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.