The Best License Plate?

The singularity of this license plate was almost lost on me. I noticed it as I was pulling out of my parking space, and I thought: “Hey, ‘Civ 104’, like the 104th entry in the Civilization series! Boy, it’ll be a long time before they get to that one. Kind of like that other hypothetical Civ game, the one they’ll call ‘Civ CIV’, because it’ll be the one where the number of the game is ‘CIV’ in Roman numerals…”

Then I started pulling back into that parking space (luckily it didn’t take me too long to think all that stuff), so I could get a picture. CIV 104! It’s that number! CIV! 104! Oh my gosh! Roman numerals!

Since I saw this license plate a couple days ago, I’ve been wondering: How many license plates, in the Iowa style (3 letters, 3 numbers), have this property? And how could I answer that question with a minimum of effort? Yes. This is the kind of thing I do for fun.

The brute-forcest way to do it would be to convert every decimal number from 001 to 999 into Roman numerals, and see which Roman numerals have three “digits”. That’s doing 1000 checks, though! A slightly less brutish method would be to convert every possible three-digit sequence of I, V, X, L, C, and D into decimals—that would be only 216 checks. Because M couldn’t appear in any of these license plates, right?

Well, no. For example: CMI 901. And what about the sequence IM? Is IM a legal Roman numeral? That’d be how you write 999, right?

Well, no. 999 is CMXCIX. I figured this out when I checked this big list of Roman numerals. And then I had yet another idea.

I pasted that list into Inform, and then I wrote a quick little rule to pull out the 3-digit Roman numerals. The complete Inform 7 source for my project is here, but this is the important part:

This returns a list of 62 possible license plates, which without further ado I reproduce here:

III 003
VII 007
XII 012
XIV 014
XVI 016
XIX 019
XXI 021
XXV 025
XXX 030
XLI 041
XLV 045
LII 052
LIV 054
LVI 056
LIX 059
LXI 061
LXV 065
LXX 070
XCI 091
XCV 095
CII 102
CIV 104
CVI 106
CIX 109
CXI 111
CXV 115
CXX 120
CXL 140
CLI 151
CLV 155
CLX 160
CXC 190
CCI 201
CCV 205
CCX 210
CCL 250
CCC 300
CDI 401
CDV 405
CDX 410
CDL 450
DII 502
DIV 504
DVI 506
DIX 509
DXI 511
DXV 515
DXX 520
DXL 540
DLI 551
DLV 555
DLX 560
DXC 590
DCI 601
DCV 605
DCX 610
DCL 650
DCC 700
CMI 901
CMV 905
CMX 910
CML 950

If you don’t think the series III 003 – XCV 095 should count, because you hate initial zeroes, the number of license plates that meets your fascist standards is 41 instead.

Thank you for your interest.