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February 15, 2012 11:24 AM   Subscribe

Is there any sequence of letters that spells out a number on a telephone keypad as well as indicating that same number in Roman numerals?

I was thinking recently about what my phone number would be in Roman numerals -- shut up, I think about these things -- and pondering how one could also dial the Roman numerals on the keypad directly. 867-5309 would be DCCCLXVII-MMMMMCCCIX, which would ring 322259844-6666622249 if dialed. So is it possible to have a phone number which would reach the same line if dialled as Arabic numerals (the numbers on the keypad) or Roman numerals (the letters)?

I suspect there is not, but the vagaries of different countries having different formats of phone numbers leave me unsure as to how to confirm this. Bonus points if someone's proof can account for things like the orphan number hanging off the end of advertising mnemonics (North American companies, with their seven-digit numbers, occasionally have phone numbers that read 212-GOOD-LUCK or 416-SNOWPLOW or somesuch, where the final character is there only to complete the mnemonic).
posted by ricochet biscuit to Grab Bag (15 answers total) 2 users marked this as a favorite
 
505 0505? LLL?
posted by Sphinx at 11:29 AM on February 15, 2012


505-0505 would be DV-DV.
posted by ricochet biscuit at 11:39 AM on February 15, 2012


Maybe 0 for the operator?

On the phone keypad I'm looking at, the '0' button has no letters on it, and Roman numerals do not include a zero.
posted by box at 11:43 AM on February 15, 2012


(Does 867-5309 have to be eight-hundred-sixty-seven five-thousand-three-hundred-nine, or can it also be, e.g., eighty-six seventy-five three-hundred-nine?)
posted by box at 11:44 AM on February 15, 2012


Since there's no symbol for anything over 1,000, other than using a bar over another symbol, there's no way to get a 7 digit number to be only 7 Roman numerals long. Unless you look at something like box is asking.
posted by monkeymadness at 11:50 AM on February 15, 2012 [1 favorite]


To clarify: M, a thousand, is over the 6. For the last four digits anything other than a 0 in the first position would require an M, making it over 6,000, which isn't possible. A 0 in the first position, like xxx-0xxx, and a non-zero in the second position would mean a C, which is over the 2. This would give at least 200, requiring two Cs in a row, making it xxx-022x, and nothing would work for the final digit. This is a cool idea, so maybe someone can come up with a neat way to make it work.
posted by monkeymadness at 11:56 AM on February 15, 2012


(Does 867-5309 have to be eight-hundred-sixty-seven five-thousand-three-hundred-nine, or can it also be, e.g., eighty-six seventy-five three-hundred-nine?)

I hadn't even considered that, but sure.


Since there's no symbol for anything over 1,000, other than using a bar over another symbol, there's no way to get a 7 digit number to be only 7 Roman numerals long. Unless you look at something like box is asking.


As stated in the question, it does not have to be seven digits long; the fist seven digits should produce the number (the rest can be orphans).
posted by ricochet biscuit at 11:56 AM on February 15, 2012


Excel has a handy roman() function, so I ran 0-999 and eliminated results that didn't match the number of numbers, then calculated keypad equivalents. 0-999 don't work. If we focus exclusively on dialing, and not appearance, and 0 counts since it has no letters, then there are some possibilities. 050 and 0L0 would be equal. But 050 isn't a valid area or country code. If we totally abuse 0, 500 (L00) would reach the Falkland Islands, and some Personal Communication Service numbers apparently. Depending on their phone systems, they might have a 050/500-0050/0500/5000 equivalent.
posted by jwells at 12:16 PM on February 15, 2012


Are we allowing 1's and 0's to represent the null string when converted to letters?

If we disallow 1s and 0s from appearing at all, it's fairly simply to show that no number is possible: The arabic numeral cannot end in 2, 3, 6, 7, or 8, because the equivalent Roman numeral would end in I, which corresponds to 4. The arabic numeral cannot end in 4 or 5, because the equivalent Roman numeral would end in V, which corresponds to 8. The only possibility is a number ending in 9, which in Roman numerals ends in IX. This means the arabic number has to end in 49. Which means the Roman numeral has to end in XLIX. Which means the arabic numeral has to end in 9549. And now you get into the problem of large numbers. If you use the method listed in the Wikipedia article that means the Roman numeral has to end in IXDXLIX (using an underline where there should be an overbar, because I don't know how to make an overbar in HTML), which means that the arabic numeral has to end in 4939549... you see the problem. The Roman numeral has more characters than the arabic numeral has digits, and this is true for nearly all numberrs.

How about if we allow 1s and 0s to be present, and represent the null string? Consider the following table:
# R L D
- - - -
0 0 0 0
1 1 0 1
2 2 1 1
3 3 1 2
4 2 1 1
5 1 1 0
6 2 1 1
7 3 1 2
8 4 1 3
9 2 1 1
# - an arabic digit

R - the number of characters needed to represent this digit in a Roman numeral. Conveniently, this is independent of what place the arabic digit occupies. E.g., a "4" in an arabic number always requires two characters in the Roman numeral, even though that might be "IV," "XL," or "CD" depending on where the 4 is.

L - the number of letters the arabic digit contributes via the keypad conversion (0 for 0 and 1, 1 for all other digits)

D - the difference between them, i.e., R-L

Note that D is never negative; all arabic digits require at least as many characters to represent in a Roman numeral as they contribute via the keypad. With the exception of 0 and 5, all digits require more characters to represent in a Roman numeral than the number of letters they contribute via the keypad.

Therefore, the arabic numeral's only digits can be 5 and 0. Which in turn means L can be the only character in the Roman numeral, since D or V would bring in a 3 or 8, respectively.

L=50 works, except that 50 is not a telephone number. Still, you could have 0050 as the second half of the telephone number, but since US telephone numbers do not begin with 0, 050-0050 doesn't work as a telephone number either.

If you treat a US telephone number as a single 7-digit number rather than a three-digit and four-digit number, and using the "large number" conventions from the Wikipedia article as above, 505-0050 = |L|LL works, assuming you don't require the vertical bars to be represented in the number. Likewise, if you are on a 5-digit system (some large organizations require you dial only the last 5 digits for internal numbers), 5-0050 = LL also works.

it does not have to be seven digits long; the fist seven digits should produce the number

I'm not sure I follow this; the Roman numeral is always at least as long as the keypad conversion of the arabic number, so taking just the first seven digits of a number longer than seven digits doesn't help, if I'm understanding your meaning correctly. (If you allow the reverse, as I noted above, 493-9549 = IXDXLIX gives a Roman numeral which represents the last four digits of the number.)
posted by DevilsAdvocate at 1:52 PM on February 15, 2012 [1 favorite]


If we totally abuse 0, 500 (L00)...

roman(500) = D, not L.
posted by DevilsAdvocate at 2:13 PM on February 15, 2012


it does not have to be seven digits long; the fist seven digits should produce the number

I think I see now what you were getting at: convert a seven-digit number (or a three-digit number followed by a four-digit number) to Roman numerals. The corresponding Roman numeral(s) will have more than seven letters: dial that on a telephone keypad, which will ignore everything after the first seven button presses, and that should dial the original number.

In that case, I find the following solutions (bold letters represent those accepted by the telephone when dialing them):

(As seven-digit numbers):
548-5999 = |LIV|LXXXVCMXCIX
584-4495 = |LVIII|XLIVCDXCV
924-4954 = |XCII|XLIVCMLIV

(As three- and four-digit numbers):
229-9494 = CCXXIX-IXCDXCIV

I believe these are the only solutions for the given numbers of digits, assuming the conventions for Roman numerals I have used ("thousands" digits of 4 or more must be notated using the "overbar" notation rather than M's; 99 is XCIX, not IC; etc.)

Outline of proof: in the seven-digit case, 5 and 9 are the only possible first digits, and in the case of 5, either 4 or 8 may be the second digit. In the three-and-four-digit case, only 2 can be the initial digit. Everything after that is forced.
posted by DevilsAdvocate at 2:42 PM on February 15, 2012


First, I love your name ricochet biscuit. Second you're right, maybe 505 050 5050? LLLL? Third, I knew the math studs would come in and crush my post.

It really depends on how you are breaking the numbers down.
posted by Sphinx at 2:53 PM on February 15, 2012


And a three-, three-, four-digit case (assuming you do not have to dial an initial 1, as on cell phones):

229-949-2695 = CCXXIX-CMXLIX-MMDCXCV

(The 229 area code is in southwest Georgia.)
posted by DevilsAdvocate at 3:14 PM on February 15, 2012


Upon further thought, that last one may not work as I'm not certain cell phones discard extra digits the same way landline systems do. Which raises the interesting question:

(North American companies, with their seven-digit numbers, occasionally have phone numbers that read 212-GOOD-LUCK or 416-SNOWPLOW or somesuch, where the final character is there only to complete the mnemonic).

Are these companies missing out due to potential customers who try to dial the entire thing from cell phones?
posted by DevilsAdvocate at 3:34 PM on February 15, 2012


Upon further thought, that last one may not work as I'm not certain cell phones discard extra digits the same way landline systems do.

I tested this out on a Sprint Blackberry and a Verizon Android phone, and both ignored the eleventh digit when I dialed a phone number.
posted by Nonsteroidal Anti-Inflammatory Drug at 7:04 PM on February 15, 2012


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