Sunday Times Teaser 2657 – Puzzling Book

by Danny Roth

A couple mantain a book of puzzles numbered from 1 to 30, some of which they have solved. The solutions are also numbered from 1 to 30 but these solution numbers do not necessarily match the numbers of the puzzle that they solve.

For each puzzle solved the sum of the puzzle and solution numbers is a perfect power of their difference. The sum of the puzzle numbers of solved puzzles has three digits and is less than the sum of the related solution numbers. In fact the digits of both are all non-zero and are the same but in a different order.

What, in increasing order, are the numbers of the solved puzzles?