Replace each letter by a positive rational number such that the following is true. Do it for rationals that can be written in the same denominator that is as small as possible.

O + N + E          =  1
T + W + O          =  2
T + H + R + E + E  =  3
F + O + U + R      =  4
F + I + V + E      =  5
S + I + X          =  6

pcbouhid,

Sorry...I didn't mean to offend.  Your problem caught my imagination, and it struck me that a slightly trickier variant might be an interesting tangent, while perhaps not being sufficiently different from the original problem to justify its own post.  While I'm relatively new here and am still learning how things work, in my defense, I've read enough threads to recognize that I'm hardly the first to do such a thing.

Anyway, here's my solution, now that you've clarified that unique values for all 13 variables are required.  (I don't think that goes necessarily goes without saying: it's still a perfectly good problem even without that constraint, though obviously one with many more solutions.)

Shared denominator of 6, with numerators as follows: O=2, N=3, E=1, T=4, W=6, H=7, R=5, F=8, U=9, and one of these four possibilities:

I=10, V=11, S=12, X=14
I=10, V=11, S=14, X=12
I=11, V=10, S=12, X=13
I=11, V=10, S=13, X=12

As I mentioned, if reducible fractions are disallowed, we have to move to a shared denominator of 7, in which case there are about a zillion solutions (though only two with the lowest possible sum of the 13 numerators, which is 101).

Posted by Ethan on 2005-12-30 12:42:52