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 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.
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
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.
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