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

Nifty, though...so many solutions if we allow duplicate values and/or reducible fractions.  What about if we disallow both and add one more additional constraint, i.e.:

(a) As before, the shared denominator must be as small as possible.
(b) Reducible fractions are not allowed.
(c) All 13 numerators must be unique.
(d) The sum of those 13 numerators must also be as small as possible.

By my count, there are 56 solutions which satisfy these conditions, all with the same demoninator and many of the same numerators.  Can you find one?  I'll post further hints in a followup comment.

Posted by Ethan on 2005-12-30 02:12:37