I remember learning recursion twice: once for Fibonacci and once for Hanoi. It did take a while to click but it unlocked recursion schemes and dynamic programming.

To me it was an example that there are some problems only solved by recursion but that they also overblown in their relevancy

Determining the correct algorithm is a game for children

Well the children don't have to move one piece at a time, they just take all the rings off then put them on in the right order. So the optimal moves for a child is 4.

Once you have it figured out, just mechanically doing it all correctly for 1023 steps is also annoying.

I did it up to the 511 moves for the 9-pice game on Android once, and according to the stats page that took me 6 min 18 seconds. The 10-piece version does not have a time, so I think I gave up :-)

Most people? 100% not. Me who has probably autism and has internalised the algorithm, I can.

I love and hate my brain.

Caxhe poisoned. By the time next child of 5 is brought, child turns 65 years old

These instructions seem incomplete.

Consider a tower of three, moving from the left to right:

321 / - / -

32 / - / 1

3 / 2 / 1

Next, 1 should be moved to the center position on top of two for the optimal solution, but your ruleset has no indicator of which choice should be made.