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Title Proposer Category
The n-Fractions Puzzle
Alan Frisch
Christopher Jefferson
Ian Miguel
Toby Walsh
Combinatorial mathematics
Games and puzzles

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: $ 1 = \sum_{i \in 1..n} x_i / 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 $ceil(n/3)$.