Number cycles

PART 1:
Find a cycle of five 4-digit numbers such that the last 2 digits of each number are equal to the first 2 digits of the next number in the cycle. Each of the 5 numbers must be exactly one of the following types (with each of the 5 types being represented exactly once): Square, Cube, Triangular, Prime, Fibonacci. The solution is unique.

PART 2:
Find a cycle of six 4-digit numbers such that the last 2 digits of each number are equal to the first 2 digits of the next number in the cycle. Each of the 6 numbers must be exactly one of the following types (with each of the 6 types being represented exactly once): Square, Cube, Triangular, Prime, Fibonacci, Power-of-Two. The solution is unique.

Does "cycle" mean that the last 2 digits of the last number are the first 2 digits of the first number?

Interesting puzzles. I think a good place to start is finding the Fibonacci (and/or the power of two) as there are only four of each with four digits.

A program could be written to check if each of those was possible (could be contained in the cycle) and then leave you with a much smaller sample to search for the cycle.

Originally posted by David113
Does "cycle" mean that the last 2 digits of the last number are the first 2 digits of the first number?

Originally posted by XanthosNZ
I guess it does. The solution isn't unique if we don't have a cycle formed (first two equal last two).

Part 1:
2850, 5041, 4181, 8117, 1728
Or any of the other four shifted versions of this order.

Part 2:
9261, 6133, 3364, 6441, 4181, 8192
Or as above.

I approached this using Matlab.