104 lines
4.5 KiB
Markdown
104 lines
4.5 KiB
Markdown
For a number written in Roman numerals to be considered valid there are basic rules which must be followed.
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Even though the rules allow some numbers to be expressed in more than one way there is always a "best" way
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of writing a particular number.
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For example, it would appear that there are at least six ways of writing the number sixteen:
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~~~
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IIIIIIIIIIIIIIII
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VIIIIIIIIIII
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VVIIIIII
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XIIIIII
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VVVI
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XVI
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~~~
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However, according to the rules only XIIIIII and XVI are valid,
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and the last example is considered to be the most efficient, as it uses the least number of numerals.
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The 11K text file, roman.txt, contains one thousand numbers written in valid,
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but not necessarily minimal, Roman numerals;
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see **About... Roman Numerals** for the definitive rules for this problem.
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Find the number of characters saved by writing each of these in their minimal form.
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*Note:* You can assume that all the Roman numerals in the file contain no more
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than four consecutive identical units.
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About... Roman Numerals
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=======================
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> How do you read and write Roman numerals?
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Traditional Roman numerals are made up of the following denominations:
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~~~
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I = 1
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V = 5
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X = 10
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L = 50
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C = 100
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D = 500
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M = 1000
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~~~
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In order for a number written in Roman numerals to be considered valid there are three basic rules
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which must be followed.
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Numerals must be arranged in descending order of size.
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`M`, `C`, and `X` cannot be equalled or exceeded by smaller denominations.
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`D`, `L`, and `V` can each only appear once.
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For example, the number sixteen could be written as `XVI` or `XIIIIII`,
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with the first being the preferred form as it uses the least number of numerals.
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We could not write `IIIIIIIIIIIIIIII` because we are making `X` (ten) from smaller denominations,
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nor could we write VVVI because the second and third rule are being broken.
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The "descending size" rule was introduced to allow the use of subtractive combinations.
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For example, four can be written `IV` because it is one before five.
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As the rule requires that the numerals be arranged in order of size it should be clear to
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a reader that the presence of a smaller numeral out of place, so to speak,
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was unambiguously to be subtracted from the following numeral rather than added.
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For example, nineteen could be written `XIX` = `X (ten) + IX (nine)`.
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Note also how the rule requires `X` (ten) be placed before `IX` (nine),
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and `IXX` would not be an acceptable configuration (descending size rule).
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Similarly, `XVIV` would be invalid because `V` can only appear once in a number.
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Generally the Romans tried to use as few numerals as possible when displaying numbers.
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For this reason, `XIX` would be the preferred form of nineteen over other valid combinations,
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like `XIIIIIIIII` or `XVIIII`.
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By mediaeval times it had become standard practice to avoid more than three consecutive
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identical numerals by taking advantage of the more compact subtractive combinations.
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That is, `IV` would be written instead of `IIII`,
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`IX` would be used instead of `IIIIIIIII` or `VIIII`, and so on.
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In addition to the three rules given above,
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if subtractive combinations are used then the following four rules must be followed.
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- Only one `I`, `X`, and `C` can be used as the leading numeral in part of a subtractive pair.
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- `I` can only be placed before `V` and `X`.
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- `X` can only be placed before `L` and `C`.
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- `C` can only be placed before `D` and `M`.
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Which means that `IL` would be considered to be an invalid way of writing forty-nine,
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and whereas `XXXXIIIIIIIII`, `XXXXVIIII`, `XXXXIX`, `XLIIIIIIIII`, `XLVIIII`, and `XLIX`
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are all quite legitimate, the latter is the preferred (minimal) form.
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However, minimal form was not a rule and there still remain in Rome many examples
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where economy of numerals has not been employed.
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For example, in the famous Colosseum the numerals above the forty-ninth entrance
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is written `XXXXVIIII` rather than `XLIX`.
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It is also expected, but not required, that higher denominations should be used whenever possible;
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for example, `V` should be used in place of `IIIII`, `L` should be used in place of `XXXXX`,
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and `D` should be used in place of `CCCCC`.
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However, in the church of Sant'Agnese fuori le Mura (St Agnes' outside the walls),
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found in Rome, the date, `MCCCCCCVI` (1606), is written on the gilded and coffered wooden ceiling;
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I am sure that many would argue that it should have been written `MDCVI`.
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So if we believe the adage, "when in Rome do as the Romans do,"
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and we see how the Romans write numerals, then it clearly gives us much more freedom
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than many would care to admit. |