Problem of the Month (May 2016)

This month we investigate various problems involving number names:

1. What is the smallest non-negative integer whose name contains exactly n of a given letter?

2. Consider the following list of spelled-out numbers and letters:

TWO T
THREE E
THREE H
FOUR R

Notice that each is a true statement of how many of that letter have appeared up to and including in that line, and the numbers in the statements are non-decreasing. Given the last lines are n occurrences of the same number, what are the shortest lists that accomplish this? (In breaking ties, we want the shortest number of statements, and then the numerically smallest list of numbers, and then the earliest alphabetical list of letters.) (This problem is a variation of a problem of Eric Angelini.)

3. It is well-known that TWELVE + ONE = ELEVEN + TWO is a true mathematical statement as well as an anagram. Other mathematical anagrams are not true mathematically, such as SIXTY + FIFTEEN being 9 more than FIFTY + SIXTEEN. Given an integer n, what is the collection of number names that use the same letters, but differ in value by n, so that the values are as small as possible?

4. If we link the names for the first n numbers crossword style, what is the smallest rectangle (by area) they will fit into?

5. Is there a positive integer n with the following property: if digits are substituted for letters in the name of n (with spaces being ignored), and then the digits are interpreted in some integer base, the result is n. (I think the smallest number for which this evaluation can give a number smaller than n is the number ONE BILLION = 10234554106 = 10,816,350 < 1,000,000,000.) If no such number exists, then how good an approximation can we find (using relative error)?


ANSWERS

1.
Smallest Number Name Containing a Letter Exactly n Times
n1234
AONE THOUSANDONE QUADRILLION
ONE THOUSAND
ONE QUATTUORDECILLION
ONE QUADRILLION
ONE THOUSAND
---
BONE BILLION---------
CONE OCTILLIONONE DECILLION
ONE OCTILLION
ONE UNDECILLION
ONE DECILLION ONE OCTILLION
ONE DUODECILLION ONE UNDECILLION
ONE DECILLION ONE OCTILLION
DONE THOUSANDONE HUNDREDONE THOUSAND ONE HUNDREDONE HUNDRED MILLION
ONE HUNDRED
EONETHREEELEVENSEVENTEEN
FFOURFIFTEENFIFTY-FOURFOUR HUNDRED FIFTY-FOUR
GEIGHTEIGHTY-EIGHTEIGHT HUNDRED EIGHTY-EIGHTEIGHT THOUSAND
EIGHT HUNDRED EIGHTY-EIGHT
HTHREETHIRTY-THREEONE HUNDRED THIRTY-THREETHREE HUNDRED THIRTY-THREE
IFIVETHIRTY-FIVEFIVE HUNDRED THIRTY-FIVEFIVE THOUSAND
FIVE HUNDRED THIRTY-FIVE
J------------
K------------
LELEVENELEVEN THOUSAND
ELEVEN
ONE MILLION ELEVENELEVEN MILLION ELEVEN
MONE MILLIONONE NOVEMDECILLION
ONE MILLION
------
NONENINENINETEENNINETY-NINE
OONEFORTY-ONEONE HUNDRED FORTY-ONEONE THOUSAND FORTY-ONE
PONE SEPTILLIONONE SEPTENDECILLION
ONE SEPTILLION
------
QONE QUADRILLIONONE QUINTILLION
ONE QUADRILLION
ONE QUATTUORDECILLION
ONE QUINTILLION
ONE QUADRILLION
ONE QUINDECILLION
ONE QUATTUORDECILLION
ONE QUINTILLION ONE QUADRILLION
RTHREETHIRTY-THREEONE HUNDRED THIRTY-THREETHREE HUNDRED THIRTY-THREE
SSIXSIXTY-SIXSIX HUNDRED SIXTY-SIXONE THOUSAND
SIX HUNDRED SIXTY-SIX
TTWOTHIRTEENTWENTY-TWOTWO HUNDRED TWENTY-TWO
UFOURONE HUNDRED FOURFOUR HUNDRED FOURONE THOUSAND
FOUR HUNDRED FOUR
VFIVESEVENTY-FIVEFIVE HUNDRED SEVENTY-FIVEFIVE THOUSAND
FIVE HUNDRED SEVENTY-FIVE
WTWOTWENTY-TWOTWO HUNDRED TWENTY-TWOTWO THOUSAND
TWO HUNDRED TWENTY-TWO
XSIXSIXTY-SIXSIX HUNDRED SIXTY-SIXSIX THOUSAND
SIX HUNDRED SIXTY-SIX
YTWENTYTWENTY THOUSAND
TWENTY
TWENTY MILLION
TWENTY THOUSAND TWENTY
TWENTY BILLION TWENTY MILLION
TWENTY THOUSAND TWENTY
ZZERO---------


2.
Shortest Known List Ending with n of Same Number
nONETWOTHREEFOURFIVESIXSEVENEIGHTNINETENELEVEN
1 ONE A TWO O THREE E TWO O
FOUR O
THREE E
FIVE E
THREE E
THREE H
SIX E
THREE E
FOUR E
SEVEN E
THREE E
THREE H
SIX E
EIGHT E
THREE E
FOUR E
SEVEN E
NINE E
THREE E
THREE H
SIX E
EIGHT E
TEN E
THREE E
FOUR E
SEVEN E
ELEVEN E
2 ONE A
ONE B
ONE A
TWO A
TWO E
THREE E
THREE H
THREE E
FOUR E
FOUR R
ONE A
ONE B
FOUR O
FIVE O
FIVE E
ONE I
TWO I
THREE I
THREE H
SIX E
SIX I
ONE A
ONE B
ONE C
ONE D
SEVEN E
SEVEN N
ONE H
THREE H
THREE R
FIVE H
EIGHT E
EIGHT H
ONE A
ONE B
THREE N
SEVEN E
NINE E
NINE N
ONE T
THREE T
THREE H
FIVE T
SIX T
SEVEN T
TEN E
TEN T
ONE V
TWO V
THREE V
FIVE V
SEVEN V
EIGHT V
ELEVEN E
ELEVEN V
3 ONE A
ONE B
ONE C
ONE A
TWO A
TWO E
TWO N
TWO O
THREE E
THREE H
THREE O
THREE E
FOUR E
FOUR R
FOUR F
ONE A
ONE B
FOUR O
FIVE O
FIVE E
FIVE F
ONE A
ONE B
ONE C
ONE D
ONE F
SIX E
SIX O
SIX N
ONE S
TWO S
FOUR O
FIVE O
SIX O
SEVEN O
SEVEN E
SEVEN S
TWO O
THREE T
THREE H
THREE O
EIGHT E
EIGHT T
EIGHT H
ONE V
TWO V
THREE V
FIVE V
SEVEN V
EIGHT V
NINE E
NINE V
NINE N
TWO O
THREE T
THREE H
FOUR H
FIVE H
SIX H
EIGHT H
EIGHT E
TEN E
TEN H
TEN T
4 ONE A
ONE B
ONE C
ONE D
ONE A
TWO T
TWO A
TWO E
TWO N
ONE A
ONE H
THREE H
THREE R
THREE N
THREE O
ONE H
THREE H
FOUR E
FOUR O
FOUR F
FOUR H
ONE A
ONE B
FOUR O
FIVE O
FIVE E
FIVE F
FIVE I
ONE A
ONE B
ONE C
ONE D
ONE I
SIX E
SIX N
SIX O
SIX I
ONE A
ONE B
ONE C
ONE S
TWO A
TWO S
SEVEN E
SEVEN N
SEVEN O
SEVEN S
ONE G
ONE H
TWO T
TWO G
THREE H
THREE G
EIGHT E
EIGHT T
EIGHT H
EIGHT G
ONE H
ONE I
THREE H
FOUR H
FIVE H
SIX H
EIGHT H
NINE H
NINE E
NINE N
NINE I
5 ONE A
ONE B
ONE C
ONE D
ONE F
ONE A
ONE B
ONE C
ONE D
TWO T
TWO A
TWO B
TWO C
TWO D
ONE W
TWO N
THREE T
THREE H
THREE N
THREE O
THREE W
ONE A
TWO A
THREE A
FOUR O
FOUR R
FOUR F
FOUR A
FOUR E
ONE A
ONE F
ONE I
TWO F
FIVE E
FIVE F
FIVE I
FIVE V
FIVE O
ONE A
ONE B
ONE C
ONE D
ONE I
SIX E
SIX N
SIX O
SIX I
SIX S
ONE A
ONE B
ONE S
ONE V
TWO B
TWO S
SEVEN E
SEVEN N
SEVEN O
SEVEN S
SEVEN V
ONE A
ONE B
ONE G
ONE H
TWO A
TWO G
TWO H
THREE A
EIGHT E
EIGHT O
EIGHT T
EIGHT H
EIGHT G
6 ONE A
ONE B
ONE C
ONE D
ONE F
ONE G
ONE A
ONE B
ONE C
ONE D
ONE F
TWO T
TWO A
TWO B
TWO C
TWO D
TWO F
ONE A
ONE B
ONE C
ONE H
TWO W
TWO A
TWO B
TWO C
THREE H
THREE R
THREE A
THREE B
THREE C
THREE W
ONE H
TWO H
THREE T
FOUR O
FOUR R
FOUR F
FOUR E
FOUR H
FOUR T
ONE A
ONE F
ONE I
FOUR N
FIVE E
FIVE F
FIVE I
FIVE V
FIVE N
FIVE O
ONE A
ONE B
ONE C
ONE S
TWO S
FIVE N
SIX E
SIX N
SIX S
SIX I
SIX X
SIX O
ONE A
ONE F
FOUR O
FOUR F
FOUR R
SIX F
SEVEN F
SEVEN E
SEVEN O
SEVEN N
SEVEN S
SEVEN V
7 ONE A
ONE B
ONE C
ONE D
ONE F
ONE G
ONE H
ONE A
ONE B
ONE C
ONE D
ONE F
ONE G
TWO T
TWO A
TWO B
TWO C
TWO D
TWO F
TWO G
ONE A
ONE B
ONE C
ONE D
ONE H
TWO W
TWO A
TWO B
TWO C
TWO D
THREE H
THREE R
THREE A
THREE B
THREE C
THREE D
THREE W
ONE F
ONE R
ONE T
TWO W
TWO F
FOUR F
FOUR R
FOUR U
FOUR E
FOUR N
FOUR T
FOUR W
ONE I
TWO T
TWO I
THREE W
FOUR W
FIVE E
FIVE I
FIVE F
FIVE V
FIVE O
FIVE T
FIVE W
ONE A
ONE F
ONE S
TWO T
FOUR N
FIVE N
FIVE F
SIX E
SIX F
SIX I
SIX S
SIX X
SIX N
SIX O


3.
Smallest Known Anagrams Differing in Value n
nValuesAnagramsAuthor
013 – 13 TWELVE
ONE
ELEVEN
TWO
 
1110 – 109 FIFTY
TWENTY-ONE
THIRTEEN
THIRTEEN
THIRTEEN
THIRTY-THREE
THIRTY-TWO
NINETEEN
FIFTEEN
TEN
Joe DeVincentis
2115 – 113 FIFTY-SIX
EIGHTEEN
EIGHTEEN
THIRTEEN
TEN
SIXTY-THREE
NINETEEN
FIFTEEN
EIGHT
EIGHT
Joe DeVincentis
3165 – 162 EIGHTY-SEVEN
TWENTY-ONE
EIGHTEEN
THIRTEEN
THIRTEEN
THIRTEEN
SEVENTY-THREE
THIRTY-THREE
NINETEEN
NINETEEN
EIGHT
EIGHT
TWO
Joe DeVincentis
4158 – 154 THIRTY-FOUR
EIGHTEEN
EIGHTEEN
EIGHTEEN
EIGHTEEN
FIFTEEN
THIRTEEN
TEN
TEN
FOUR
FIFTY-THREE
NINETEEN
NINETEEN
FOURTEEN
FOURTEEN
EIGHT
EIGHT
EIGHT
EIGHT
THREE
Joe DeVincentis
5150 – 145 SIXTY-THREE
NINETEEN
NINETEEN
FOURTEEN
EIGHT
EIGHT
EIGHT
EIGHT
THREE
THIRTY-FOUR
EIGHTEEN
EIGHTEEN
EIGHTEEN
EIGHTEEN
THIRTEEN
TEN
TEN
SIX
Joe DeVincentis
6106 – 100 THIRTY-TWO
NINETEEN
NINETEEN
FOURTEEN
EIGHT
EIGHT
THREE
THREE
TWENTY-ONE
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
THIRTEEN
FOUR
Joe DeVincentis
7102 – 95 FIFTY-THREE
NINETEEN
FOURTEEN
EIGHT
EIGHT
THIRTY-FOUR
EIGHTEEN
EIGHTEEN
FIFTEEN
TEN
Joe DeVincentis
8130 – 122 FIFTY-THREE
THIRTY-TWO
NINETEEN
SIXTEEN
TEN
SIXTY
TWENTY-ONE
FIFTEEN
THIRTEEN
THIRTEEN
Joe DeVincentis
975 – 66 SIXTY
FIFTEEN
FIFTY
SIXTEEN
 
10120 – 110 SIXTY
TWENTY-ONE
THIRTEEN
THIRTEEN
THIRTEEN
THIRTY-THREE
THIRTY-TWO
NINETEEN
SIXTEEN
TEN
THREE
TWO
Joe DeVincentis
1163 – 52 EIGHTEEN
EIGHTEEN
THIRTEEN
TEN
FOUR
NINETEEN
FOURTEEN
EIGHT
EIGHT
THREE
Joe DeVincentis
12163 – 152 FIFTY-FOUR
TWENTY-ONE
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
THIRTEEN
THIRTEEN
THIRTY-THREE
THIRTY-TWO
NINETEEN
NINETEEN
FIFTEEN
FOURTEEN
EIGHT
EIGHT
THREE
Joe DeVincentis
13178 – 165 FIFTY-TWO
NINETEEN
NINETEEN
NINETEEN
FOURTEEN
FOURTEEN
EIGHT
EIGHT
EIGHT
EIGHT
THREE
THREE
THREE
TWENTY-ONE
EIGHTEEN
EIGHTEEN
EIGHTEEN
EIGHTEEN
FIFTEEN
THIRTEEN
THIRTEEN
THIRTEEN
TEN
FOUR
FOUR
Joe DeVincentis
14160 – 146 SEVENTY-THREE
NINETEEN
NINETEEN
FOURTEEN
EIGHT
EIGHT
EIGHT
EIGHT
THREE
THIRTY-FOUR
EIGHTEEN
EIGHTEEN
EIGHTEEN
EIGHTEEN
THIRTEEN
TEN
TEN
SEVEN
Joe DeVincentis
15167 – 152 SIXTY-THREE
THIRTY-THREE
NINETEEN
NINETEEN
FIFTEEN
EIGHT
EIGHT
TWO
FIFTY-SIX
TWENTY-ONE
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
THIRTEEN
Joe DeVincentis
1698 – 82 SIXTY-THREE
NINETEEN
EIGHT
EIGHT
THIRTY-SIX
EIGHTEEN
EIGHTEEN
TEN
Joe DeVincentis
1764 – 47 THIRTY-TWO
NINETEEN
TEN
THREE
TWENTY-ONE
THIRTEEN
THIRTEEN
Joe DeVincentis
1863 – 45 FIFTY
THIRTEEN
THIRTY
FIFTEEN
 
19130 – 111 SEVENTY
TWENTY-ONE
THIRTEEN
THIRTEEN
THIRTEEN
THIRTY-THREE
THIRTY-TWO
NINETEEN
SEVENTEEN
Joe DeVincentis
20128 – 108 EIGHTY-SEVEN
EIGHTEEN
THIRTEEN
TEN
SEVENTY-THREE
NINETEEN
EIGHT
EIGHT
Joe DeVincentis
21173 – 152 SIXTY-FOUR
TWENTY-ONE
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
THIRTEEN
THIRTEEN
THIRTY-THREE
THIRTY-THREE
NINETEEN
NINETEEN
SIXTEEN
FOURTEEN
EIGHT
EIGHT
TWO
Joe DeVincentis
22126 – 104 EIGHTEEN
EIGHTEEN
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
TEN
TEN
FOUR
FOUR
NINETEEN
NINETEEN
FOURTEEN
FOURTEEN
EIGHT
EIGHT
EIGHT
EIGHT
THREE
THREE
Joe DeVincentis
23170 – 147 THIRTY-TWO
THIRTY-TWO
NINETEEN
NINETEEN
NINETEEN
FOURTEEN
TEN
EIGHT
EIGHT
THREE
THREE
THREE
TWENTY-ONE
TWENTY-ONE
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
THIRTEEN
THIRTEEN
THIRTEEN
FOUR
Joe DeVincentis
24126 – 102 FIFTY-TWO
NINETEEN
NINETEEN
FOURTEEN
EIGHT
EIGHT
THREE
THREE
TWENTY-FOUR
EIGHTEEN
EIGHTEEN
FIFTEEN
THIRTEEN
THIRTEEN
ONE
Joe DeVincentis
25108 – 83 SEVENTY-THREE
NINETEEN
EIGHT
EIGHT
THIRTY-SEVEN
EIGHTEEN
EIGHTEEN
TEN
Joe DeVincentis
nValuesAnagramsAuthor
26129 – 103 SIXTY-THREE
THIRTY-TWO
FIFTEEN
TEN
NINE
FIFTY-SIX
TWENTY-ONE
THIRTEEN
THIRTEEN
Joe DeVincentis
2773 – 46 SIXTY
THIRTEEN
THIRTY
SIXTEEN
 
2892 – 64 THIRTY-TWO
EIGHTEEN
EIGHTEEN
TEN
TEN
FOUR
TWENTY-ONE
FOURTEEN
THIRTEEN
EIGHT
EIGHT
Joe DeVincentis
29126 – 97 FIFTY-FOUR
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
TEN
THIRTY-THREE
NINETEEN
FIFTEEN
FOURTEEN
EIGHT
EIGHT
Joe DeVincentis
30183 – 153 SEVENTY-FOUR
TWENTY-ONE
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
THIRTEEN
THIRTEEN
THIRTY-THREE
THIRTY-THREE
NINETEEN
NINETEEN
SEVENTEEN
FOURTEEN
EIGHT
EIGHT
TWO
Joe DeVincentis
31184 – 153 SEVENTY-TWO
NINETEEN
NINETEEN
NINETEEN
FOURTEEN
EIGHT
EIGHT
EIGHT
EIGHT
THREE
THREE
THREE
TWENTY-ONE
EIGHTEEN
EIGHTEEN
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
THIRTEEN
TEN
SEVEN
FOUR
Joe DeVincentis
32175 – 143 NINETY-THREE
NINETEEN
FOURTEEN
FOURTEEN
EIGHT
EIGHT
EIGHT
EIGHT
THREE
THIRTY-FOUR
EIGHTEEN
EIGHTEEN
EIGHTEEN
EIGHTEEN
THIRTEEN
TEN
TEN
FOUR
Joe DeVincentis
33122 – 89 SIXTY-TWO
NINETEEN
NINETEEN
EIGHT
EIGHT
THREE
THREE
TWENTY-ONE
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
SIX
Joe DeVincentis
34128 – 94 THIRTY-TWO
THIRTY-TWO
NINETEEN
NINETEEN
TEN
TEN
THREE
THREE
TWENTY-ONE
TWENTY-ONE
THIRTEEN
THIRTEEN
THIRTEEN
THIRTEEN
Joe DeVincentis
3584 – 49 FIFTY-TWO
NINETEEN
TEN
THREE
TWENTY-ONE
FIFTEEN
THIRTEEN
Joe DeVincentis
3683 – 47 SEVENTY
THIRTEEN
THIRTY
SEVENTEEN
 
37150 – 113 NINETY
TWENTY-ONE
THIRTEEN
THIRTEEN
THIRTEEN
THIRTY-THREE
THIRTY-TWO
NINETEEN
NINETEEN
Joe DeVincentis
38136 – 98 SIXTY-FOUR
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
TEN
THIRTY-THREE
NINETEEN
SIXTEEN
FOURTEEN
EIGHT
EIGHT
Joe DeVincentis
39142 – 103 THIRTY-TWO
EIGHTEEN
EIGHTEEN
EIGHTEEN
EIGHTEEN
TEN
TEN
TEN
FOUR
FOUR
TWENTY-ONE
NINETEEN
FOURTEEN
FOURTEEN
EIGHT
EIGHT
EIGHT
EIGHT
THREE
Joe DeVincentis
40189 – 149 FIFTY-FOUR
EIGHTEEN
EIGHTEEN
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
THIRTEEN
TEN
TEN
FOUR
THIRTY-THREE
NINETEEN
NINETEEN
FIFTEEN
FOURTEEN
FOURTEEN
EIGHT
EIGHT
EIGHT
EIGHT
THREE
Joe DeVincentis
41190 – 149 FIFTY-TWO
THIRTY-TWO
NINETEEN
NINETEEN
NINETEEN
FOURTEEN
TEN
EIGHT
EIGHT
THREE
THREE
THREE
TWENTY-ONE
TWENTY-ONE
EIGHTEEN
EIGHTEEN
FIFTEEN
THIRTEEN
THIRTEEN
THIRTEEN
THIRTEEN
FOUR
Joe DeVincentis
42132 – 90 SEVENTY-TWO
NINETEEN
NINETEEN
EIGHT
EIGHT
THREE
THREE
TWENTY-ONE
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
SEVEN
Joe DeVincentis
43123 – 80 NINETY-THREE
FOURTEEN
EIGHT
EIGHT
THIRTY-FOUR
EIGHTEEN
EIGHTEEN
TEN
Joe DeVincentis
4484 – 40 SIXTY-THREE
TEN
NINE
TWO
TWENTY-ONE
THIRTEEN
SIX
Joe DeVincentis
4593 – 48 EIGHTY
THIRTEEN
THIRTY
EIGHTEEN
 
46112 – 66 FIFTY-TWO
EIGHTEEN
EIGHTEEN
TEN
TEN
FOUR
TWENTY-ONE
FIFTEEN
FOURTEEN
EIGHT
EIGHT
Joe DeVincentis
47146 – 99 SEVENTY-FOUR
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
TEN
THIRTY-THREE
NINETEEN
SEVENTEEN
FOURTEEN
EIGHT
EIGHT
Joe DeVincentis
48203 – 155 NINETY-FOUR
TWENTY-ONE
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
THIRTEEN
THIRTEEN
THIRTY-THREE
THIRTY-THREE
NINETEEN
NINETEEN
NINETEEN
FOURTEEN
EIGHT
EIGHT
TWO
Joe DeVincentis
49199 – 150 NINETY-TWO
NINETEEN
NINETEEN
FOURTEEN
FOURTEEN
EIGHT
EIGHT
EIGHT
EIGHT
THREE
THREE
THREE
TWENTY-ONE
EIGHTEEN
EIGHTEEN
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
THIRTEEN
TEN
FOUR
FOUR
Joe DeVincentis
50186 – 136 SIXTY-TWO
THIRTY-TWO
NINETEEN
NINETEEN
NINETEEN
TEN
EIGHT
EIGHT
THREE
THREE
THREE
TWENTY-ONE
TWENTY-ONE
EIGHTEEN
EIGHTEEN
THIRTEEN
THIRTEEN
THIRTEEN
THIRTEEN
SIX
Joe DeVincentis

Joe DeVincentis also sent these solutions for 51-100.


4.
Smallest Known Crosswords for Numbers 1 to n
ONE
1 × 3 = 3
TWO
  N
  E
3 × 3 = 9
THREE
W    
ONE  
3 × 5 = 15
 T T
 W H
FOUR
   E
 ONE
5 × 4 = 20
   TWO
   H N
FOUR E
   E  
FIVE  
5 × 6 = 30
    F 
   SIX
  F V 
  ONE 
  U   
THREE 
W     
O     
8 × 6 = 48
   TWO 
   H   
FOUR   
I  E  O
V SEVEN
E I   E
  X    
7 × 7 = 49
 S     
EIGHT T
 X  W H
   FOUR
 O I  E
 N V  E
SEVEN  
7 × 7 = 49
(Jean-Charles Meyrignac)
 E SIX  TWO
FIVE    H  
 G V FOUR  
 H E  N E  
 T NINE E  
5 × 11 = 55
 S       
EIGHT TEN
 X  W H I
   FOUR N
 O I  E E
 N V  E  
SEVEN    
7 × 9 = 63
   SIX N    T
FIVE  EIGHT H
O  V T N  W R
U  ELEVEN ONE
R  N N      E
5 × 13 = 65
 E TWO  T SIX 
FIVE    H E  N
 G N FOUR V  I
 H    N ELEVEN
 TWELVE E N  E
5 × 14 = 70
(Mark Mammel)
SIX N TWELVE TWO
E  FIVE I    H N
V   N N G FOUR E
ELEVEN  H    E  
N       THIRTEEN
5 × 16 = 80
(Jean-Charles Meyrignac)
    FOURTEEN TWELVE
 F         I H    I
SIX THIRTEEN R  O G
 V  E   W  ELEVEN H
SEVEN  FOUR  E  E T
5 × 19 = 95
(Jean-Charles Meyrignac)
  FOURTEEN
 F N  W L
NINE SEVEN
 V    L V 
 E S  V E 
E FIFTEEN 
I  X H   F
G    R TWO
H    E E U
THIRTEEN R
10 × 10 = 100
(Jean-Charles Meyrignac)
 THIRTEEN  E
 W   E I F L
FOUR N G ONE
I   F  H U V
F S I  THREE
TWELVE   T N
E V E S  E  
E E  SIXTEEN
NINE  X  N  
9 × 12 = 108
(Jean-Charles Meyrignac)
FIVE  SIX
I   ONE  
F N   V  
THIRTEEN 
E N W N  
ELEVEN  F
N   L TWO
    V E U
SIXTEEN R
   H I  T
FOUR G  E
   E H  E
SEVENTEEN
13 × 9 = 117
(Jean-Charles Meyrignac)
  SIXTEEN   F
E I  W L  TWO
I X SEVEN   U
G  T L V FOUR
H  H V E I  T
THIRTEEN FIVE
   E     T  E
O SEVENTEEN N
N      E E   
EIGHTEEN NINE
10 × 13 = 130
(Jean-Charles Meyrignac)
 EIGHTEEN 
F    W L O
O NINETEEN
U    L V E
R F  V E  
THIRTEEN
E F H I F 
E T R G I 
N E E H V
 SEVENTEEN
TEN      I
 V SIXTEEN
 E I  W  E
 N X FOUR 
14 × 10 = 140
(Jean-Charles Meyrignac)
FIVE TWENTY TWELVE
I    H   E  H    I
F FOURTEEN EIGHT G
TWO  E  L S R    H
E U  E SEVENTEEN T
E R S O V V E  I E
N   I N E E E  N E
  SIXTEEN NINETEEN
8 × 18 = 144
(Jean-Charles Meyrignac)
TWO   FOUR  S
W   F  N    E
E THIRTEEN  V
N W F W I   E
TWENTYEIGHT N
Y N E N H H T
T TWENTYTHREE
W Y N Y   E E
O N  FOURTEEN
 NINE N  W   
  N   E SEVEN
TWENTY E N  I
W   W SIXTEEN
E  TEN G Y  E
N F N  H S  T
THIRTY TWELVE
Y V Y  E V  E
S E F  ELEVEN
I  SIX N N   
X   V        
  TWENTYFOUR 
21 × 13 = 273
(Jean-Charles Meyrignac)
T TWENTYNINE  S    TWENTYEIGHT 
W     W       E    H   W I   E 
EIGHTEEN  S   V  THIRTYEIGHT N 
N   H N  FIFTEEN W R H N H W  T
T THIRTYSIX H N  E THIRTYTWO  H
Y W R Y  V SIXTEEN E R Y    F I
TWENTYFIVE  R E  TWENTYSIX FOUR
H N Y O     TWENTY N Y E  T U T
R T T U FORTY N  T   S V THIRTY
E Y H R     F    W  SEVEN R T O
E O R     THIRTYFOUR V NINETEEN
  N E       V        E  N E E E
ELEVEN TWELVE   THRTYNINE  ONE 
13 × 31 = 403
(Jean-Charles Meyrignac)
FOURTEEN  T FIFTEEN    TWENTYFOUR FIFTY 
O   H     W    H L     H     O    O    F
R THIRTYONE THIRTEEN THIRTYFOUR FORTYTWO
T W R W   N    E V   W R   I R  O T  W R
Y E TWENTYTWO SEVEN  E TWELVE FORTYONE T
F N Y N H Y   E  N TEN Y I E  O T F  N Y
O T T THIRTYFIVE  S  T S G  FORTYEIGHT S
U Y H Y R H   E THIRTYEIGHT O T S V  Y I
R N R E T R   N W X  F X THIRTYSEVEN SIX
  I E I Y E   TWENTYSIX   I T N V  I E  
  N EIGHTEEN  E N E  V THIRTYNINE  N V O
  E   H W     E T E  E W  T   N NINETEEN
      T O  TWENTYONE  FORTYTHREE     N E
13 × 40 = 520
(Jean-Charles Meyrignac)

Bryce Herdt thought these grids should contain ZERO as well, and here are those best results.


5.

The best approximations found so far are shown below.

NumberRepresentationBaseRelative ErrorAuthor
TEN (1,0,2) 3 .1  
ONE BILLION (7,3,4,6,5,2,2,5,7,3) 8 9.35-6 Bryce Herdt
TEN BILLION (4,2,7,1,8,0,0,8,10,7) 11 2.02×10-6 Bryce Herdt
FIVE TRILLION (6,8,9,0,4,10,8,1,1,8,2,3) 12 4.72×10-7 Berend Van Der Zwaag
ONE QUADRILLION (9,4,1,10,6,8,5,0,2,3,3,2,9,4) 12 1.05×10-8 Berend van der Zwaag
TWO QUADRILLION (1,0,6,3,4,8,12,10,5,14,14,5,6,2) 15 1.14×10-11 Bryce Herdt
FOUR QUINTILLION (1,4,18,16,10,18,3,6,14,3,0,0,3,4,6) 21 7.58×10-12 Berend van der Zwaag
SIXTY QUINTILLION (8,16,1,7,14,11,6,16,4,7,16,17,17,16,15,4) 18 4.28×10-12 Berend van der Zwaag
ONE SEXTILLION (6,32,42,17,42,26,25,47,12,12,47,6,32) 48 1.17×10-15 Berend van der Zwaag
FOUR SEXTILLION (8,11,2,14,38,5,10,22,16,17,17,16,11,38) 39 8.62×10-17 Berend van der Zwaag
FOUR SEPTILLION (10,53,15,12,41,20,62,43,56,13,13,56,53,0) 65 3.58×10-20 Berend van der Zwaag
TWO DECILLION (248,524,264,191,614,72,122,643,643,122,264,0) 645 1.90×10-25 Berend van der Zwaag
FOUR DECILLION (45,340,101,200,442,332,241,43,282,282,43,340,458) 459 3.33×10-26 Berend van der Zwaag
FORTY DECILLION (49,101,171,130,126,292,297,196,259,224,224,259,101,0) 340 3.11×10-27 Berend van der Zwaag
TWO UNDECILLION (5,257,240,18,299,304,61,492,426,204,204,426,240,299) 544 1.85×10-29 Berend van der Zwaag
FORTY UNDECILLION (40,113,196,152,190,126,164,11,
155,56,109,130,130,109,113,164)
251 1.44×10-31 Berend van der Zwaag
FOUR SEDECILLION (306,1577,2954,1428,3183,2799,2289,2799,
2456,2849,2085,2085,2849,1577,3222)
3223 4.07×10-39 Berend van der Zwaag
FORTY DUOVIGINTILLION (134,3574,252,3173,3554,168,102,3574,2289,1282,3701,
1282,4015,3173,1282,4014,4014,1282,3574,4015)
4016 1.36×10-39 Berend van der Zwaag
FOUR TRESVIGINTILLION (169,407,3920,4820,1464,4820,3481,1254,3036,630,
2154,630,5056,1464,630,5055,5055,630,407,5056)
5057 6.07×10-42 Berend van der Zwaag
FORTY SESVIGINTILLION (3703,1904,8460,4554,4202,13828,55,13828,1650,
1835,2839,1835,0,4554,1835,1,1,1835,1904,0)
14443 2.34×10-46 Berend van der Zwaag
FORTY-SIX NOVEMVIGINTILLION (2,4213,5349,2178,2888,4361,2724,474,
605,4213,907,2522,5657,907,2724,0,
2724,605,2178,2724,1,1,2724,4213,605)
6332 8.71×10-48 Berend van der Zwaag
SIX HUNDRED TWO TRIGINTILLION (16,3511,2905,4684,1417,4767,1086,802,1486,
1086,1001,4651,3057,1001,802,3511,6058,3511,
4767,1001,3511,6057,6057,3511,3057,4767)
6059 3.33×10-49 Berend van der Zwaag
TWO UNTRIGINTILLION (149689,125387,213175,86446,111502,149689,
19767,15092,168550,15092,111502,149689,
15092,229267,229267,15092,213175,111502)
229268 7.06×10-50 Berend van der Zwaag
ONE DUOTRIGINTILLION (129535,68183,54675,148627,156400,129535,
45169,35662,23605,59537,23605,68183,
45169,23605,0,0,23605,129535,68183)
164429 1.14×10-53 Berend van der Zwaag
SEVEN DUOTRIGINTILLION (1,90693,278,90693,34006,12903,67227,
9831,67925,62289,87949,35929,87949,
34006,67925,87949,0,0,87949,9831,34006)
94995 1.10×10-56 Berend van der Zwaag


If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 5/19/16.