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

Since the numbers don't explicitly have to be unique, and each expression contains a letter which is not repeated we can set the numerators of all other letters to 1. Since they are all positive, 3 is the smallest possible denominator which solves O+N+E=1

One solution is:

1/3 + 1/3 + 1/3 = 1

1/3 + 4/3 + 1/3  = 2

1/3 + 5/3 + 1/3 + 1/3 + 1/3 = 3

1/3 + 1/3 + 9/3 + 1/3 = 4

1/3 + 1/3 + 12/3 + 1/3 = 5

1/3 + 1/3 + 16/3  = 6

If reducable fractions like 9/3 and 12/3 aren't allowed, then we can just shift toward the letter F:

1/3 + 1/3 + 1/3 = 1

1/3 + 4/3 + 1/3  = 2

1/3 + 5/3 + 1/3 + 1/3 + 1/3 = 3

2/3 + 1/3 + 8/3 + 1/3 = 4

2/3 + 1/3 + 11/3 + 1/3 = 5

1/3 + 1/3 + 16/3  = 6

If the variables have to be unique, then 4 is the smallest denominator which allows T+H+R+E+E=3 (i.e. 3/4+2/4+5/4+1/4+1/4=3) but then we run into the problem of reducable fractions (not necessarily illegal).

Posted by Eric on 2005-12-29 14:53:05