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

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Discussion Overview

The discussion revolves around a variant of the Tower of Hanoi puzzle, specifically a 9-level "step pyramid" version. Participants explore the implications of special rules regarding certain levels that affect the number of moves required to solve the puzzle. The focus is on the theoretical aspects of solving this puzzle efficiently under the new conditions.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant describes the structure of the step pyramid puzzle and outlines the special rules for levels 2 and 5, which allow for additional moves without counting them.
  • Another participant questions whether moving levels 2 and 5 in succession would count as a single move, indicating a need for clarification on the rules.
  • A later reply suggests that the puzzle should be solved using the fewest moves possible and asserts that levels 2 and 5 would not follow each other in the optimal solution.
  • One participant argues that the special nature of levels 2 and 5 does not significantly alter the shortest solution, proposing that the puzzle can be solved normally with adjustments made for the special levels.
  • Another participant proposes that if chaining moves of the special levels is not allowed, it could introduce complexity, suggesting a modification to make levels 2 and 3 special instead.
  • One participant notes a preference for levels 2 and 5 based on their impact on the total number of moves required to complete the puzzle.

Areas of Agreement / Disagreement

Participants express differing views on the impact of the special levels on the solution's complexity and the optimal strategy for solving the puzzle. No consensus is reached regarding the best approach or the implications of the special rules.

Contextual Notes

Participants discuss the potential for optimization and the implications of the special rules, but the discussion remains open-ended regarding the exact nature of the solution and the effects of the rules on the overall strategy.

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

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