When in Rome...
 

Besides being a prime year, 2011 is also special because it requires the same number of "digits" in its Arabic decimal and its Roman numeral representations, to wit:

2011 = MMXI.

In the (relatively) near future, 2015 = MMXV and 2020 = MMXX are also such years. 2012 = MMXII and 2013 = MMXIII, however, are not.

Since the year 1 AD, what other years had (or will have) this distinction? (This is sort of an open ended puzzle - I'm not exactly sure of the answer myself - but it is something interesting I've been thinking about).

Are you asking only for "traditional" Roman numerals (no more than 3 of a single digit, etc., i.e. would you consider the following valid? MXM = 1990, MIM = 1999, MIIII = 1004, MXMI = 1991)?

[QUOTE=Uncwilly;245200]Are you asking only for "traditional" Roman numerals (no more than 3 of a single digit, etc., i.e. would you consider the following valid? MXM = 1990, MIM = 1999, MIIII = 1004, MXMI = 1991)?[/QUOTE]

Those are the new-style Roman numerals, not the traditional ones. Traditional would be like VIIII for 9 or (()) for 10,000.

[QUOTE=CRGreathouse;245203]Those are the new-style Roman numerals, not the traditional ones. Traditional would be like VIIII for 9 or (()) for 10,000.[/QUOTE]Correct, I was trying to find out the bounds of the puzzle. One can write a prog to do the traditionals. The new-style is much harder to do.

[QUOTE=Uncwilly;245200]Are you asking only for "traditional" Roman numerals (no more than 3 of a single digit, etc., i.e. would you consider the following valid? MXM = 1990, MIM = 1999, MIIII = 1004, MXMI = 1991)?[/QUOTE]

Yes, I was originally thinking of the traditional Roman numerals rules (in fact, I wasn't aware that new rules had been adopted!), and the usual set of symbols with the usual values:

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

So, in particular, 1990 would be MCMXC, 1999 would be MCMXCIX, 1004 would be MIV, and 1991 would be MCMXCI.

(Obviously, this admits fewer candidates - if we allow MXMI for 1991, we have an example of what we're looking for, but if we instead have 1991 = MCMXCIX, we don't.)

Incidentally, one of the most interesting math tutoring experiences I ever had was with a frustrated algebra student who insisted that X could not change from problem to problem, since it had a fixed value! The fixed value? You guessed it...10.

[QUOTE=CRGreathouse;245203]Those are the new-style Roman numerals, not the traditional ones. Traditional would be like VIIII for 9 or (()) for 10,000.[/QUOTE]

Actually, I don't think IIII would be allowed in the "traditional" rules. IIII as a construct seems to have come about from Louis XIV's desire to have more symmetric clock faces, well after the Romans.

Ever hear the one about the captain on the Roman sailing vessel? Couldn't figure out why the first mate always asked for two of every order the captain gave. :smile:

I've seen (M) and M-with-an-overbar for 10,000, but never (()).

[QUOTE=Uncwilly;245208]Correct, I was trying to find out the bounds of the puzzle. One can write a prog to do the traditionals. The new-style is much harder to do.[/QUOTE]

OK, so let's consider both.

If we little ol' human beans make it to 4000, I'm sure McWendy'sKing (a division of Taco Bell formed after the Great Fast Food Schism of 3347) will somehow capitalize off the year of MMMM.

[QUOTE=NBtarheel_33;245248]
Incidentally, one of the most interesting math tutoring experiences I ever had was with a frustrated algebra student who insisted that X could not change from problem to problem, since it had a fixed value! The fixed value? You guessed it...10.[/QUOTE]

When I first encounted the BASIC instruction "LET x = x + 1" I found
it a bit troublesome. Funny how quickly you can get over these things!

David

[QUOTE=CRGreathouse;245203]Those are the new-style Roman numerals, not the traditional ones. Traditional would be like VIIII for 9 or (()) for 10,000.[/QUOTE]

any links to learn this if so I can try and make a PARI code for this puzzle.

never mind I found the OEIS entries lol.

for the "old style"
iiii->v = -3 in length
viiii->x = - 4 in length
xxx->xl = -1 in length
xl-> l = -1 in length
lxxx - > xc = -2 in length
xc-> c = -1 in length
ccc-> cd = -1 in length
cd->d = -1 in length
dccc-> cm = -2 in length

though I guess we should say add the x and then convert it to xl and such just to help out more.

new style might be easier not sure.

[QUOTE=science_man_88;245281]for the "old style"
iiii->v = -3 in length
viiii->x = - 4 in length
xxx->xl = -1 in length
xl-> l = -1 in length
lxxx - > xc = -2 in length
xc-> c = -1 in length
ccc-> cd = -1 in length
cd->d = -1 in length
dccc-> cm = -2 in length

though I guess we should say add the x and then convert it to xl and such just to help out more.

new style might be easier not sure.[/QUOTE]

my mistake it also allows xxxx and cccc if I read more correctly lol.