# Archaeological alphametic

Enigmarchaeologists uncovered an ancient Roman precursor to our familiar, and comparatively dignified,   SEND + MORE = MONEY  alphametic.

TEE + HEE + HEE + HO + HO + HEH + OH + ME + OH + MY + HMM   =   OH + HEY + AHA

What is that equation in substituted Roman numerals?

And just for giggles, what does   TEE + HEE  by itself equal with the same substitutions?

Each letter— A, E, H, M, O, T and Y—stands for a different Roman digit—  I, V, X, L, C, D or M — in a correspondence to be deduced.   Only standard Roman numerals are in play, not alternative forms.

For example . . .

. . . If the puzzle were   GIN + FIG   =   FUN,   its letters could be substituted to become   XLV + DLX   =   DCV,   which amounts to   45 + 560 = 605   in Arabic numerals.

             Example        Substitutions        Roman               Arabic
F   -->   D
GIN          U   -->   C           XLV                   45
+ FIG          I   -->   L         + DLX       =        + 560
-------         G   -->   X        -------              -------
FUN          N   -->   V           DCV                  605

• are they all valid roman numerals? for example. CDC is not valid. – Marius Apr 10 '17 at 6:40
• maybe you should add that to the question to avoid confusion. – Marius Apr 10 '17 at 6:41
• Just to be pedantic, currently the incorrect IXC is allowed but the correct XCIX is not. – boboquack Apr 10 '17 at 6:54
• Thanks for nudges, Marius and @Boboquack, they even helped me catch an omission in the instructions. Guess I'll just refer to the Wikipedia page. – humn Apr 10 '17 at 6:56

## 1 Answer

E and M are doubled with a letter on front and can therefore be just I and X. Suppose E = I, M = X. Then we have HEH, which makes it impossible already - XIX would work, but M is X already.

So

E is X (and M is I), and then H needs to be C to have a valid HEH. (LXL is not valid). This means A needs to be M to have valid AHA (MCM). HO would then need to be CD as there is no other option. MY requires Y to be V. Which leads to a single remaining letter for T = L.

Lets substitute back:

LXX + CXX + CXX + CD + CD + CXC + DC + IX + DC + IV + CII   =   DC + CXV + MCM. 70 + 120 + 120 + 400 + 400 + 190 + 600 + 9 + 600 + 4 + 102 = 600 + 115 + 1900. 2615 = 2615

Looks good. Now the TEE+HEE part:

TEE + HEE = LXX + CXX = 70 + 120 = 190 = CXC = HEH