWxhYyIgGqnYJLEL7BEjBf2v04SCJyxvWBtT01MIpKVRWVxlBv9rI3ZjQuZ0_axcZ89tNZHIo-U_1RsQD7JOQpG0Lw3G4mlSbUDGqCEvw8r3e97DDI7zQqDm1WFtwu_3xFojBmk5YnRnMSRAMmPdscWv2dTcgbhVHe6OiLTVbNsiCtIE18YtQTbT5nTV856LAwNwvB7V6eC6XEo0-p0AvAZk_nsIDG0rcGFA3XdswwIAnmFCRcmZYA32eJhLEtuoBjh7OWlURtZ0dyte9ASD7_5GcspOYFz2qOF0T-YTuJKqdaG5BPbg7MXbeoeLG_FO-U5IWyHkdgVIdS7ghRH8jFhH7dX3mYDgV0JiBCpITIDeQxWAFHjApHSbFTZBfc5CQD9d2xpOVodrV4Ps3_r5QhHyqime02ntBbBEGrRHGi9MnwQC54ZtLeoo3To–ekcIn_V_iYFi0PEfjpGKoNagqAYBId7IQ_iG69MOWDMQ10mCF5yxBb3MwHk8iFUY-0vp7bV8C3Skc4WOCG4YRn7qkWoKp1DOog8tXELnkCCuppDB221flev6RPO9FEvkuA2uGuMskaUGdHq0kG8gIdttOyRDpI6-No4RncRtksNFoooV03dElEOcrwuP3ZL7YF0054NEIEVULnWZfnrajegvnxZmf0QTIv5IZhn0-FzPwQzG1cOn_0QjJoQchugdtu4fIT38J7EnMidxdPFo6YcZC3Grd2uQzRrxIMgbcXrzs0dCacMEmgVOFdNLtpCiwsmalzBcw_IDGTfojptQm7F6LLXMhOSRXI__n5xyRUazgPkdPKtlQecWITpBQlWGaH6ogezuoyXnUOs4LwkQQKriLRNTnQQQuEolwiC5VKOsA_0yV7KGAtukSQn_wUNUEFFAFODoPInqzFwTjt4YMWagutk2Oe3ijouWQUjSXe4Qq1Ot3hIleNIK6hMmnQodo581FqWk4JTVAkXmPcU0px-0w-f_CN9QiImIyHQfxALTbzlqLIyF3I6FR3CzST_2srh9lgROSncmdOQnVw2zfXTMveogiuBAksspd6CkHNmkF5whINvCCaSwRYXAWUmczZTGCzEEzSYM9s0iaR8B1BZKguZ07bouu23ii0eFnOKwHXp3NbkBy_sYdeIpCEORGs7BYykOUyNDzIGiSGnVyiSdDVeP62axkFlDjCzBwwyfpTUaCVEwNtgD9NvNQlU58gERczEY0gmEEuW_Cwz5pwoAVfJTq-bNTKlgSniEAqSfZ2rXoiYgDGJg6Tivp8G644rfzkmtuiq9YvPEmJCpq-a6LJw9vUCT3NKEJDwlLLlCiHbSCTIjNSrW5QmT0zFKTeH6NW4XPgO1UpijBAqN14NES5c9QiXT0bg5m21n_atETgazuTHIeeTu6noi6yCM7W6hby7hdglD0Mym9upQkUO4PiYw14oSxk0HmkoTnBwZOdkaRUqA6kACQVSeNwY63KK20-rU8gcEvBOqbmIFsRCNXocIFDuOPbT6SJIoDcqeE1LmYLfSPaTikgeQzJbcHO6Bb6WJ9CYAwfLh8xQq5RJgqqqk92XBd7vWeg9kEwC7cVdNweTxMWRBpeCXApwFoL5qh7SkHaIFgeVLfC2LnKshdCFKlkwSfZLJzqLB6-FPBNeI0Ez0ZpPTZnQkCALJlmVMZ7KxBb4eQ_cRoH1kU0OrIq6WRuC5UJ-CfKrb6OsxvHsr-gSDHDLJ9k1zlSV5s39ejkFXrnIrCmznYyQDJi13-JIMrZyBZKgLYAQq_cQEhSUIOtyw1JOHx-Auen3I6NcnyYTGrSXlINwqroJWtywl7mlCoLXGrHymlRiJdP0zznCHJAkMctIC3DKLiwMC_xcHknbVqjLtW–dkg74ShYilwqSepCIq3ynGC-WhieGOABt8p4lfjvzQ55d4tMaNxk0wYZXcFRNRrkHoQfkFiRaBg5zZW_W0IIh9TMQ8qMDLgtNpU4c2EXBsgRu9kT8u4Wl2RqlVkBU4GH09ikBqTGdG4WaBVZ1fQW5bqHYCZHC1Y1AzJGakCOyCQjI-DDZXn1bciVC5nWOAlwaAlQIabKWIKb4DU5jACBvROgxglMLHWWkIugVbawqdpNUDln5XpYOOr1gfPU2AQbzSkgSEjVC5FpLuoVJOMMBjL3wsb-lANpl6pEpQ683l28gZPeLHM5a0EzJPern3PBjQU3hE9MveZcgc2aFFpymiCUJR9FyJ5B2jbbOGNVPIQgqGUjhE6AFrdELkg1pFKsAp8RK16qqHBZK4-UvUQCxXG8aQ9uobJFfAJSzWAH4AWYudiScCKFAeYXybTE3ChPQHIZYx4C1YEVKmtHk2_RNtVCdWJaRVoG25SBaJODjSQNmO4c5ZOvyZG2iMpxJvaHvI-IJwFZ7G6hE_Dj3qrFJpsBwlIdFuXziGYyyVklWEo651zXAmwI0rjxBJ7VnHbTD6mi4s3BDxIOtwcpToQDnaElzIv1OZcKVBWncSARdVYWWnEjaaQ8HeNpEidZoXkUNwHHVQgmcgbANToXm4BeSrAHaK-Jsew9aaCcVaRDViSmMoZMcapk0n6TuO6Phnx-WqAI2SI5IRAg1VSzqh8WLd4vXWmU1bgQKxNsxA8QpErBcAmiZrODYakG_O29cbWQ5dwSlIMpru4n-RgPNWmA4BQiX43VguW3X4g6uugmG0mne1iM8oUvRJ19GepukRXd9U16yPP12xEtZsSz5rWWUd7v9YfZ2f6gVDeZfrFq-b_xOc-GR6PB2vRff1g0psatJMejPDlI-sOk3N3rNqZmzEydHObZ4OCse7Q_maospdyobtx9Y-v_kkVN5bzJ6Oy86I-Gd9_62tpcK_dddPfLkSdFoyK1nJGd3czeL06yYn82uHVYCriT9UNFz7V50-n764f7lyfdYTl-NvHoTu_61DxG9uNH47xEIHlSepQX_WEvOc5HgyRLZgglt0auJlPhSelplneTmYLVpDgpX19DMjpOSjWj_KBfJP3JL0njqpuPkkE3G06S7kU5c9zNB-fl0NKJlSe3WhnLp07dmHU39Mkfpa5SxeDapamOu0O3Xs5Nmw4uPTid2lVUJyZPShgnRd64nTjNg9PGygqrNx9dJpNpdjHyqpaslRP2ryc0eKEtwZmW5ExLcKbl4ExLcqZVceZfH6MHwg==&lang=sage. You should be able to edit the script and re-run it. The matrices are hardcoded in it.



It’s the first time I use Sage and I don’t know how to do some stuff. In the script I randomly generate some scramble move, find the solution to solve the 3C orbits by solving a linear system, and apply all the moves to 2C orbits to see if they are solved. So far it’s true. It’s enough for practical purpose, but it’s not a rigorous proof. It’ll be good if Sage can let me directly verify that the kernel of one matrix is a subspace of the kernel of another.



Nan