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Athenian democracy is a cornerstone of egalitarianism. One person, one vote. Everyone.

Here's something, though, Athenians mightn't've imagined:

                      I =    II =    III
        IV =   V =   VI =   VII =   VIII
        IX =   X =   XI =   XII =   XIII
       XIV =  XV =  XVI =  XVII =  XVIII =  IL =  L =  LI =  LII =  LIII
       XIX =  XX =  XXI =  XXII =  XXIII =  ... ?
      XXIV = XXV = XXVI = XXVII = XXVIII =  ... ?
      XXIX = XXX = XXXI = XXXII = XXXIII =  ... ?


Can you imagine it?

Please fill in the right side ... ?s and explain.

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  • $\begingroup$ Apology for the lack of more specific tags: They would give away the solution. $\endgroup$ – humn Mar 24 '19 at 19:52
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Equal numbers are those with

equal products of Roman numerals

so the blanks are

IC = C = CI = CII = CIII
ID = D = DI = DII = DIII
IM = M = MI = MII = MIII

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  • 2
    $\begingroup$ @humn Is IL a valid Roman numeral? $\endgroup$ – noedne Mar 24 '19 at 20:04
  • $\begingroup$ Got me. @noedne! Formally it would be XLIX, which would overflow. $\endgroup$ – humn Mar 24 '19 at 20:08
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    $\begingroup$ The ancient Romans never bothered (as far as the surviving sources tell us) to agree on a particular authoritative specification for how their notation for numbers worked -- and in the absence of that, speaking about "valid" representations misses the point somewhat. It's certainly true that the modern tradition tends to frown on notations such as IL, but the actual Roman practice was less rigid than that. (For example there is apparently one attested use of XXCIIII for 84). $\endgroup$ – hmakholm left over Monica Mar 24 '19 at 23:39
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    $\begingroup$ @HenningMakholm Very interesting! $\endgroup$ – noedne Mar 24 '19 at 23:45

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