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This continues the spirit of Mo-roman numerals started by humn

  1. What is/are the natural number(s) that has/have the most different representations in mo-Roman numerals and what is the number of representations in mo-Roman numerals where all the roman numerals are used once and only once?
  2. What is/are the natural number(s) that has/have the fewest (at least 1) representations in mo-Roman numerals where all the roman numerals are used once and only once?

An answer not containing an explanation is not a valid answer.
("Because the code I wrote said so" is not a valid answer. So no computers).

Explanations:

What is a mo-Roman numeral?
It's a number formed from Roman numerals but with more permissive rules.
Any roman numeral that has a larger numeral on his right side is subtracted, otherwise is added.
Example:
90 can be written in Roman numerals as XC. In Mo-roman numerals it can be written also as
$XCVVIXI = -10 + 100 - 5 - 5 - 1 + 10 + 1$

What are the roman numerals and their values?

$I = 1$
$V = 5$
$X = 10$
$L = 50$
$C = 100$
$D = 500$
$M = 1000$

Note: This is not meant to be a difficult puzzle. It is here to lure in more people to the mo-roman numerals questions.

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  • 4
    $\begingroup$ If this keeps going, maybe we'll need mo-tags ;) $\endgroup$ – n_plum Apr 25 '17 at 12:09
  • 3
    $\begingroup$ That was my secret agenda. $\endgroup$ – Marius Apr 25 '17 at 12:10
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    $\begingroup$ <sarcasm>I love how everyone is kind enough to explain a downvote</sarcasm> $\endgroup$ – Marius Apr 25 '17 at 13:16
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Minimum:

MDCLXVI (1666) is obvious. Then you have DMCLXVI as well where any other order of making 666 doesn't work - you need subtraction 1000-500 and this means D needs to be before M and is uniquely positioned. Any additional subtractions, say 466 or 1466 (where you subtract C) can be seen that C can come before M or before D, so you have 2 options. It continues the same way with other numbers (say I can come before any other number).

So, maximum is similar:

IVXLCDM gives 1000-666 = 334. This number can be written by arbitrary order of letters before M, which have 6! positions. The other number of this property has the form of IVXLCMD. D is the last number and M (as well as other numbers) come at any spot before it. This number is 1334 (M+D - rest).

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  • $\begingroup$ Close. Very close. There is one more number that qualifies for maximum. Same number of representations as the one you provided. $\endgroup$ – Marius Apr 25 '17 at 13:18
  • $\begingroup$ @Marius yeah you are right. I had the correct idea at minimum but managed to forget the same reasoning with maximum. Fixed now. $\endgroup$ – Zizy Archer Apr 25 '17 at 13:37
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Partial Answer for part 1

The natural number that has the most representations is:

Any number

consider the number 4 as an example.
This can be represented as IV or IIVI or IIIVII or IIIIVIII or...
the number of representations of the number 4 is countably infinite and therefore, unless a number has an uncountably infinite number of representations, it has the most representations, just like every other number

Edit: This does not answer the second part of question 1, which asks about representations using only one of each digit.

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  • $\begingroup$ I think you missed this part in the question: "where all the roman numerals are used once and only once" $\endgroup$ – Marius Apr 25 '17 at 13:27
  • $\begingroup$ In regards, to your edit. Your answer does not even answer the first part of the question. You missed a very important line. See my comment above. $\endgroup$ – Marius Apr 25 '17 at 13:39

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