prob041: The n-Fractions Puzzle

proposed by Alan Frisch, Christopher Jefferson, Ian Miguel, Toby Walsh
frisch@cs.york.ac.uk, caj@cs.york.ac.uk, ianm@cs.york.ac.uk, tw@4c.ucc.ie

Original Specification

The original fractions puzzle is specified as follows. Find 9 distinct non-zero digits that satisfy: 
A    D    G
-- + -- + -- == 1
BC   EF   HI
where BC is shorthand for 10B+C, EF for 10E+F and HI for 10H+I.

n-Fractions

A simple generalisation is as follows. Find 3n non-zero digits satisfying: 
                  x_i
sum_{i in 1 .. n} ------ == 1
                  y_iz_i
where y_iz_i is shorthand for 10y_i+z_i and the number of occurrences of each digit in 1..9 is between 1 and ceil(n/3).

Since each fraction is at least 1/99, this family of problems has solutions for at most n <= 99. An interesting problem would be to find the greatest n such that at least one solution exists. A further generalisation might specify that the fractions sum to ceil(n/3).