{"text_format":"1","workbook":"xlsx","n_sheets":9,"sheets":["Instructions","N=100","DataLogN100","N=169","DataLogN169","N=256","DataLogN256","RvsSqrtNplot","Notes"],"not_rendered":{"charts":3,"images":1,"macros":1,"external_links":0,"defined_names":0,"comments":0,"pivot_tables":0},"unopened":[],"bounds":{"document_bytes":16777216,"sheet_cells":100000,"rows_first":2000,"rows_last":500,"rows_between":2000,"halvings":0,"cell_chars":16384,"cell_head_chars":4096},"cells_cut":0,"cells_raw":0,"read":"whole","date_system":"1900"}

{"sheet":"Instructions","index":0,"n_rows":12,"n_cols":3,"rows_kept":12,"cut":false,"hidden":false,"merged":[[6,2,9,10]]}
["Note: There are nine worksheets in this workbook."]
[]
["Run Random Walks with different number of steps N (100, 169, 256): use corresponding worksheet for different N."]
[]
["Log N, ΔX, ΔY, and R for each walk into the corresponding DataLog worksheet."]
["Note the button:"]
[null,"Walk Again","Loads new random sequence and therefore makes a new random walk and then Pastes values of N, ΔX, ΔY, and R into the Log table and brings you back to \"Walk\" data."]
[]
[]
[]
[]
["Make a plot of R vs √N in the RvsSqrtNplot worksheet. You will have three points (N=100, 169, 256). Do you get a straight line?"]

{"sheet":"N=100","index":1,"n_rows":104,"n_cols":16,"rows_kept":104,"cut":false,"hidden":false,"merged":[]}
[null,null,null,null,null,null,null,null,null,null,null,null,"N","ΔX","ΔY","R"]
[null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"A104","v":100},{"$f":"F104","v":-2.4310979714736032},{"$f":"G104","v":-7.6500427784888041},{"$f":"SQRT(F104^2+G104^2)","v":8.0270412892679062}]
[]
["N","Random Seq","Random Angles","Δx","Δy","ΔX","ΔY"]
[1,0,{"$f":"B5*2*3.14159265359","v":0},0,{"$f":"SIN(C5)","v":0},0,{"$f":"E5","v":0}]
[2,{"$f":"RAND()","v":0.49565275447887425},{"$f":"B6*2*3.14159265359","v":3.1142781044049586},{"$f":"COS(C6)","v":-0.99962698089426394},{"$f":"SIN(C6)","v":2.7311152817465115E-2},{"$f":"F5+D6","v":-0.99962698089426394},{"$f":"G5+E6","v":2.7311152817465115E-2}]
[3,0.72510331136299133,4.5559584721435122,-0.15579329948812926,-0.98778967793483352,{"$f":"F6+D7","v":-1.1554202803823932},{"$f":"G6+E7","v":-0.96047852511736842}]
[4,4.8060779634110928E-2,0.30197478444866155,0.95475103799912331,0.29740621284633018,{"$f":"F7+D8","v":-0.20066924238326989},{"$f":"G7+E8","v":-0.66307231227103824}]
[5,1.9748744990432887E-2,0.12408502435913255,0.99231132623328877,0.12376684462783845,0.79164208385001889,{"$f":"G8+E9","v":-0.53930546764319975}]
[6,0.75008594228634928,4.7129289716958551,5.399912849225276E-4,-0.9999998542046955,0.79218207513494143,-1.5393053218478951]
[7,0.35128365042748044,2.2071802710185007,-0.59429112026723652,0.80425000116351442,0.19789095486770492,-0.73505532068438073]
[8,0.27765352130445975,1.7445485255469706,-0.17287925936448492,0.98494302458649208,2.5011695503219994E-2,0.24988770390211135]
[9,0.76377488172747232,4.7989391148631961,8.6442118171305649E-2,-0.99625687460918333,{"$f":"F12+D13","v":0.11145381367452564},{"$f":"G12+E13","v":-0.74636917070707198}]
[10,0.56927476146677292,3.576858816996427,-0.90675784614996391,-0.42165176205665084,-0.79530403247543824,-1.1680209327637228]
[11,0.95414189830661966,5.9950503564049864,0.95877552359805718,-0.28416455681395469,0.16347149112261894,-1.4521854895776776]
[12,0.24840281693831867,1.5607609296489671,1.0035228703981594E-2,0.99994964582465795,0.17350671982660054,-0.45223584375301962]
[13,4.6861497034780708E-2,0.29443946964139328,0.95696496051405344,0.29020348782937833,1.130471680340654,-0.16203235592364129]
[14,0.37313847405703759,2.3444981777387444,-0.6987880610385061,0.71532876759574338,0.43168361930214794,0.55329641167210208]
[15,0.11210034282500314,0.70434722696790064,0.76203440847368287,0.64753653202129391,1.1937180277758308,1.200832943693396]
[16,0.65835053469362415,4.1365384065610762,-0.54454838887733947,-0.83872942727144961,0.64916963889849133,0.36210351642194638]
[17,0.88338984713186941,5.5505021080109485,0.74338237798024576,-0.66886668335957289,1.3925520168787371,-0.30676316693762651]
[18,0.24997461864192194,1.5706368510188478,1.5947577537288351E-4,0.9999999872837384,1.39271149265411,0.69323682034611189]
[19,0.84751506511751151,5.3250942047600498,0.57508269300388615,-0.81809528553060251,1.9677941856579961,-0.12485846518449062]
[20,0.79317058935768847,4.9836377931395299,0.26793479777587076,-0.96343704731591218,2.2357289834338667,-1.0882955125004028]
[21,0.74018773465743903,4.6507366989544696,-6.1613232092903543E-2,-0.9981001000055385,2.1741157513409632,-2.0863956125059415]
[22,0.56475966760133101,3.5484896455605437,-0.9183532934587354,-0.39576158023928221,1.255762457882228,-2.4821571927452237]
[23,0.43135555946681292,2.7102869134122876,-0.90842065000016381,0.41805731981784494,0.34734180788206415,-2.064099872927379]
[24,4.2956002432963047E-2,0.26990052334198178,0.96379742514920863,0.26663556266138899,1.3111392330312728,-1.79746431026599]
[25,0.9844782751055402,6.1856594333810406,0.99524812013976427,-9.7371347727477978E-2,2.3063873531710373,-1.894835657993468]
[26,0.59130526687218321,3.7152805648694502,-0.83990513934359912,-0.54273322811875935,1.466482213827438,-2.4375688861122273]
[27,2.22924583217462E-2,0.14006764658811824,0.99020655433139892,0.13961009905855037,2.4566887681588367,-2.2979587870536768]
[28,0.81747724638200203,5.1363610234213599,0.4113840554502135,-0.91146209955287538,2.86807282360905,-3.2094208866065523]
[29,0.95845957046597641,6.0221790906778772,0.96613081026339587,-0.258052819127779,3.8342036338724457,-3.4674737057343314]
[30,0.47327282437792462,2.9736608564189564,-0.98593256217268854,0.16714359948139693,2.8482710716997572,-3.3003301062529347]
[31,0.6870716496612288,4.3169984941313571,-0.3851685769316604,-0.92284623169000346,2.4631024947680968,-4.2231763379429381]
[32,0.31800092009708325,1.9980587088237147,{"$f":"COS(C36)","v":-0.41438084243694268},{"$f":"SIN(C36)","v":0.91010357510628959},2.048721652331154,-3.3130727628366485]
[33,0.73196973799363763,4.5991015030620179,-0.11304530986880788,{"$f":"SIN(C37)","v":-0.99358983384325406},1.9356763424623462,-4.306662596679903]
[34,0.53846807830937049,3.3832947180188864,-0.9709319828681916,-0.2393555611295497,0.96474435959415461,-4.5460181578094527]
[35,0.82120911322617551,5.1598090343450229,0.43264098416179636,-0.90156629197386939,1.3973853437559509,-5.4475844497833226]
[36,0.48638369105642398,3.0560388612976994,-0.99634250603301155,8.5449462677411839E-2,0.40104283772293936,-5.362134987105911]
[37,0.77425113932332612,4.864763382663698,0.1517854493185376,-0.98841346478848091,{"$f":"F40+D41","v":0.55282828704147691},{"$f":"G40+E41","v":-6.350548451894392}]
[38,0.1411300655012635,0.8867463539588899,0.63193700698600408,0.77501975407183721,1.1847652940274811,-5.5755286978225547]
[39,0.36861396248614309,2.3160698331143341,-0.6781728362817715,0.73490244531470816,0.5065924577457096,-4.8406262525078461]
[40,0.93122847398265685,5.8510810653554826,0.90808654097068675,-0.41878256184790369,1.4146789987163964,-5.2594088143557496]
[41,0.1480977923867417,0.93052587315016955,0.59741234923990538,0.80193421486781402,2.0120913479563018,-4.4574745994879352]
[42,0.45312782700408638,2.8470861049064764,-0.95694549178916288,0.2902676794691364,1.055145856167139,-4.1672069200187991]
[43,0.50491660116184145,3.1724845697713464,-0.9995228827022965,-3.0887003002739735E-2,5.5622973464842462E-2,-4.1980939230215393]
[44,0.46607731861363733,2.9284501603230577,-0.97737100281795064,0.21153232105433306,-0.92174802935310818,-3.9865616019672063]
[45,0.38427527330252498,2.4144727511270041,-0.74709195868852707,0.66472069718261373,-1.6688399880416354,-3.3218409047845925]
[46,0.43414560748221265,2.7278173021089742,-0.91560940009323633,0.40206893247415171,-2.5844493881348716,-2.9197719723104409]
[47,0.85510430236971802,5.3727787887558165,0.61342475096887195,-0.789753173402157,-1.9710246371659996,-3.7095251457125977]
[48,0.80960307575216217,5.0868861502137221,0.36580459549531225,-0.93069167714904966,-1.6052200416706874,-4.6402168228616478]
[49,0.47036556895334869,2.9553940318510419,-0.98271506227991312,0.18512456986631065,-2.5879351039506004,-4.4550922529953372]
[50,0.62663163985449888,3.9372427125478966,-0.69982056468048104,-0.71431868045732405,-3.2877556686310814,-5.169410933452661]
[51,0.80917800644541804,5.0842153609910543,{"$f":"COS(C55)","v":0.36331761248710209},{"$f":"SIN(C55)","v":-0.93166534359536624},-2.9244380561439791,-6.1010762770480271]
[52,0.60690285568601943,3.8132831057319811,-0.78277079935348171,{"$f":"SIN(C56)","v":-0.62231011214627652},-3.7072088554974609,-6.723386389194304]
[53,0.96612063120439817,6.0703149549469417,0.97742853324942458,-0.21126633046909868,-2.7297803222480361,-6.9346527196634025]
[54,0.34353309855336245,2.158482117360506,-0.55443656405699226,0.83222598880152543,-3.2842168863050283,-6.1024267308618771]
[55,0.92052413753021722,5.7838237358344022,0.87788846202384008,-0.47886516708298665,-2.4063284242811882,-6.5812918979448636]
[56,7.3343119031006321E-2,0.46082840787837276,0.8956844199958881,0.44469025149718477,-1.5106440042853002,-6.1366016464476791]
[57,0.16593840314162345,1.0426217365163599,0.50395752341644129,0.86372843798960741,-1.0066864808688589,-5.2728732084580718]
[58,0.20601203992377504,1.2944118223512429,0.27287915823553999,0.96204831739401897,-0.73380732263331894,-4.310824891064053]
[59,0.67050975253812739,4.2129370254684595,-0.47894451182108627,-0.8778451768942297,-1.2127518344544053,-5.1886700679582827]
[60,0.62334035716359504,3.9165629735026339,-0.71444180137536195,-0.69969487095985483,-1.9271936358297672,-5.8883649389181372]
[61,0.89588640888732329,5.6290203212230834,0.79355630808975108,-0.60849682488157986,-1.1336373277400162,-6.4968617637997168]
[62,1.8455939112917208E-2,0.11596208546449009,0.99328392846298119,0.11570236582346649,-0.14035339927703505,-6.3811593979762504]
[63,0.97050001278581211,6.0978314209538169,0.98287109293786001,-0.18429436960237403,0.84251769366082496,-6.5654537675786244]
[64,0.32066740895694046,2.0148127524497288,-0.42956988753345354,0.9030336160545166,0.41294780612737142,-5.6624201515241079]
[65,0.86326248194118349,5.4240381427725843,0.65308354592904638,-0.75728586546742249,1.0660313520564177,-6.4197060169915305]
[66,{"$f":"RAND()","v":0.90337880375819335},{"$f":"B70*2*3.14159265359","v":5.6760964265913252},{"$f":"COS(C70)","v":0.82131222797468884},-0.57047894280091771,1.8873435800311067,-6.9901849597924484]
[67,0.56100635560011525,3.5249068907412426,-0.9274302191275845,-0.3739962414904201,0.95991336090352219,-7.364181201282868]
[68,0.38196260880904009,2.3999418515611026,-0.73735443716513305,0.67550605770258632,0.22255892373838915,-6.6886751435802818]
[69,0.12932404022655786,0.81256690941666365,0.68763698386558536,0.72605466627537119,0.91019590760397451,-5.9626204773049105]
[70,0.84365883951841425,5.3008648247346297,0.55509389424498301,-0.83178769441002165,1.4652898018489575,-6.7944081717149318]
[71,0.91498388855413737,5.7490133248697788,0.86069049172056122,-0.50912854708984701,2.3259802935695189,-7.3035367188047786]
[72,0.74527901949019004,4.6827261850102788,-2.9658445608646895E-2,-0.99956009154231384,2.296321847960872,-8.3030968103470926]
[73,0.26365165515244426,1.6565722058675261,-8.5670735067776352E-2,0.99632350426603244,2.2106511128930957,-7.3067733060810598]
[74,0.52930983969756318,3.3257518077335302,-0.983090573813861,-0.18311997072502348,1.2275605390792346,-7.4898932768060833]
[75,0.3311769149414785,2.0808459260374983,-0.48822053374897501,0.87272029335054768,0.73934000533025956,-6.6171729834555357]
[76,0.59855349139227942,3.7608225026772608,-0.8143257000085582,-0.58040817904779018,-7.4985694678298631E-2,-7.1975811625033259]
[77,0.51863199821268779,3.2586609510033639,-0.99315532940634288,-0.11680107735709698,-1.0681410240846416,-7.3143822398604232]
[78,0.60555426246059629,3.8048096445926403,{"$f":"COS(C82)","v":-0.78801575567936177},{"$f":"SIN(C82)","v":-0.61565507291102894},-1.8561567797640035,-7.9300373127714519]
[79,0.91846631885098395,5.7708940797442034,{"$f":"COS(C83)","v":0.87162369269049556},{"$f":"SIN(C83)","v":-0.49017561989615976},-0.98453308707350795,-8.4202129326676118]
[80,0.79684854640067393,5.0067470787924551,0.29012561744932946,-0.95698857156166983,-0.69440746962417843,-9.3772015042292818]
[81,0.49447655428712456,3.1068878206418549,-0.9993978477259603,3.4697866803570933E-2,-1.6938053173501388,-9.3425036374257111]
[82,0.16306278601563451,1.0245537012412711,0.51948029300480925,0.85448243117084488,-1.1743250243453296,-8.4880212062548654]
[83,0.91584663986505088,5.754434151230261,0.86343772983442557,-0.50445543579029217,-0.31088729451090402,-8.9924766420451583]
[84,2.4035109995103054E-3,1.5101704997768667E-2,0.99988597142023228,1.5101130984747333E-2,0.68899867690932826,-8.9773755110604103]
[85,0.46857694691459029,2.9441557881370164,-0.98057257415921273,0.19615663844686812,-0.29157389724988447,-8.7812188726135414]
[86,8.2539691891214795E-2,0.518612179350045,0.86850792424237766,0.49567528234540437,0.57693402699249319,-8.2855435902681371]
[87,0.38532651933448847,2.4210779247492682,-0.75146622525434326,0.65977156069505505,-0.17453219826185007,-7.6257720295730822]
[88,0.14451479959105362,0.90801326546057037,0.61531307154373693,0.78828283248299402,0.44078087328188686,-6.837489197090088]
[89,0.48410913628192009,3.0417474121581605,-0.99501960343931806,9.9679430031789085E-2,-0.55423873015743119,-6.737809767058299]
[90,0.37415265285627353,2.3508704510689569,-0.70333211321823463,0.71086140598289083,-1.2575708433756658,-6.0269483610754087]
[91,0.56463269184401477,3.5476918333478062,-0.9186687445793803,-0.39502878089190568,-2.1762395879550462,-6.4219771419673144]
[92,0.69299099841084411,4.3541908592230145,-0.35058728963010827,-0.9365300595014634,-2.5268268775851546,-7.358507201468778]
[93,0.43348688224802834,2.7236784093960784,-0.91393744228822127,0.40585508692594235,-3.4407643198733759,-6.9526521145428353]
[94,0.81508811983918938,5.1213496986305662,0.39765596594826957,-0.91753459484956135,-3.0431083539251063,-7.8701867093923967]
[95,0.78418678589266522,4.9271908912055027,0.21315389566422996,-0.97701863685559376,-2.8299544582608762,-8.8472053462479909]
[96,0.47192945107849293,2.9652201930419095,-0.98448665499111876,0.17545947151521293,-3.8144411132519949,-8.6717458747327782]
[97,0.36945390716296289,2.321347361166572,-0.68204184223304032,0.73131315142239883,-4.4964829554850354,-7.9404327233103791]
[98,0.18320485775977546,1.1511100704802231,0.40747395710613532,0.91321682763748246,-4.0890089983789002,-7.0272158956728967]
[99,1.9707637390113497E-2,0.12382673768879233,0.99234326045959231,0.12351053971392741,-3.0966657379193077,-6.9037053559589694]
[100,0.86590527121807814,5.4406432775271414,0.66556776644570426,-0.74633742252983437,{"$f":"F103+D104","v":-2.4310979714736032},{"$f":"G103+E104","v":-7.6500427784888041}]

{"sheet":"DataLogN100","index":2,"n_rows":6,"n_cols":12,"rows_kept":6,"cut":false,"hidden":false,"merged":[]}
["√N","AvgΔX","AvgΔY","AvgR",null,"GO TO Worksheet:","N=100","DataLogN100","N=169","DataLogN169","N=256","DataLogN256"]
[{"$f":"SQRT(A5)","v":10},{"$f":"AVERAGE(B5:B1048576)","v":-3.9738954496938899},{"$f":"AVERAGE(C5:C1048576)","v":-6.0025361138819253},{"$f":"AVERAGE(D5:D1048576)","v":7.8784199707677391}]
[]
["N","ΔX","ΔX","R"]
[100,-2.6942678421443911,-9.614699210639797,9.9850648528835766]
[100,-5.2535230572433891,-2.3903730171240536,5.7717750886519026]

{"sheet":"N=169","index":3,"n_rows":173,"n_cols":16,"rows_kept":173,"cut":false,"hidden":false,"merged":[]}
[null,null,null,null,null,null,null,null,null,null,null,null,"N","ΔX","ΔY","R"]
[null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"A173","v":169},{"$f":"B173","v":0.94712922813963141},5.9509884502476664,{"$f":"SQRT(F173^2+G173^2)","v":1.9719641341655905}]
[]
["N","Random Seq","Random Angles","Δx","Δy","ΔX","ΔY"]
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{"sheet":"DataLogN256","index":6,"n_rows":6,"n_cols":12,"rows_kept":6,"cut":false,"hidden":false,"merged":[]}
["√N","AvgΔX","AvgΔY","AvgR",null,"GO TO Worksheet:","N=100","DataLogN100","N=169","DataLogN169","N=256","DataLogN256"]
[{"$f":"SQRT(A5)","v":16},{"$f":"AVERAGE(B5:B1048576)","v":0.40212839371488523},{"$f":"AVERAGE(C5:C1048576)","v":2.5266472149892611},{"$f":"AVERAGE(D5:D1048576)","v":14.851407073812812}]
[]
["N","ΔX","ΔX","R"]
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{"sheet":"RvsSqrtNplot","index":7,"n_rows":0,"n_cols":0,"rows_kept":0,"cut":false,"hidden":false,"merged":[]}

{"sheet":"Notes","index":8,"n_rows":1,"n_cols":1,"rows_kept":1,"cut":false,"hidden":false,"merged":[]}
["F9 - to recalculate the formulas"]