Replace each variable with a positive rational number (i.e. fraction) such that all of the following equations simultaneously hold true. Note that 0/n is not considered positive.

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
S + E + V + E + N = 7
E + I + G + H + T = 8

Additional constraints:

1) All fractions must have the same denominator, which must be as small as possible.
2) All variables must be distinct.
3) Reducible fractions, i.e. those in which the numerator and denominator share a common divisor greater than 1, are not allowed.

Hint: there are four distinct solutions. For extra credit, find the one with the lowest possible sum of the 14 variables.

This problem was inspired by a similar submission by pcbouhid (see http://perplexus.info/show.php?pid=3872)

I tackled this first by eliminating numbers that could not work as denominators. If each variable must be distinct we can eliminate many denominators.
If the denominator is divisible by 2, the numerators cannot be divisible by 2 otherwise the fraction could be simplified. However in the first equation, O + N + E = 1, three variables must sum to the value of the denonimator in order to be equated to 1. However, the sum of any three odd integers is odd, preventing simplification of the final fraction to an integer.

Using the first three integers 1+2+3 = 6 the lowest denominator is 6, (or 7 using only odd numbers).
I attempted to find a solution using 7 as the denominator and I was unable. I did include one solution for 7.
Both 9 and 15 are not possible since there is no combination of three positive integers that will sum to these totals without at least one having a common factor.
The following are the best solutions I achieved for the lowest odd numbers (7, 11, 13, 17, 19, and 23)

                DENOMINATORS
      7     11     13     17     19     23

                 NUMERATORS
O     1      1      2      4      4      4
N     4      4      5      6      7      9
E     2      6      6      7      8     10
T     6      8      9     11     12     14
W    *7*    13     15     19     22     28
H     3      3     10     14     15     18
R     8     10      8     12     14     17
F     9      9     12     16     17     22
U    10     24     30     36     41     49
I     5     14     16     20     24     29
V    19     26     31     42     46     54
S    22     35     43     57     64     78
X    15     17     19     25     26     31
G    40     57     63     84     93    113

Edited on February 22, 2006, 7:46 pm

Posted by Leming on 2006-02-22 19:44:55