To solve the puzzle, observe some of the entries as the staircase starts:
ONE
FOUR
SEVEN
ELEVEN
FIFTEEN
EIGHTEEN
TWENTYONE
TWENTYFOUR
TWENTYSEVEN
SEVENTYTHREE - Smallest 2-digit integer with longest name.
ONEHUNDREDONE - What's right of ONEHUNDRED starts over at this step.
ONEHUNDREDFOUR
ONEHUNDREDSEVEN
ONEHUNDREDELEVEN
ONEHUNDREDFIFTEEN
ONEHUNDREDEIGHTEEN
ONEHUNDREDTWENTYONE
ONEHUNDREDTWENTYFOUR
ONEHUNDREDTWENTYSEVEN
ONEHUNDREDSEVENTYTHREE - Top of staircase repeats down to here.
THREEHUNDREDTWENTYTHREE - Hundreds went from ONE to THREE, not to FOUR.
THREEHUNDREDSEVENTYTHREE - Smallest 3-digit integer with longest name.
ONETHOUSANDONEHUNDREDFOUR - Right of 1000 starts at 104 - all others too short.
ONETHOUSANDONEHUNDREDSEVEN
ONETHOUSANDONEHUNDREDELEVEN
...
Note the right side repeats the same sequence over and over, but starts at a different point depending on what's to its left.
[ - - - - cut here down to closing square bracket below for short answer - - - -
The staircase can be thought of as an "odometer" but with special wheels. Its wheels each have a part of an integer's name just as an odometer has 0-9 for each place.
Here the "odometer" will have eight wheels:
Wheel 1 - (blank) or ONETRILLION
Wheel 2 - Sequence B below
Wheel 3 - (blank) or BILLION
Wheel 4 - Sequence B below
Wheel 5 - (blank) or MILLION
Wheel 6 - Sequence B below
Wheel 7 - (blank) or THOUSAND
Wheel 8 - Sequence A below.
Sequence A is the start of the staircase above, stopping before THOUSAND:
(blank)
ONE
FOUR
SEVEN
ELEVEN
FIFTEEN
EIGHTEEN
TWENTYONE
TWENTYFOUR
TWENTYSEVEN
SEVENTYTHREE
ONEHUNDREDONE
ONEHUNDREDFOUR
ONEHUNDREDSEVEN
ONEHUNDREDELEVEN
ONEHUNDREDFIFTEEN
ONEHUNDREDEIGHTEEN
ONEHUNDREDTWENTYONE
ONEHUNDREDTWENTYFOUR
ONEHUNDREDTWENTYSEVEN
ONEHUNDREDSEVENTYTHREE
THREEHUNDREDTWENTYTHREE
THREEHUNDREDSEVENTYTHREE
Sequence A assures each step will be one letter longer than the previous step.
Sequence B is similar to Sequence A. It is a list of integer names of increasing length, not constrained to just one letter increases:
(blank)
ONE
THREE (Not FOUR because 3 is smaller than 4. SEVEN is skipped.)
ELEVEN
THIRTEEN (Replacing larger 15 and same-length 18.)
TWENTYONE
TWENTYTHREE (Replacing 24 and 27.)
SEVENTYTHREE
ONEHUNDREDONE
ONEHUNDREDTHREE (Replacing 104 and 107.)
ONEHUNDREDELEVEN
ONEHUNDREDTHIRTEEN (Replacing 115 and 118.)
ONEHUNDREDTWENTYONE
ONEHUNDREDTWENTYTHREE (Replacing 124 and 127.)
ONEHUNDREDSEVENTYTHREE
THREEHUNDREDTWENTYTHREE
THREEHUNDREDSEVENTYTHREE
Sequence B assures each step is the smallest integer with that name length. Note the step from ONEHUNDREDSEVENTYTHREE to THREEHUNDREDTWENTYTHREE in the staircase above.
The "odometer" algorithm for this puzzle works similar to an ordinary odometer, but it doesn't show any number names which are not exactly one letter longer than the previous step. For example, note ONETHOUSANDONEHUNDREDFOUR as first THOUSAND above.
No matter where any wheel starts, it increments until reaching its maximum.
Just before turning Wheel 1 from (blank) to ONETRILLION each of the other wheels reads its maximum value.
- - - - - - end cut out portion here - - - - ]
Hence we know the answer is:
THREEHUNDREDSEVENTYTHREE
BILLION
THREEHUNDREDSEVENTYTHREE
MILLION
THREEHUNDREDSEVENTYTHREE
THOUSAND
THREEHUNDREDSEVENTYTHREE
(373,373,373,373)
This has 118 letters; the first entry with TRILLION must have 119.
Since the staircase starts with a 3-letter step, TRILLION is the 117th step.
Answer to the second (missing integers) puzzle
______________________________________________
Replace the start of the staircase:
[Sequence A] [Sequence C] <- cut for short answer
(blank) (blank)
ONE TWO
FOUR FIVE
SEVEN EIGHT
ELEVEN TWELVE
FIFTEEN SIXTEEN
EIGHTEEN NINETEEN
TWENTYONE TWENTYTWO
TWENTYFOUR TWENTYFIVE
TWENTYSEVEN TWENTYEIGHT
SEVENTYTHREE SEVENTYSEVEN
ONEHUNDREDONE ONEHUNDREDTWO
ONEHUNDREDFOUR ONEHUNDREDFIVE
ONEHUNDREDSEVEN ONEHUNDREDEIGHT
ONEHUNDREDELEVEN ONEHUNDREDTWELVE
ONEHUNDREDFIFTEEN ONEHUNDREDSIXTEEN
ONEHUNDREDEIGHTEEN ONEHUNDREDNINETEEN
ONEHUNDREDTWENTYONE ONEHUNDREDTWENTYTWO
ONEHUNDREDTWENTYFOUR ONEHUNDREDTWENTYFIVE
ONEHUNDREDTWENTYSEVEN ONEHUNDREDTWENTYEIGHT
ONEHUNDREDSEVENTYTHREE ONEHUNDREDSEVENTYSEVEN
THREEHUNDREDTWENTYTHREE THREEHUNDREDTWENTYSEVEN
THREEHUNDREDSEVENTYTHREE THREEHUNDREDSEVENTYSEVEN
Using these integer names in the place of the corresponding original names
[putting this sequence on Wheel 8] <- cut for short answer
creates a staircase using all different integers from the first staircase. It ends with 373,373,373,377. It is the same size.
Answer to the third (missing vowels) puzzle
___________________________________________
This staircase starts:
(blank)
N
TW
THR
TWLV
THRTN
TWNTTW
TWNTTHR
NHNDRDTW
NHNDRDTHR
NHNDRDTWLV
NHNDRDTHRTN
NHNDRDTWNTTW
NHNDRDTWNTTHR
TWHNDRDTWNTTHR
THRHNDRDTWNTTHR
The answer to the puzzle is derived as above
[Just before turning Wheel 1 from (blank) to NTRLLN the odometer reads] <- cut
:
THRHNDRDTWNTTHR
BLLN
THRHNDRDTWNTTHR
MLLN
THRHNDRDTWNTTHR
THSND
THRHNDRDTWNTTHR
This has 73 letters; the first entry with TRLLN must have 74.