Description
This problem is documented in many places, including here:
https://archive.nytimes.com/wordplay.blogs.nytimes.com/2015/04/13/finkel-4/
and here (one person version only):
https://en.wikipedia.org/wiki/Taxman_(mathematical_game)
The problem was originally conceived as a way to get junior high students interested in mathematics. But the problem goes way beyond a fun classroom problem.
As the value of the starting integer N increases, finding the optimum solution becomes more and more difficult.
This is a zero sum game, so the optimum solution is the one that minimizes the total amount taken as tax.
For starting values up to 1000, the OEIS database has the optimum solutions:
https://oeis.org/A019312
I have written a Python algorithm that is robust. But it deviates from optimum as N increases since there is no known algorithm ,except brute force, that returns the optimum solution. It would be great to see other solutions.