(* 082404 *)
(* googleFinA.nb *)
(* ouch.  I spelled forty incorrectly (as the classic "fourty") *)
(* correct the other versions *)
(* This seems to do the trick!  *)
(* Part I: f(f(f(x)))=60097, x=2465 *)
(* Part II: f(f(x))=1, x=6808 *)

(* new hypothesis testing approach *)
(* adding "positive" to + numbers didn't help *)
(* there simply may be no solution to f(f(x))=1 *)
(* are there any other synonyms for "negative"? *)
(* double check your numbers, code, and data base *)
(* can you come up with a plausible argument not to go higher than 200 
    characters? *)
(* however, I've tried up to 1000 and it didn't help *)

(* 082104 *)
(* google6.nb *)
(* my silly attempt to solve the "google challenge" in Physics Today *)
(* what if the solution to part II f(f(z))=1 is a real fax number? *)
(* It would be just like them to do that *)
(* they call z the "machine extension number" *)
(* this implies a 4 digit number to me, 
  but unless my algorithm is screwed up *)
(* there are no 4 digit numbers that meet this criterion *) 

(* but they could be hinting at a *)
(* more generic term without using the word "phone number" *)
(* which would be to "obvious" *)

(* Form 1: 1-aaa-ppp-ssss or Form 2: aaa-ppp-ssss *)
(* aaa might be 800, 888, 
  or the real area code for Google Inc. in Mt. View *)
(* ppp could be cycled over or look up some real prefixes at Google Inc. *)

(* names up to 999999999999 i.e. (10^12)-1 *)
(* still need negative numbers *)
(* add more clever # creator using mathematica string functions *)


(* 082204 *)
(* I've managed to solve the first part of the google challenge *)
(* f(f(f(x)))=60097, x=2465 *)
(* using google3.nb *) 
(* but the second part *)
(* f(f(x))=1 find x *)
(* has proved more ellusive even including negative numbers 
      from +/-999999999 *)

(* I hesitate to increase the range beyond 999999999 (just takes too long) *)
(* but perhaps compile the functions? *)

(* There may be a number theory trick that can be used *)

(* e.g. for large numbers;  however, unless the large number *)
(* has a very long name, finding even an f(x)=1 will prove unlikely *)

(* google5.nb adds the "um-teen hundred" standard *)
(* however, it may have to do with a naming convention thing *)
(* go back to the "no um-teen hundred" standard *)
(* a weaker one is "minus" versus "negative" *)
(* are there any others? *)
(* is the code working? *)

(* try removing spaces for StringLength comparison *)

(* google4.nb works fine, and has up to +/-999999999 *)
(* google3.nb works fine, but has limited range and only positive numbers *)
(* with negative numbers, 
  there is a language ambiguity "minus" or "negative" *)

(* standard: use"negative" rather than "minus" *)
(* this can be changed *)

(* assumption: everything is in base 10, positive, and real *)

(* pre-google5.nb *)
(* standard: for the hundreds place, "um-teenth hundred" is not used *)
(* rather it is written out "one thousand x hundred blah blah" *)
(* however, the "um-teenth thousand" is used for the thousands place *)
(* this double standard could cause trouble and is an intrinsic language 
    ambiguity/degeneracy *)
(* should be easy enough to modify to include both at some point *)


(* create "data base" of numbers and their number of characters *)
(* written in American English *)
(* this is sort of a "minimal" set, 
  so other numbers can be constructed from them *)
(* regardless, fairly unelegent and brute force *)
	
sp=" ";
(* try a "minus" convention instead *)
(* ng=" negative "; eng=8; *)
(* ng= "minus ";eng=5; *)
ng=" negative "; eng=8;

(* add the very silly "positive" switch *)
(* to turn it off, just make it ="" *)
(* ps=" positive"; eps=8; *)
ps= "";eps=0;

x0=" zero ";e0=4;
x1=" one "; e1=3;
x2=" two "; e2=3;
x3=" three "; e3=5;
x4=" four "; e4=4;
x5=" five "; e5=4;
x6=" six "; e6=3;
x7=" seven "; e7=5;
x8=" eight "; e8=5;
x9=" nine "; e9=4;
x10=" ten "; e10=3;
x11=" eleven "; e11=6;
x12=" twelve "; e12=6;
x13=" thirteen "; e13=8;
x14=" fourteen "; e14=8;
x15=" fifteen "; e15=7;
x16=" sixteen "; e16=7;
x17=" seventeen "; e17=9;
x18=" eighteen "; e18=8;
x19=" nineteen "; e19=8;
x20=" twenty "; e20=6;
x30=" thirty "; e30=6;
x40=" forty "; e40=5;
x50=" fifty "; e50=5;
x60=" sixty "; e60=5;
x70=" seventy "; e70=7;
x80=" eighty "; e80=6;
x90=" ninety "; e90=6;
x100=" hundred "; e100=7;
x1000=" thousand "; e1000=8;
xM=" million "; eM=7;
xB= " billion "; eB=7;

num={"",x1,x2,x3,x4,x5,x6,x7,x8,x9};
teen={x10,x11,x12,x13,x14,x15,x16,x17,x18,x19};
tens={x10,x20,x30,x40,x50,x60,x70,x80,x90};

enum={0,e1,e2,e3,e4,e5,e6,e7,e8,e9};
eteen={e10,e11,e12,e13,e14,e15,e16,e17,e18,e19};
etens={e10,e20,e30,e40,e50,e60,e70,e80,e90};

(* working in blocks of 100, 
  create the name for sanity checks at QA later on *)
NumToWordHund[bb_,cc_,dd_]:=
		
	If[(bb==0)&&(cc==0),num[[dd+1]],
				
	If[(bb==0),
	If[(cc>0)&&(dd==0),tens[[cc]],
	If[cc==1,teen[[dd+1]],
	If[(cc>1)&&(dd>0),tens[[cc]]<>num[[dd+1]]]]],
	
	If[(bb>0),
	If[(cc==0),num[[bb+1]]<>x100<>num[[dd+1]],
	If[(cc>0)&&(dd==0),num[[bb+1]]<>x100<>tens[[cc]],
	If[cc==1,num[[bb+1]]<>x100<>teen[[dd+1]],
	If[(cc>1)&&(dd>0),num[[bb+1]]<>x100<>tens[[cc]]<>num[[dd+1]]
																			]]]]]]]

NumToWordTens[cc_,dd_]:=
		
	If[(cc==0),num[[dd+1]],
	If[(cc>0)&&(dd==0),tens[[cc]],
	If[cc==1,teen[[dd+1]],
	If[(cc>1)&&(dd>0),tens[[cc]]<>num[[dd+1]]]]]]
	
(* removes um-teen hundred standard *)
(* comment out below and add bracket to add "um-teen" hundred standard for 
    1*00 numbers *)
(* convets a number >1000, in blocks of 100, 
  into a name for sanity checks and QA later on *)
NumToWord[ssn_,a10_,a11_,a12_,a1_,a2_,a3_,a4_,a5_,a6_,a7_,a8_,a9_]:=
If[ssn==0,
If[(a10==0)&&(a11==0)&&(a12==0)&&(a1==0)&&(a2==0)&&(a3==0)&&(a4==0)&&(
          a5==0)&&(a6==0)&&(a7==0)&&(a8==0)&&(a9==0),"zero",
If[(a10==0)&&(a11==0)&&(a12==0)&&(a1==0)&&(a2==0)&&(a3==0)&&(a4==0)&&(
            a5==0)&&(a6==0),ps<>NumToWordHund[a7,a8,a9],
(* If[(a10==0)&&(a11==0)&&(a12==0)&&(a1==0)&&(a2==0)&&(a3==0)&&(a4==0)&&(
                a5==0)&&(a6==1)&&(a7>0),
            ps<>teen[[a7+1]]<>x100<>NumToWordTens[a8,a9], *)
If[(a10==0)&&(a11==0)&&(a12==0)&&(a1==0)&&(a2==0)&&(a3==0),
          ps<>NumToWordHund[a4,a5,a6]<>x1000<>NumToWordHund[a7,a8,a9],
If[(a10==0)&&(a11==0)&&(a12==0),
            ps<>NumToWordHund[a1,a2,a3]<>xM<>NumToWordHund[a4,a5,a6]<>x1000<>
              NumToWordHund[a7,a8,a9],
ps<>NumToWordHund[a10,a11,a12]<>xB<>NumToWordHund[a1,a2,a3]<>xM<>
              NumToWordHund[a4,a5,a6]<>x1000<>NumToWordHund[a7,a8,a9]]]]],
If[ssn==1,		
If[(a10==0)&&(a11==0)&&(a12==0)&&(a1==0)&&(a2==0)&&(a3==0)&&(a4==0)&&(
            a5==0)&&(a6==0)&&(a7==0)&&(a8==0)&&(a9==0),"zero",
If[(a10==0)&&(a11==0)&&(a12==0)&&(a1==0)&&(a2==0)&&(a3==0)&&(a4==0)&&(
              a5==0)&&(a6==0),ng<>NumToWordHund[a7,a8,a9],
(* If[(a10==0)&&(a11==0)&&(a12==0)&&(a1==0)&&(a2==0)&&(a3==0)&&(a4==0)&&(
                  a5==0)&&(a6==1)&&(a7>0),
              ng<>teen[[a7+1]]<>x100<>NumToWordTens[a8,a9], *)
If[(a10==0)&&(a11==0)&&(a12==0)&&(a1==0)&&(a2==0)&&(a3==0),
            ng<>NumToWordHund[a4,a5,a6]<>x1000<>NumToWordHund[a7,a8,a9],
If[(a10==0)&&(a11==0)&&(a12==0),
              ng<>NumToWordHund[a1,a2,a3]<>xM<>NumToWordHund[a4,a5,a6]<>
                x1000<>NumToWordHund[a7,a8,a9],
ng<>NumToWordHund[a10,a11,a12]<>xB<>NumToWordHund[a1,a2,a3]<>xM<>
                NumToWordHund[a4,a5,a6]<>x1000<>
                NumToWordHund[a7,a8,a9]]]]]]]		
							
hbil[u4_]:=If[Abs[u4]>10^11-1,Floor[Abs[u4]/10^11],0];
tbil[u5_]:=If[Abs[u5]>10^10-1,Floor[(Abs[u5]-hbil[u5] 10^11)/10^10],0];
bil[u6_]:=
  If[Abs[u6]>10^9-1,Floor[(Abs[u6]-hbil[u6] 10^11-tbil[u6] 10^10)/10^9],0];
hmil[u1_]:=
  If[Abs[u1]>99999999,
    Floor[(Abs[u1]-hbil[u1] 10^11-tbil[u1] 10^10-bil[u1] 10^9)/100000000],0];
tmil[u2_]:=
  If[Abs[u2]>9999999,
    Floor[(Abs[u2]-hbil[u2] 10^11-tbil[u2] 10^10-bil[u2] 10^9-hmil[u2] 10^8)/
        10000000],0]
mil[u3_]:=
  If[Abs[u3]>999999,
    Floor[(Abs[u3]-hbil[u3] 10^11-tbil[u3] 10^10-bil[u3] 10^9-hmil[u3] 10^8-
            tmil[u3] 10^7)/1000000],0]
hthou[uu_]:=
  If[Abs[uu]>99999,
    Floor[(Abs[uu]-hbil[uu] 10^11-tbil[uu] 10^10-bil[uu] 10^9-hmil[uu] 10^8-
            tmil[uu] 10^7-mil[uu] 10^6)/100000],0]
tthou[vv_]:=
  If[Abs[vv]>9999,
    Floor[(Abs[vv]-hbil[vv] 10^11-tbil[vv] 10^10-bil[vv] 10^9-hmil[vv] 10^8-
            tmil[vv] 10^7-mil[vv] 10^6-hthou[vv] 10^5)/10000],0];
thou[xx_]:=
  If[Abs[xx]>999,
    Floor[(Abs[xx]-hbil[xx] 10^11-tbil[xx] 10^10-bil[xx] 10^9-hmil[xx] 10^8-
            tmil[xx] 10^7-mil[xx] 10^6-hthou[xx] 10^5-tthou[xx] 10^4)/1000],
    0];
hund[yy_]:=
  If[Abs[yy]>99,
    Floor[(Abs[yy]-hbil[yy] 10^11-tbil[yy] 10^10-bil[yy] 10^9-hmil[yy] 10^8-
            tmil[yy] 10^7-mil[yy] 10^6-hthou[yy] 10^5-tthou[yy] 10^4-
            thou[yy] 10^3)/100],0];
tensp[zz_]:=
  If[Abs[zz]>9,
    Floor[(Abs[zz]-hbil[zz] 10^11-tbil[zz] 10^10-bil[zz] 10^9-hmil[zz] 10^8-
            tmil[zz] 10^7-mil[zz] 10^6-hthou[zz] 10^5-tthou[zz] 10^4 -
            thou[zz] 10^3-hund[zz] 10^2)/10],0];
ones[qq_]:=(
    Abs[qq]-hbil[qq] 10^11-tbil[qq] 10^10-bil[qq] 10^9-hmil[qq] 10^8-
      tmil[qq] 10^7-mil[qq] 10^6-hthou[qq] 10^5-tthou[qq] 10^4-thou[qq] 10^3-
      hund[qq] 10^2-tensp[qq] 10);
(* takes a number from user input and gives it a name in American English *)
Nm[xxx_]:=
  NumToWord[If[xxx<0,1,0],hbil[xxx],tbil[xxx],bil[xxx],hmil[xxx],tmil[xxx],
    mil[xxx],hthou[xxx],tthou[xxx],thou[xxx],hund[xxx],tensp[xxx],ones[xxx]]

(* number of characters in a number's name using American English *)
(* spaces not included *)
NumLett[myNumb_]:=StringLength[StringReplace[Nm[myNumb]," "->""]]

(* Google function *)
f[p_]:=3 NumLett[p]^3-p

(* QA *)
abc=1441;
Nm[abc]
NumLett[abc]
f[abc]
Nm[f[abc]]
test=15105282689
hbil[test]
tbil[test]
bil[test]
hmil[test]
tmil[test]
mil[test]
hthou[test]
tthou[test]
thou[test]
hund[test]
tensp[test]
ones[test]

" one  thousand  four  hundred  forty  one "
30
79559
" seventy  nine  thousand  five  hundred  fifty  nine "
15105282689
0
1
5
1
0
5
2
8
2
6
8
9
(* hypothesis testing rather than brute force to invert f(x) *)
(* if you are given f(x)=a, this may be faster *)

(* since the number of letters in just about any number is going to be less 
    than 200 *)
(* it is much faster in principle to iterate over the number of letters 
    rather than *)
(* the numbers themselves *)

(* 0. Given f(x)=a, a is >0 and real *)
(* I. Pick Ex e.g. in iteration *)
(* II. generate x hypothesis from f(x)=a *)
(* i.e. xhypoth=3 Ex^3-a *)
(* III. test to see if Ex represents the number of letters in xhypoth *)
(* IV. if it does, then you have inverted f(x)=a *)
(* V. if not, then continue *)

hypoth[a_]:=
  Do[If[NumLett[3 nn^3-a]==nn,
      Print[nn," ",3 nn^3-a," ",Nm[3 nn^3-a]," ",f[3 nn^3-a]]],{nn,0,200}]
Thypoth[a_,nn_]:=
  Print[nn," ",NumLett[3 nn^3-a]," ",3 nn^3-a," ",Nm[3 nn^3-a]," ",
    f[3 nn^3-a]]
(* Do the problem part I and II by brute force *)
Do[If[f[f[f[i]]]==60097,Print[i," ",f[i]," ",f[f[i]]," ",f[f[f[i]]]]],{i,0,
    7000}]
2465\[InvisibleSpace]" "\[InvisibleSpace]86908
  \[InvisibleSpace]" "\[InvisibleSpace]20903
  \[InvisibleSpace]" "\[InvisibleSpace]60097
Do[If[f[f[j]]==1,Print[j," ",f[j]," ",f[f[j]]]],{j,0,7000}]
6808\[InvisibleSpace]" "\[InvisibleSpace]59048
  \[InvisibleSpace]" "\[InvisibleSpace]1
hypoth[1]

hypoth[59048]
hypoth[206762]
hypoth[311468]

f[f[6808]]
f[f[387583]]
27\[InvisibleSpace]" "\[InvisibleSpace]59048
  \[InvisibleSpace]" "\[InvisibleSpace]" fifty  nine  thousand  forty  eight \
"\[InvisibleSpace]" "\[InvisibleSpace]1
41\[InvisibleSpace]" "\[InvisibleSpace]206762
  \[InvisibleSpace]" "\[InvisibleSpace]" two  hundred  six  thousand  seven  \
hundred  sixty  two "\[InvisibleSpace]" "\[InvisibleSpace]1
47\[InvisibleSpace]" "\[InvisibleSpace]311468
  \[InvisibleSpace]" "\[InvisibleSpace]" three  hundred  eleven  thousand  \
four  hundred  sixty  eight "\[InvisibleSpace]" "\[InvisibleSpace]1
28\[InvisibleSpace]" "\[InvisibleSpace]6808
  \[InvisibleSpace]" "\[InvisibleSpace]" six  thousand  eight  hundred  eight \
"\[InvisibleSpace]" "\[InvisibleSpace]59048
53\[InvisibleSpace]" "\[InvisibleSpace]387583
  \[InvisibleSpace]" "\[InvisibleSpace]" three  hundred  eighty  seven  \
thousand  five  hundred  eighty  three "\[InvisibleSpace]" \
"\[InvisibleSpace]59048
1
1
hypoth[60097]

hypoth[20903]
hypoth[91862]
hypoth[337856]

hypoth[11041]
hypoth[86908]
hypoth[-26006]
hypoth[163690]
hypoth[283138]
hypoth[-64481]

f[f[f[244511]]]
f[f[f[320735]]]
f[f[f[363959]]]
f[f[f[2465]]]
f[f[f[224561]]]
f[f[f[85055]]]
f[f[f[-35065]]]
f[f[f[375950]]]
30\[InvisibleSpace]" "\[InvisibleSpace]20903
  \[InvisibleSpace]" "\[InvisibleSpace]" twenty  thousand  nine  hundred  \
three "\[InvisibleSpace]" "\[InvisibleSpace]60097
37\[InvisibleSpace]" "\[InvisibleSpace]91862
  \[InvisibleSpace]" "\[InvisibleSpace]" ninety  one  thousand  eight  \
hundred  sixty  two "\[InvisibleSpace]" "\[InvisibleSpace]60097
51\[InvisibleSpace]" "\[InvisibleSpace]337856
  \[InvisibleSpace]" "\[InvisibleSpace]" three  hundred  thirty  seven  \
thousand  eight  hundred  fifty  six "\[InvisibleSpace]" \
"\[InvisibleSpace]60097
22\[InvisibleSpace]" "\[InvisibleSpace]11041
  \[InvisibleSpace]" "\[InvisibleSpace]" eleven  thousand  forty  one "\
\[InvisibleSpace]" "\[InvisibleSpace]20903
33\[InvisibleSpace]" "\[InvisibleSpace]86908
  \[InvisibleSpace]" "\[InvisibleSpace]" eighty  six  thousand  nine  hundred \
 eight "\[InvisibleSpace]" "\[InvisibleSpace]20903
28\[InvisibleSpace]" "\[InvisibleSpace]-26006
    \[InvisibleSpace]" "\[InvisibleSpace]" negative  twenty  six  thousand  \
six "\[InvisibleSpace]" "\[InvisibleSpace]91862
44\[InvisibleSpace]" "\[InvisibleSpace]163690
  \[InvisibleSpace]" "\[InvisibleSpace]" one  hundred  sixty  three  thousand \
 six  hundred  ninety "\[InvisibleSpace]" "\[InvisibleSpace]91862
50\[InvisibleSpace]" "\[InvisibleSpace]283138
  \[InvisibleSpace]" "\[InvisibleSpace]" two  hundred  eighty  three  \
thousand  one  hundred  thirty  eight "\[InvisibleSpace]" \
"\[InvisibleSpace]91862
45\[InvisibleSpace]" "\[InvisibleSpace]-64481
    \[InvisibleSpace]" "\[InvisibleSpace]" negative  sixty  four  thousand  \
four  hundred  eighty  one "\[InvisibleSpace]" "\[InvisibleSpace]337856
44\[InvisibleSpace]" "\[InvisibleSpace]244511
  \[InvisibleSpace]" "\[InvisibleSpace]" two  hundred  forty  four  thousand  \
five  hundred  eleven "\[InvisibleSpace]" "\[InvisibleSpace]11041
48\[InvisibleSpace]" "\[InvisibleSpace]320735
  \[InvisibleSpace]" "\[InvisibleSpace]" three  hundred  twenty  thousand  \
seven  hundred  thirty  five "\[InvisibleSpace]" "\[InvisibleSpace]11041
50\[InvisibleSpace]" "\[InvisibleSpace]363959
  \[InvisibleSpace]" "\[InvisibleSpace]" three  hundred  sixty  three  \
thousand  nine  hundred  fifty  nine "\[InvisibleSpace]" \
"\[InvisibleSpace]11041
31\[InvisibleSpace]" "\[InvisibleSpace]2465
  \[InvisibleSpace]" "\[InvisibleSpace]" two  thousand  four  hundred  sixty  \
five "\[InvisibleSpace]" "\[InvisibleSpace]86908
47\[InvisibleSpace]" "\[InvisibleSpace]224561
  \[InvisibleSpace]" "\[InvisibleSpace]" two  hundred  twenty  four  thousand \
 five  hundred  sixty  one "\[InvisibleSpace]" "\[InvisibleSpace]86908
27\[InvisibleSpace]" "\[InvisibleSpace]85055
  \[InvisibleSpace]" "\[InvisibleSpace]" eighty  five  thousand  fifty  five \
"\[InvisibleSpace]" "\[InvisibleSpace]-26006
35\[InvisibleSpace]" "\[InvisibleSpace]-35065
    \[InvisibleSpace]" "\[InvisibleSpace]" negative  thirty  five  thousand  \
sixty  five "\[InvisibleSpace]" "\[InvisibleSpace]163690
47\[InvisibleSpace]" "\[InvisibleSpace]375950
  \[InvisibleSpace]" "\[InvisibleSpace]" three  hundred  seventy  five  \
thousand  nine  hundred  fifty "\[InvisibleSpace]" "\[InvisibleSpace]-64481
60097
60097
60097
60097
60097
60097
60097
60097

(* QA on problem one;  looks like the answer is Y=2465 *)
f[f[f[2465]]]
Nm[2465]
NumLett[2465]

Nm[NumLett[2465]]
NumLett[NumLett[2465]]

f[2465]==3 NumLett[2465]^3-2465
f[2465]

Nm[f[2465]]
NumLett[f[2465]]

f[86908]==3 NumLett[86908]^3-86908
f[86908]

Nm[f[86908]]
NumLett[f[86908]]

f[20903]==3 NumLett[20903]^3-20903
(* plots *)
xmax=4000;
ef=Table[NumLett[x],{x,0,xmax}];
tf=Table[f[x],{x,0,xmax}];
p1=ListPlot[ef]
p3=ListPlot[tf]
Show[p1,p3]
GraphicsData["PostScript", "\<\
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.92025 .46689 Mdot
.92049 .42688 Mdot
.92073 .50932 Mdot
.92097 .50932 Mdot
.9212 .46688 Mdot
.92144 .3892 Mdot
.92168 .50931 Mdot
.92192 .5093 Mdot
.92216 .60173 Mdot
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.9243 .46683 Mdot
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.92501 .50925 Mdot
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.92573 .55417 Mdot
.92596 .50924 Mdot
.9262 .35374 Mdot
.92644 .46679 Mdot
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.92692 .55415 Mdot
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.92835 .5092 Mdot
.92858 .18535 Mdot
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.9293 .32052 Mdot
.92954 .28949 Mdot
.92977 .28948 Mdot
.93001 .26052 Mdot
.93025 .3205 Mdot
.93049 .3205 Mdot
.93073 .28947 Mdot
.93096 .2605 Mdot
.9312 .35366 Mdot
.93144 .35365 Mdot
.93168 .42669 Mdot
.93192 .42669 Mdot
.93215 .38902 Mdot
.93239 .38902 Mdot
.93263 .46669 Mdot
.93287 .42667 Mdot
.93311 .42667 Mdot
.93334 .35362 Mdot
.93358 .46667 Mdot
.93382 .46667 Mdot
.93406 .55404 Mdot
.9343 .5091 Mdot
.93453 .50909 Mdot
.93477 .46665 Mdot
.93501 .55402 Mdot
.93525 .55402 Mdot
.93549 .50908 Mdot
.93572 .35358 Mdot
.93596 .46663 Mdot
.9362 .46663 Mdot
.93644 .554 Mdot
.93668 .50906 Mdot
.93691 .50905 Mdot
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.93739 .55398 Mdot
.93763 .55398 Mdot
.93787 .50904 Mdot
.9381 .32037 Mdot
.93834 .42658 Mdot
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.93882 .50902 Mdot
.93906 .46658 Mdot
.93929 .46658 Mdot
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.93977 .50901 Mdot
.94001 .509 Mdot
.94025 .46656 Mdot
.94049 .32033 Mdot
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.94096 .42653 Mdot
.9412 .50898 Mdot
.94144 .46654 Mdot
.94168 .46654 Mdot
.94191 .42652 Mdot
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.94239 .50896 Mdot
.94263 .46652 Mdot
.94287 .3203 Mdot
.9431 .4265 Mdot
.94334 .4265 Mdot
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.94382 .4665 Mdot
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.96 .42622 Mdot
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.97 .5085 Mdot
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.97286 .426 Mdot
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% End of Graphics
MathPictureEnd
\
\>"]
\[SkeletonIndicator]Graphics\[SkeletonIndicator]
Do[Print[x," ",Nm[x]," ",NumLett[x]," ",f[x]," ",NumLett[f[x]]," ",f[f[x]]],{
    x,1000,9000,100}]
printhypoth[a_]:=
  Do[Print[NumLett[3 nn^3-a]==nn," ",nn," ",NumLett[3 nn^3-a]," ",3 nn^3-a,
      " ",Nm[3 nn^3-a]," ",f[3 nn^3-a]],{nn,0,100}]
diffnum[a_]:=Table[nn-NumLett[3 nn^3-a],{nn,0,1000}]
(* this gives a sense of the increasing nature of the difference between *)
(* a number and its required number of letters to make f(x)=a true *)
(* it is interesting that the zero crossing, 
  where a solution potentially exists *)
(* is near 40-50 in all graphs *)

(* it seems very unlikely that a freak number with >
    100 letters in it could suddenly *)
(* buck the trend and drive the difference to zero *)
(* but I can't prove it, except by my "data" here *)
(* if I take the trend line and assume a gaussian smear, 
  I could probably put a confidence *)
(* level on the non-existence of f(f(x))=1 *)
(* i.e. if it does exist in the sweet spot (and it needs to be tested there), 
  then it doesn't *)
(* exist for numbers with greater than 100 letters with X percent CL *)
ListPlot[diffnum[311468]]
ListPlot[diffnum[1]]
ListPlot[diffnum[206762]]