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

n	1	2	3	4	5
A	ONE THOUSAND	ONE QUADRILLION ONE THOUSAND	ONE QUATTUORDECILLION ONE QUADRILLION ONE THOUSAND	ONE QUATTUORVIGINTILLION ONE QUATTUORDECILLION ONE QUADRILLION ONE THOUSAND (Alex Rower)	ONE QUATTUORTRIGINTILLION ONE QUATTUORVIGINTILLION ONE QUATTUORDECILLION ONE QUADRILLION ONE THOUSAND (Alex Rower)
B	ONE BILLION	---	---	---	---
C	ONE OCTILLION	ONE DECILLION ONE OCTILLION	ONE UNDECILLION ONE DECILLION ONE OCTILLION	ONE DUODECILLION ONE UNDECILLION ONE DECILLION ONE OCTILLION	ONE TREDECILLION ONE DUODECILLION ONE UNDECILLION ONE DECILLION ONE OCTILLION (Alex Rower)
D	ONE THOUSAND	ONE HUNDRED	ONE THOUSAND ONE HUNDRED	ONE HUNDRED MILLION ONE HUNDRED	ONE HUNDRED THOUSAND ONE HUNDRED
E	ONE	THREE	ELEVEN	SEVENTEEN	ONE HUNDRED ELEVEN (Alex Rower)
F	FOUR	FIFTEEN	FIFTY-FOUR	FOUR HUNDRED FIFTY-FOUR	FOUR THOUSAND FOUR HUNDRED FIFTY-FOUR (Alex Rower)
G	EIGHT	EIGHTY-EIGHT	EIGHT HUNDRED EIGHTY-EIGHT	EIGHT THOUSAND EIGHT HUNDRED EIGHTY-EIGHT	EIGHTY-EIGHT THOUSAND EIGHT HUNDRED EIGHTY-EIGHT (Alex Rower)
H	THREE	THIRTY-THREE	ONE HUNDRED THIRTY-THREE	THREE HUNDRED THIRTY-THREE	ONE THOUSAND THREE HUNDRED THIRTY-THREE (Alex Rower)
I	FIVE	THIRTY-FIVE	FIVE HUNDRED THIRTY-FIVE	FIVE THOUSAND FIVE HUNDRED THIRTY-FIVE	THIRTY-FIVE THOUSAND FIVE HUNDRED THIRTY-FIVE (Alex Rower)
J	---	---	---	---	---
K	---	---	---	---	---
L	ELEVEN	ELEVEN THOUSAND ELEVEN	ONE MILLION ELEVEN	ELEVEN MILLION ELEVEN	ELEVEN MILLION ELEVEN THOUSAND ELEVEN (Alex Rower)
M	ONE MILLION	ONE NOVEMDECILLION ONE MILLION	ONE NOVEMVIGINTILLION ONE NOVEMDECILLION ONE MILLION (Alex Rower)	ONE NOVEMTRIGINTILLION ONE NOVEMVIGINTILLION ONE NOVEMDECILLION ONE MILLION (Alex Rower)	ONE NOVEMQUADRAGINTILLION ONE NOVEMTRIGINTILLION ONE NOVEMVIGINTILLION ONE NOVEMDECILLION ONE MILLION (Alex Rower)
N	ONE	NINE	NINETEEN	NINETY-NINE	ONE HUNDRED NINETEEN (Alex Rower)
O	ONE	FORTY-ONE	ONE HUNDRED FORTY-ONE	ONE THOUSAND FORTY-ONE	ONE THOUSAND ONE HUNDRED FORTY-ONE (Alex Rower)
P	ONE SEPTILLION	ONE SEPTENDECILLION ONE SEPTILLION	ONE SEPTENVIGINTILLION ONE SEPTENDECILLION ONE SEPTILLION (Alex Rower)	ONE SEPTENTRIGINTILLION ONE SEPTENVIGINTILLION ONE SEPTENDECILLION ONE SEPTILLION (Alex Rower)	ONE SEPTENQUADRAGINTILLION ONE SEPTENTRIGINTILLION ONE SEPTENVIGINTILLION ONE SEPTENDECILLION ONE SEPTILLION (Alex Rower)
Q	ONE QUADRILLION	ONE QUINTILLION ONE QUADRILLION	ONE QUATTUORDECILLION ONE QUINTILLION ONE QUADRILLION	ONE QUINDECILLION ONE QUATTUORDECILLION ONE QUINTILLION ONE QUADRILLION	ONE QUATTUORVIGINTLLION ONE QUINDECILLION ONE QUATTUORDECILLION ONE QUINTILLION ONE QUADRILION (Alex Rower)
R	THREE	THIRTY-THREE	ONE HUNDRED THIRTY-THREE	THREE HUNDRED THIRTY-THREE	THREE THOUSAND THREE HUNDRED THIRTY-THREE (Alex Rower)
S	SIX	SIXTY-SIX	SIX HUNDRED SIXTY-SIX	ONE THOUSAND SIX HUNDRED SIXTY-SIX	SIX THOUSAND SIX HUNDRED SIXTY-SIX (Alex Rower)
T	TWO	THIRTEEN	TWENTY-TWO	TWO HUNDRED TWENTY-TWO	ONE THOUSAND TWO HUNDRED TWENTY-TWO (Alex Rower)
U	FOUR	ONE HUNDRED FOUR	FOUR HUNDRED FOUR	ONE THOUSAND FOUR HUNDRED FOUR	FOUR THOUSAND FOUR HUNDRED FOUR (Alex Rower)
V	FIVE	SEVENTY-FIVE	FIVE HUNDRED SEVENTY-FIVE	FIVE THOUSAND FIVE HUNDRED SEVENTY-FIVE	SEVENTY-FIVE THOUSAND FIVE HUNDRED SEVENTY-FIVE (Alex Rower)
W	TWO	TWENTY-TWO	TWO HUNDRED TWENTY-TWO	TWO THOUSAND TWO HUNDRED TWENTY-TWO	TWENTY-TWO THOUSAND TWO HUNDRED TWENTY-TWO (Alex Rower)
X	SIX	SIXTY-SIX	SIX HUNDRED SIXTY-SIX	SIX THOUSAND SIX HUNDRED SIXTY-SIX	SIXTY-SIX THOUSAND SIX HUNDRED SIXTY-SIX (Alex Rower)
Y	TWENTY	TWENTY THOUSAND TWENTY	TWENTY MILLION TWENTY THOUSAND TWENTY	TWENTY BILLION TWENTY MILLION TWENTY THOUSAND TWENTY	TWENTY TRILLION TWENTY BILLION TWENTY MILLION TWENTY THOUSAND TWENTY (Alex Rower)
Z	ZERO	---	---	---	---

Alex Rower sent solutions for more than 5 letters here

2.
Shortest Known List Ending with n of Same Number

n	ONE	TWO	THREE	FOUR	FIVE	SIX	SEVEN	EIGHT	NINE	TEN	ELEVEN
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

n	Values	Anagrams		Author
0	13 – 13	TWELVE ONE	ELEVEN TWO
1	110 – 109	FIFTY TWENTY-ONE THIRTEEN THIRTEEN THIRTEEN	THIRTY-THREE THIRTY-TWO NINETEEN FIFTEEN TEN	Joe DeVincentis
2	115 – 113	FIFTY-SIX EIGHTEEN EIGHTEEN THIRTEEN TEN	SIXTY-THREE NINETEEN FIFTEEN EIGHT EIGHT	Joe DeVincentis
3	165 – 162	EIGHTY-SEVEN TWENTY-ONE EIGHTEEN THIRTEEN THIRTEEN THIRTEEN	SEVENTY-THREE THIRTY-THREE NINETEEN NINETEEN EIGHT EIGHT TWO	Joe DeVincentis
4	158 – 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
5	150 – 145	SIXTY-THREE NINETEEN NINETEEN FOURTEEN EIGHT EIGHT EIGHT EIGHT THREE	THIRTY-FOUR EIGHTEEN EIGHTEEN EIGHTEEN EIGHTEEN THIRTEEN TEN TEN SIX	Joe DeVincentis
6	106 – 100	THIRTY-TWO NINETEEN NINETEEN FOURTEEN EIGHT EIGHT THREE THREE	TWENTY-ONE EIGHTEEN EIGHTEEN THIRTEEN THIRTEEN THIRTEEN FOUR	Joe DeVincentis
7	102 – 95	FIFTY-THREE NINETEEN FOURTEEN EIGHT EIGHT	THIRTY-FOUR EIGHTEEN EIGHTEEN FIFTEEN TEN	Joe DeVincentis
8	130 – 122	FIFTY-THREE THIRTY-TWO NINETEEN SIXTEEN TEN	SIXTY TWENTY-ONE FIFTEEN THIRTEEN THIRTEEN	Joe DeVincentis
9	75 – 66	SIXTY FIFTEEN	FIFTY SIXTEEN
10	120 – 110	SIXTY TWENTY-ONE THIRTEEN THIRTEEN THIRTEEN	THIRTY-THREE THIRTY-TWO NINETEEN SIXTEEN TEN THREE TWO	Joe DeVincentis
11	63 – 52	EIGHTEEN EIGHTEEN THIRTEEN TEN FOUR	NINETEEN FOURTEEN EIGHT EIGHT THREE	Joe DeVincentis
12	163 – 152	FIFTY-FOUR TWENTY-ONE EIGHTEEN EIGHTEEN THIRTEEN THIRTEEN THIRTEEN THIRTEEN	THIRTY-THREE THIRTY-TWO NINETEEN NINETEEN FIFTEEN FOURTEEN EIGHT EIGHT THREE	Joe DeVincentis
13	178 – 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
14	160 – 146	SEVENTY-THREE NINETEEN NINETEEN FOURTEEN EIGHT EIGHT EIGHT EIGHT THREE	THIRTY-FOUR EIGHTEEN EIGHTEEN EIGHTEEN EIGHTEEN THIRTEEN TEN TEN SEVEN	Joe DeVincentis
15	167 – 152	SIXTY-THREE THIRTY-THREE NINETEEN NINETEEN FIFTEEN EIGHT EIGHT TWO	FIFTY-SIX TWENTY-ONE EIGHTEEN EIGHTEEN THIRTEEN THIRTEEN THIRTEEN	Joe DeVincentis
16	98 – 82	SIXTY-THREE NINETEEN EIGHT EIGHT	THIRTY-SIX EIGHTEEN EIGHTEEN TEN	Joe DeVincentis
17	64 – 47	THIRTY-TWO NINETEEN TEN THREE	TWENTY-ONE THIRTEEN THIRTEEN	Joe DeVincentis
18	63 – 45	FIFTY THIRTEEN	THIRTY FIFTEEN
19	130 – 111	SEVENTY TWENTY-ONE THIRTEEN THIRTEEN THIRTEEN	THIRTY-THREE THIRTY-TWO NINETEEN SEVENTEEN	Joe DeVincentis
20	128 – 108	EIGHTY-SEVEN EIGHTEEN THIRTEEN TEN	SEVENTY-THREE NINETEEN EIGHT EIGHT	Joe DeVincentis
21	173 – 152	SIXTY-FOUR TWENTY-ONE EIGHTEEN EIGHTEEN THIRTEEN THIRTEEN THIRTEEN THIRTEEN	THIRTY-THREE THIRTY-THREE NINETEEN NINETEEN SIXTEEN FOURTEEN EIGHT EIGHT TWO	Joe DeVincentis
22	126 – 104	EIGHTEEN EIGHTEEN EIGHTEEN EIGHTEEN THIRTEEN THIRTEEN TEN TEN FOUR FOUR	NINETEEN NINETEEN FOURTEEN FOURTEEN EIGHT EIGHT EIGHT EIGHT THREE THREE	Joe DeVincentis
23	170 – 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
24	126 – 102	FIFTY-TWO NINETEEN NINETEEN FOURTEEN EIGHT EIGHT THREE THREE	TWENTY-FOUR EIGHTEEN EIGHTEEN FIFTEEN THIRTEEN THIRTEEN ONE	Joe DeVincentis
25	108 – 83	SEVENTY-THREE NINETEEN EIGHT EIGHT	THIRTY-SEVEN EIGHTEEN EIGHTEEN TEN	Joe DeVincentis

n	Values	Anagrams		Author
26	129 – 103	SIXTY-THREE THIRTY-TWO FIFTEEN TEN NINE	FIFTY-SIX TWENTY-ONE THIRTEEN THIRTEEN	Joe DeVincentis
27	73 – 46	SIXTY THIRTEEN	THIRTY SIXTEEN
28	92 – 64	THIRTY-TWO EIGHTEEN EIGHTEEN TEN TEN FOUR	TWENTY-ONE FOURTEEN THIRTEEN EIGHT EIGHT	Joe DeVincentis
29	126 – 97	FIFTY-FOUR EIGHTEEN EIGHTEEN THIRTEEN THIRTEEN TEN	THIRTY-THREE NINETEEN FIFTEEN FOURTEEN EIGHT EIGHT	Joe DeVincentis
30	183 – 153	SEVENTY-FOUR TWENTY-ONE EIGHTEEN EIGHTEEN THIRTEEN THIRTEEN THIRTEEN THIRTEEN	THIRTY-THREE THIRTY-THREE NINETEEN NINETEEN SEVENTEEN FOURTEEN EIGHT EIGHT TWO	Joe DeVincentis
31	184 – 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
32	175 – 143	NINETY-THREE NINETEEN FOURTEEN FOURTEEN EIGHT EIGHT EIGHT EIGHT THREE	THIRTY-FOUR EIGHTEEN EIGHTEEN EIGHTEEN EIGHTEEN THIRTEEN TEN TEN FOUR	Joe DeVincentis
33	122 – 89	SIXTY-TWO NINETEEN NINETEEN EIGHT EIGHT THREE THREE	TWENTY-ONE EIGHTEEN EIGHTEEN THIRTEEN THIRTEEN SIX	Joe DeVincentis
34	128 – 94	THIRTY-TWO THIRTY-TWO NINETEEN NINETEEN TEN TEN THREE THREE	TWENTY-ONE TWENTY-ONE THIRTEEN THIRTEEN THIRTEEN THIRTEEN	Joe DeVincentis
35	84 – 49	FIFTY-TWO NINETEEN TEN THREE	TWENTY-ONE FIFTEEN THIRTEEN	Joe DeVincentis
36	83 – 47	SEVENTY THIRTEEN	THIRTY SEVENTEEN
37	150 – 113	NINETY TWENTY-ONE THIRTEEN THIRTEEN THIRTEEN	THIRTY-THREE THIRTY-TWO NINETEEN NINETEEN	Joe DeVincentis
38	136 – 98	SIXTY-FOUR EIGHTEEN EIGHTEEN THIRTEEN THIRTEEN TEN	THIRTY-THREE NINETEEN SIXTEEN FOURTEEN EIGHT EIGHT	Joe DeVincentis
39	142 – 103	THIRTY-TWO EIGHTEEN EIGHTEEN EIGHTEEN EIGHTEEN TEN TEN TEN FOUR FOUR	TWENTY-ONE NINETEEN FOURTEEN FOURTEEN EIGHT EIGHT EIGHT EIGHT THREE	Joe DeVincentis
40	189 – 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
41	190 – 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
42	132 – 90	SEVENTY-TWO NINETEEN NINETEEN EIGHT EIGHT THREE THREE	TWENTY-ONE EIGHTEEN EIGHTEEN THIRTEEN THIRTEEN SEVEN	Joe DeVincentis
43	123 – 80	NINETY-THREE FOURTEEN EIGHT EIGHT	THIRTY-FOUR EIGHTEEN EIGHTEEN TEN	Joe DeVincentis
44	84 – 40	SIXTY-THREE TEN NINE TWO	TWENTY-ONE THIRTEEN SIX	Joe DeVincentis
45	93 – 48	EIGHTY THIRTEEN	THIRTY EIGHTEEN
46	112 – 66	FIFTY-TWO EIGHTEEN EIGHTEEN TEN TEN FOUR	TWENTY-ONE FIFTEEN FOURTEEN EIGHT EIGHT	Joe DeVincentis
47	146 – 99	SEVENTY-FOUR EIGHTEEN EIGHTEEN THIRTEEN THIRTEEN TEN	THIRTY-THREE NINETEEN SEVENTEEN FOURTEEN EIGHT EIGHT	Joe DeVincentis
48	203 – 155	NINETY-FOUR TWENTY-ONE EIGHTEEN EIGHTEEN THIRTEEN THIRTEEN THIRTEEN THIRTEEN	THIRTY-THREE THIRTY-THREE NINETEEN NINETEEN NINETEEN FOURTEEN EIGHT EIGHT TWO	Joe DeVincentis
49	199 – 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
50	186 – 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.

The best approximations found so far are shown below.

Number Representation Base Relative Error Author
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

Number	Representation	Base	Relative Error	Author
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.