I believe the answer is,
4085 (four thousand and eighty-five)
962000 (nine hundred and sixty-two thousand)
My approach
First I made a table of all words and vowel appearences.
[word] aeiou
a 10000
one 01010
... full table can be found at https://paste.ubuntu.com/p/X8sfQnhVFK/
First question,
From the table it's pretty clear no number <100 could work. We show why numbers in 100~999 couldn't work.
Those numbers have format: [] hundred (and []-[]/[])
(here [] stands for a word)
hundred 01001
and 10000
[need] 11221
We need 7 vowels, however one word (except the -teen case, where 2 words can be used then) contains no more than 2 vowels and we only have 3 slots.
Now 4 digit numbers. Plain [] thousand
certainly won't work. Notice that thousand
and and
both contain a
, so we can at most have one and
.
Two cases left: [] thousand and [] hundred
and [] thousand and []-[]/[]
.
First case:
hundred 01001
thousand 10011
and 10000
[need] 01210
Looking through all digits, sadly there aren't two digits matching this need.
Second case looks more promising.
thousand 10011
and 10000
[need] 02211
As before, we need two or three words here with 6 vowels in total. Our only bet is to have three words, all with 2 vowels.
The only -ty's with 2 vowels:
seventy 02000
eighty 01100
ninety 01100
If we choose seventy
, we're left with 00211
and sadly no match again. If we choose eighty
instead, we're left with 01111
. four
and five
is such a match.
Second question,
6 digits, so a few hundred thousands. [] hundred thousand
won't work. As said, we can only have at most one and
pairing with thousand
. Three cases: [] hundred and [] thousand
, [] hundred thousand and [] hundred
, [] hundred and []-[]/[] thousand
(There's also [] hundred thousand and []-[]/[]
but is certainly smaller).
The second case has 3u
s so not valid and the first case is identical to the [] thousand and [] hundred
discussed in the first question. The only bet is the third case.
hundred 01001
thousand 10011
and 10000
[need] 01210
Two digits won't work. -teen's have two e
s. So we're going for the three word case.
To make the number as big as possible, we put the largest digit nine
in the front.
nine 01100
[need] 00110
Two vowels left. Again we choose the biggest possible number to put with one vowel, which is sixty
. We're left with the one last o
so the last digit is two
.
On the bonus question,
The answer is yes. For example, 3000000000084000 (three quadrillion and eighty-four thousand). I'm not sure if it's the smallest one (hopefully not).
one thousand and one hundred and one
(twoand
s)? Or just oneand
and one comma? $\endgroup$