# Representation of Mo-roman numerals

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?

("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.

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

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).

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

# 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.

• I think you missed this part in the question: "where all the roman numerals are used once and only once" – Marius Apr 25 '17 at 13:27
• 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. – Marius Apr 25 '17 at 13:39