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?

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:

 4 `four`

 8 `two cubed`

 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
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. Click here to go back to Math Magic. Last updated 1/10/04.