Problem of the Month (December 2003)
This month we investigate honest numbers, numbers n that can be described using exactly n letters in standard mathematical English. For example, the smallest honest numbers are 4 = "four", 8 = "two cubed", and 11 = "two plus nine". It is known that all n≥13 are honest. Can you prove it?
Define H(n) to be the honesty number of n, the number of different ways that n can be described in exactly n letters. Can you determine H(n) for some small values of n?
A number is called highly honest if H(n)=n. Are there any highly honest numbers?
Define L(n) to be the letter number of n, the minimum number of letters needed to describe n. If L(n) is less than the number of letters in the name of n, we say n is wasteful. For example, 27 is wasteful since "three cubed" is shorter than "twenty seven". What other wasteful numbers can you find?
ANSWERS
Joseph DeVincentis, Bill Clagett, and Matt King proved that all n≥13 are honest.
Joseph DeVincentis gave a nice argument that if there are any highly honest numbers, they must be smaller than 43. It revolves around the phrase "plus five plus ten" that can be appended any number of times, and rearranged a large number of times for large enough numbers. His computer program suggested that there are no highly honest numbers at all.
Joseph DeVincentis, Bill Clagett and Clinton Weaver sent many interesting examples of honest numbers. The first two of these wrote programs to find honest numbers. My favorite example was from Bill Clagett, who sent:
461 = eighteenth root of eight hundred eightyfour quattuordecillion
three hundred thirtyfour tredecillion six hundred eighty duodecillion
eight hundred twentysix undecillion six hundred fiftythree decillion
six hundred thirtyseven nonillion one hundred three octillion ninety
septillion nine hundred eightytwo sextillion five hundred eightyone
quintillion four hundred fortyeight quadrillion seven hundred
ninetyfour trillion nine hundred thirteen billion four hundred
thirtytwo million nine hundred fiftynine thousand eightyone
Here are the known descriptions of n using n letters:


10
half a score
ten over one 

11
two plus nine
five plus six 

13
one plus twelve
two plus eleven
five plus eight
the sixth prime
one plus a dozen 

14
seven plus seven
twenty minus six
forty two thirds
a score minus six
four added to ten
E in base fifteen
E in base sixteen 

15
zero plus fifteen
one plus fourteen
two plus thirteen
three plus twelve
one times fifteen
twenty minus five
forty five thirds
sixteen minus one
a score minus five
three plus a dozen
a quarter of sixty
one half of thirty
five more than ten
six more than nine 

16
minus four squared
sixteen minus zero
eighteen minus two
forty eight thirds
sixty four fourths
seven added to nine
twice five plus six
twice six plus four
four plus one dozen
four plus twice six
ninety six over six
one fifth of eighty
thirty two over two
thrice two plus ten
two fifths of forty
two times two cubed
two four in base six 

17
zero plus seventeen
three plus fourteen
one times seventeen
sixty eight fourths
twice four plus nine
twice eight plus one
twice nine minus one
one added to sixteen
two added to fifteen
five added to twelve
eight more than nine
fifty one over three
six more than eleven
thirty four over two
thrice six minus one
two plus thrice five
one plus six plus ten
five added to a dozen
two one in base eight
one seven in base ten 

18
minus two plus twenty
seven added to eleven
twice five plus eight
twice seven plus four
twice nine minus zero
twenty two minus four
fifty four over three
forty fifths plus ten
nine tenths of twenty
nine thirds times six
seventy two over four
six plus sixty fifths
six thirds times nine
sixty minus forty two
ten fifths times nine
three tenths of sixty
thrice six minus zero
thrice sixty over ten
twenty four minus six
twice nine minus zero
twice ninety over ten
two more than sixteen
two plus four squared
two cubed added to ten
six added to one dozen
six added to twice six
ten plus ten minus two
two times six plus six
minus two plus a score
one half of thirty six
three zero in base six
two four in base seven
one six in base twelve 

19
twenty two minus three
twenty four minus five
zero added to nineteen
two added to seventeen
three added to sixteen
five added to fourteen
twice two plus fifteen
twice four plus eleven
twice eight plus three
eight more than eleven
eighty minus sixty one
fifty halves minus six
fifty minus thirty one
fifty nine minus forty
fifty seven over three
five squared minus six
forty halves minus one
forty minus twenty one
four more than fifteen
nine plus fifty fifths
nine plus sixty sixths
ninety tenths plus ten
one more than eighteen
one plus ninety fifths
seven more than twelve
seven plus thrice four
six more than thirteen
sixty nine minus fifty
sixty thirds minus one
three squared plus ten
thrice seven minus two
twenty minus one cubed
twenty six minus seven
a score minus one cubed
four plus five plus ten
half of fifty minus six
half of forty minus one
one added to thrice six
one added to twice nine
one less than one score
one less than twice ten
seven more than a dozen
zero plus nine plus ten
one plus eight plus ten
one plus nine plus nine
two plus seven plus ten
three plus six plus ten
four plus five plus ten
four plus six plus nine
six plus six plus seven
one times nine plus ten
one times ten plus nine
two times nine plus one
two times ten minus one
a fourth of seventy six
three four in base five
two three in base eight
six plus a baker's dozen 

Clinton Weaver and Joseph DeVincentis improved many of my shortest descriptions of numbers. Here is a list of the small known wasteful numbers:
Small Wasteful Numbers
24  two dozen
 27  three cubed
 48  four dozen
 72  six dozen
 100  five score
 104  twice fifty two
 108  nine dozen
 112  twice fifty six
 114  twice fifty seven
 116  twice fifty eight
 117  thrice thirty nine
 118  twice fifty nine
 119  ten dozen minus one
 120  ten dozen
 121  eleven squared
 122  twice sixty one
 123  thrice forty one
 124  twice sixty two
 125  five cubed
 126  thrice forty two
 127  five cubed plus two
 128  twice sixty four
 129  thrice forty three
 130  twice sixty five
 131  five cubed plus six
 132  eleven dozen
 133  a gross minus eleven
 134  a gross minus ten
 135  a gross minus nine
 136  twice sixty eight
 137  a gross minus seven


138  twice sixty nine
 139  a gross minus five
 140  seven score
 141  a gross minus three
 142  a gross minus two
 143  a gross minus one
 144  a gross
 145  a gross plus one
 146  a gross plus two
 147  a gross plus three
 148  a gross plus four
 149  a gross plus five
 150  thrice fifty
 151  a gross plus seven
 152  twice seventy six
 153  a gross plus nine
 154  a gross plus ten
 155  a gross plus eleven
 156  thirteen dozen
 157  a gross plus thirteen
 158  twice seventy nine
 159  thrice fifty three
 160  eight score
 161  eight score plus one
 162  twice eighty one
 163  nineteen plus a gross
 164  twice eighty two
 165  thrice fifty five
 166  twice eighty three
 167  eight score plus seven
 168  fourteen dozen


169  thirteen squared
 170  twice eighty five
 171  thrice fifty seven
 172  twice eighty six
 173  the fortieth prime
 174  twice eighty seven
 175  nine score minus five
 176  twice eighty eight
 177  thrice fifty nine
 178  twice eighty nine
 179  nine score minus one
 180  nine score
 181  nine score plus one
 182  twice ninety one
 183  thrice sixty one
 184  twice ninety two
 185  nine score plus five
 186  thrice sixty two
 187  nine score plus seven
 188  twice ninety four
 189  thrice sixty three
 190  twice ninety five
 191  ten score minus nine
 192  sixteen dozen
 193  ten score minus seven
 194  fifty plus a gross
 195  thrice sixty five
 196  fourteen squared
 197  ten score minus three
 198  thrice sixty six
 199  ten score minus one
 200  ten score


Joseph DeVincentis noted that negative integers can be wasteful too. Here is the beginning of his list. Are the rest of the negative numbers wasteful?
Negative Wasteful Numbers
3  one minus four
 4  two minus six
 5  one minus six
 7  two minus nine
 8  two minus ten
 9  one minus ten
 13  two minus fifteen


14  six minus twenty
 17  one minus eighteen
 18  two minus twenty
 19  one minus twenty
 21  nine minus thirty
 22  two minus two dozen
 23  one minus two dozen


24  six minus thirty
 25  five minus thirty
 26  four minus thirty
 27  three minus thirty
 28  two minus thirty
 29  one minus thirty
 30  ten minus forty


Jeremy Galvagni suggested looking for the "most acceptable" descriptions of n in n letters for dishonest numbers. My favorites among the suggestions of Joseph DeVincentis and his are below:
0 =
1 = I
2 = II
3 = III
5 = a five
6 = one six
7 = one 'n' six
9 = just a nine
12 = eleven and one
Joseph DeVincentis defined a sequence S(n) to be the least positive integer which requires at least n letters to describe. The sequence starts 1, 1, 1, 3, 3, 11, 13, 13, 17, 23, 23, 73, 101, 103, 103, 111, 113, 157, 167.... if the above data is the best possible. What is S(20)?
If you can extend any of these results, please
email me.
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