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RandomSymmetricQuiescentRule[k_,r_:1]:=Module[{dom,quiet,rules},dom=Tuples[Reverse[Range[0,k-1]],2r+1];quiet=Range[0,k-1];rules=GatherBy[Complement[dom,ConstantArray[#,2r+1]&/@quiet],Sort[{Identity[#],Reverse[#]}]&];rules=Association[Join[ConstantArray[#,2r+1]->#&/@quiet,Catenate[Map[Thread,Thread[rules->RandomInteger[{0,k-1},Length@rules]]]]]];{FromDigits[Lookup[rules,dom],k],k}]
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RandomSymmetricQuiescentRule[3]
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{6783262123128,3}
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Clear[RandomSymmetricQuiescentRule]
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CAImage[rule_,h_:32,w_:Automatic]:=Image[255/Last[rule]CellularAutomaton[rule,{RandomInteger[Last[rule]-1,h],0},{h-1,{0,Replace[w,Automatic:>h]-1}}],"Byte"]
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ClickToCopy[CAImage[#,256],#]&/@Table[RandomSymmetricQuiescentRule[3],10]
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ResourceFunction["InteractiveListSelector"][CAImage[#,256]->#&/@Table[RandomSymmetricQuiescentRule[5],400]]
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CAImage[#,256]&/@{{6349211693517,3},{5324829516012,3},{6647539662672,3},{6507246443307,3},{6365511833685,3},{6858473386320,3},{6394903451625,3},{6774618406188,3},{6749880891483,3},{6010310658669,3}}
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CAImage[#,256]&/@{{284239410405091683209629494613710220400,4},{265529289117608431835306839985662081200,4},{329282280412628400091562363918956715324,4},{327272188097966298317230619157338197272,4},{282439373804312109724504781230526511176,4},{295220627206906196273441130900507269840,4},{333040508317036837390157938401602638456,4},{275851984344106965531423475374904843632,4},{316590779996158957340277193191987258372,4},{272631272863596451930822316251763821000,4},{295706225245720850593272827152800594488,4},{295444143622280899054014388608143011952,4},{262590932150406302301275065810423190208,4}}
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CAImage[#,256]&/@{{284239410405091683209629494613710220400,4},{265529289117608431835306839985662081200,4},{329282280412628400091562363918956715324,4},{327272188097966298317230619157338197272,4},{282439373804312109724504781230526511176,4},{295220627206906196273441130900507269840,4},{333040508317036837390157938401602638456,4},{275851984344106965531423475374904843632,4},{316590779996158957340277193191987258372,4},{272631272863596451930822316251763821000,4},{295706225245720850593272827152800594488,4},{295444143622280899054014388608143011952,4},{262590932150406302301275065810423190208,4}}
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CAImage[#,256]&/@{{2333872653738554337632156692623686834420828069422201702821657500609001657728397966359925,5},{2066791355148084243192896918596814991037842783469706282765229307937870488089149911376965,5},{2294537612021235175086075815980196710134265867179878284117062773936194139035869019904520,5},{2150054749453173174370967996359743952879718394846178350891551413068924399641259290919240,5},{1965870122514017852574892437735161627193677376532502040739100856944369280070396114141205,5},{2347142573705098324242962160903178878991953824588237396879540571320071462121551087450015,5},{2331759532437230681996676671058791594387589085466391375149900550400077696429548159853960,5},{1959067860445415223577991629067898567339244013478536489340983561993690715789326379750785,5},{2258799938494994165705872219829148103589171619656937566544206264785617119831400218806365,5},{2151102468625566576058434868839613303915323919546050685891305394292786381165378271848740,5},{2190132899532967378400595164291907304038348637119903413956649633365709292353488323976960,5}}
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CAImage[#,256]&/@{{2333872653738554337632156692623686834420828069422201702821657500609001657728397966359925,5},{2066791355148084243192896918596814991037842783469706282765229307937870488089149911376965,5},{2294537612021235175086075815980196710134265867179878284117062773936194139035869019904520,5},{2150054749453173174370967996359743952879718394846178350891551413068924399641259290919240,5},{1965870122514017852574892437735161627193677376532502040739100856944369280070396114141205,5},{2347142573705098324242962160903178878991953824588237396879540571320071462121551087450015,5},{2331759532437230681996676671058791594387589085466391375149900550400077696429548159853960,5},{1959067860445415223577991629067898567339244013478536489340983561993690715789326379750785,5},{2258799938494994165705872219829148103589171619656937566544206264785617119831400218806365,5},{2151102468625566576058434868839613303915323919546050685891305394292786381165378271848740,5},{2190132899532967378400595164291907304038348637119903413956649633365709292353488323976960,5}}
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ClickToCopy[CAImage[#],#]&/@Table[RandomSymmetricQuiescentRule[3],10]