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 eighty-four quattuordecillion
three hundred thirty-four tredecillion six hundred eighty duodecillion
eight hundred twenty-six undecillion six hundred fifty-three decillion
six hundred thirty-seven nonillion one hundred three octillion ninety
septillion nine hundred eighty-two sextillion five hundred eighty-one
quintillion four hundred forty-eight quadrillion seven hundred
ninety-four trillion nine hundred thirteen billion four hundred
thirty-two million nine hundred fifty-nine thousand eighty-one
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, Joseph DeVincentis, and Haym Hirsh 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 | eleven fifteens (HH)
| | 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 | eleven sixteens
| | 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 | eleven seventeens
| | 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
|
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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
|
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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
8 = one eight
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)?
In 2019, Alex Rower sent this list of almost honest numbers, this list of honest numbers in other languages, this list of phrases one can add to an honest number to keep it honest, this list of honest Braille numbers, this list of honest Scrabble numbers, and this list of honest Morse Code numbers.
If you can extend any of these results, please
e-mail me.
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