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:

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
24two dozen
27three cubed
48four dozen
72six dozen
100five score
104twice fifty two
108nine dozen
112twice fifty six
114twice fifty seven
116twice fifty eight
117thrice thirty nine
118twice fifty nine
119ten dozen minus one
120ten dozen
121eleven squared
122twice sixty one
123thrice forty one
124twice sixty two
125five cubed
126thrice forty two
127five cubed plus two
128twice sixty four
129thrice forty three
130twice sixty five
131five cubed plus six
132eleven dozen
133a gross minus eleven
134a gross minus ten
135a gross minus nine
136twice sixty eight
137a gross minus seven

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

169thirteen squared
170twice eighty five
171thrice fifty seven
172twice eighty six
173the fortieth prime
174twice eighty seven
175nine score minus five
176twice eighty eight
177thrice fifty nine
178twice eighty nine
179nine score minus one
180nine score
181nine score plus one
182twice ninety one
183thrice sixty one
184twice ninety two
185nine score plus five
186thrice sixty two
187nine score plus seven
188twice ninety four
189thrice sixty three
190twice ninety five
191ten score minus nine
192sixteen dozen
193ten score minus seven
194fifty plus a gross
195thrice sixty five
196fourteen squared
197ten score minus three
198thrice sixty six
199ten score minus one
200ten 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
-3one minus four
-4two minus six
-5 one minus six
-7two minus nine
-8two minus ten
-9one minus ten
-13two minus fifteen

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

-24six minus thirty
-25five minus thirty
-26four minus thirty
-27three minus thirty
-28two minus thirty
-29one minus thirty
-30ten 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 e-mail me. Click here to go back to Math Magic. Last updated 1/10/04.