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This is straightforward to solve. This Python program runs in 31ms.

Solution:The five numbers are 343, 243, 256, 216 and 512.Pretty similar in structure to Jim’s version

Without writing a computer program, here is my list:

343, 243, 256 216 512

The only discrepency I have with this solution is that the enigma specifies that proceeding numbers can have the same hundreth _or_ the same tens _or_ the same units.

E.g. 343 couldn’t be followed by 243 (since the tens match _and_ the units match)

By sticking to this stringently the puzzle is unsolvable though…

I agree that the puzzle could have been clearer on whether the “or” was “exclusive” or “inclusive”. But having studied logic and programming I was perfectly happy to use the standard inclusive meaning. And, as you point out, there are no solutions if you assume the exclusive meaning.