# One-thousand-four-hundred-seventy-four numbers get valued

Rows A through P, below, categorize the spellings of one-thousand-four-hundred-seventy-four numbers that have a straightforward numerical property to be deduced. Spelled numbers of each row share a value for that property, by which the rows are listed in increasing order. The numbers of each row are also in order.

Rows Y and Z categorize all remaining integers based on the reason for their spellings’ lack of that numerical property.

               How many numbers
Row  Value     share the value    Numbers with the value

A.  __?__        forty-six       ... minus-twenty  minus-fourteen  minus-ten
minus-four  minus-three  minus-two
minus-one  eight  nine  thirteen
fifteen  eighteen  nineteen ...
B.  __?__       thirty-eight     ... minus-nineteen  minus-eighteen
minus-fifteen  minus-thirteen
minus-nine  minus-eight ...
C.  __?__          eight         (eight numbers under minus-twenty)
D.  __?__         sixteen        ... minus-seventeen  minus-seven  five ...
E.  __?__           nine         seven  seventeen ...
F.  __?__           two          (two numbers over twenty)
G.  __?__           nine         six  sixteen ...
H.  __?__           two          eleven  twelve
I.  __?__  seventeen x sixteen   (two-hundred-seventy-two  numbers over twenty)
J.  __?__     thirty x sixteen   (four-hundred-eighty      numbers over twenty)
K.  __?__      eight x sixteen   (one-hundred-twenty-eight numbers over twenty)
L.  __?__      seven x sixteen   (one-hundred-twelve       numbers over twenty)
M.  __?__       nine x sixteen   (one-hundred-forty-four   numbers over twenty)
N.  __?__        two x sixteen   (            thirty-two   numbers over twenty)
O.  __?__       nine x sixteen   (one-hundred-forty-four   numbers over twenty)
P.  __?__        two x sixteen   (            thirty-two   numbers over twenty)

Y.  Unvalued   Infinitely many   ... minus-sixteen  minus-twelve  minus-eleven
for a                           minus-six  minus-five ...
reason

Z.  Unvalued      seventeen      zero  one  two  three  four  ten  fourteen
for a                       twenty ...
different reason


Q. What are the values for rows A through P?

R. Why are the numbers of row Z unvalued?

S. What are the first and last numbers of each row other than row Y?

T. What is the smallest positive number in row Y?

U. How long is the longest sequence of consecutive numbers in any row other than row Y ?

V. Which rows contain such a sequence and how many in each such row?

(The number-spelling system in play can be inferred from spellings demonstrated here.)

Firstly, the categorisation

Consider in the spelling of each integer the appearances of the letters I,V, X, L, C, D - the Roman numerals. Extracting the appearances of just these three from each integer and concatenating, we find that
The numbers in category A correspond to I.
The numbers in category B correspond to II
The numbers in category C correspond to III
The numbers in category D correspond to IV
The numbers in category E correspond to V
The numbers in category F correspond to VI
The numbers in category G correspond to IX
The numbers in category H correspond to LV
The numbers in category I correspond to D
The numbers in category J correspond to DI
The numbers in category K correspond to DII
The numbers in category L correspond to DIV
The numbers in category M correspond to DV
The numbers in category N correspond to DVI
The numbers in category O correspond to DIX
The numbers in category P correspond to DLV

Given that information it seems that the numbers of Y are unvalued because

They correspond to erroneous Roman numerals e.g, minus-sixteen $$\rightarrow$$ IIX.

R. Why are the numbers of row Z unvalued?

These do not contain any of I,V,X,L,C or D.

S. What are the first and last numbers of each row other than row Y?

A - minus-forty-four and ninety-four
B - minus-ninety-four and ninety-nine
C - minus-ninety-nine and minus-thirty-eight
D - minus-seventy-four and ninety-seven
E - seven and seventy-four
F - seventy-eight and seventy-nine
G - six and sixty-four
H - eleven and twelve
I - one-thousand and forty-four-thousand-forty-four
J - one-thousand-eight and forty-four-thousand-ninety-four
K - one-thousand-thirty-eight and forty-four-thousand-ninety-nine
L - one-thousand-five and forty-four-thousand-ninety-seven
M - one-thousand-seven and forty-four-thousand-seventy-four
N - one-thousand-seventy-eight and forty-four-thousand-seventy-nine
O - one-thousand-six and forty-four-thousand-sixty-four
P - one-thousand-eleven and forty-four-thousand-twelve
Z - zero and forty-four

T. What is the smallest positive number in row Y?

thirty-five

U. How long is the longest sequence of consecutive numbers in any row other than row Y?

The longest sequence is of length seven.
For example in row A we have forty-eight, forty-nine, fifty, fifty-one, fifty-two, fifty-three, fifty-four.

V. Which rows contain such a sequence and how many in each such row?

Row A has 2 sequences of length 7 (twenty-eight to thirty-four and forty-eight to fifty-four).
Row B has 2 sequences of length 7 (minus-thirty-four to minus-twenty-eight and minus-fifty-four to minus-forty-eight).
Row J has 32 sequences of length 7 (two for each of the positive values in row Z).
I don't think any other row has such a sequence.

• @humn I think I made a slight mistake before, I've just realised there are no DIII, fixed now. Also, the comment on row G was where there are gaps in the Roman numerals. I will take it out I think Apr 4, 2020 at 17:35
• Thanks @humn, that was a very nice puzzle, especially the unexpected part of U. Apr 4, 2020 at 18:38