How Long to Solve a 9-Level Step Pyramid Tower of Hanoi?

[Helios]

I have thought of a 9-level "Tower of Hanoi" mutation that I call the step pyramid puzzle because it resembles a classic Maya step pyramid with a square temple house on top of a larger terraced pyramid.

9 ______
8 ______
7 ____________
6 ______________
5 ________________
4 __________________
3 ____________________
2 ______________________
1 ________________________


Solve this puzzle just like the Hanoi puzzle except for these changes,

1) Upper stories 8 and 9, which comprise a two-storied "temple house" are identically sized parts and either can be set atop of their counterpart.
2) The 2nd and the 5th terraces are special. If either of these terraces are moved, then an extra move of the next terrace pending is immeadeatly made without being counted. Levels 2 & 5 can be indicated by a different color from the rest of the levels.

It is supposed that if one move is made each day. With these rules in effect, the puzzle will take ? days to complete.

 

[davee123]

2) The 2nd and the 5th terraces are special. If either of these terraces are moved, then an extra move of the next terrace pending is immeadeatly made without being counted. Levels 2 & 5 can be indicated by a different color from the rest of the levels.

Does this imply that you could move 2, then 5, then 2, then 5, then 3, and have it only count as a single move?

DaveE

 

[Helios]

I should have said that the puzzle should be solved in the procedure that takes the fewest number of moves. Terrace 2 & 5 would never follow each other as moves go.

 

[davee123]

I should have said that the puzzle should be solved in the procedure that takes the fewest number of moves. Terrace 2 & 5 would never follow each other as moves go.

I think 2 & 5 being special effectively makes no difference to the shortest solution. Effectively, you just have to solve the puzzle normally, and then subtract the number of moves with 2 & 5 from the total number of moves. For all intents and purposes, it's no more difficult than with 1 special level. However, if you CAN'T infinitely chain the special levels' moves together, it means that there may be some optimization possible by preferring to move one of the special levels "prematurely" so to speak. I'm not sure that's the case, given the limited number of moves at a given time, but it theoretically adds a degree of complexity. Hence, to add a degree of complexity, I'd suggest making #2 and #3 the special levels, and that their moves could not be chained together. But that could add a bit too much difficulty for the level of problem.

DaveE

 

[Helios]

I chose #2 & #5 because of a special preference for the number of moves the puzzle takes to complete.