Bob Kraus -- Another solver wanted to know how I discovered this problem. Well, I never resort to computer programming to solve or design any of my puzzles (that takes the fun out of it). However, I would hate to do them purely by hand either! The secret is that I use an MS-Excel spreadsheet with clever use of conditional formulas and macros to manipulate the lists, while still going through the same step-wise solving logic one would do by hand. (The most-excellent Excel porgram does allow you to use Visual Basic progamming while editing the macros, making it extremely powerful, but I have only used this to generate a list of primes, not for actually solving any puzzle.)

JP Ikäheimonen -- Prime 6133 Square 3364 Triangular 6441 Fibonacci 4181 Power of two 8192 Cube 9261 Do some people actually solve these manually (w/o the aid of a computer?)

Solvers incleded Bill Daly, Derek Westcott, Joseph DeVincentis, and Evgeni Lukin.

baiocchi -- The unique solution for the new cycle-problem is 3364, 6441, 4181, 8192,
9261, 6133.
Of course, in the framework of 6-cycles, my effort was minimum, because
my old program just required a few changes; I also used it to find
uniqueness in the framework of sub-cycles; 5-cycles gives uniqueness
only excluding powers of 2 (i.e. the old problem); but also in the
framework of 4-cycles there is a case (and only one) of uniqueness: it
is given by the classes [Prime, Triangular, Cube, Power of 2] and its
solution is 7381, 8192, 9261, 6173.

Roger Phillips -- The cycle of 6 numbers has this solution: 8192 (power of 2), 9261 (cube), 6133 (prime),
3364 (square), 6441 (triangular), 4181 (fibonacci) which is unique _unless_ we allow a single number
to represent more than one category, thus:
4096 (s, c, 2), 9613 (p), 1367 (p), 6765 (f), 6571 (p), 7140 (t)
4096 (s, c, 2), 9623 (p), 2341 (p), 4181 (f), 8171 (p), 7140 (t)
4096 (s, c, 2), 9631 (p), 3167 (p), 6765 (f), 6571 (p), 7140 (t)
4096 (s, c, 2), 9649 (p), 4967 (p), 6765 (f), 6571 (p), 7140 (t)
4096 (s, c, 2), 9677 (p), 7741 (p), 4181 (f), 8171 (p), 7140 (t)
4096 (s, c, 2), 9689 (p), 8941 (p), 4181 (f), 8171 (p), 7140 (t)
4096 (s, c, 2), 9697 (p), 9767 (p), 6765 (f), 6571 (p), 7140 (t)