How Many Doors Remain Open After Toggling States in the 1000-Door Puzzle?

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  • Thread starter Thread starter maze
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Discussion Overview

The discussion revolves around a puzzle involving toggling the states of 1000 doors in a sequential manner. Participants explore the mechanics of the puzzle, including the initial conditions and the process of toggling doors based on their position in the sequence.

Discussion Character

  • Puzzle-related, Mathematical reasoning

Main Points Raised

  • One participant presents the puzzle and outlines the steps involved in toggling the doors.
  • Another participant claims to have found the numerical result for the number of doors that remain open.
  • A subsequent participant confirms the numerical result and poses a hypothetical question regarding the scenario with a vastly larger number of doors (10^100000).
  • Another participant agrees with the numerical result, reinforcing the correctness of the previous claims.

Areas of Agreement / Disagreement

Participants generally agree on the numerical result derived from the puzzle, but there is no consensus on the implications or methods for larger scenarios, as the discussion remains open-ended regarding those cases.

Contextual Notes

The discussion does not delve into the mathematical derivation of the result, and assumptions regarding the toggling process are not explicitly stated. The implications of scaling the problem to a larger number of doors remain unexplored.

maze
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Hey i posted this puzzle a few days ago in another forum, but the stickied thread is completely dead and I don't think anyone even reads it anymore, so I'm going to post it here. Plus I think this is the more proper forum as it is more of a puzzle than a math problem (though it is both).

PUZZLE:
In a very long hallway, there are 1000 doors all initially open.
First, you close every door.
Second, you open every other door.
Next, you toggle the state of every 3rd door (open it if it is closed and close it if it is open),
Next, you toggle every 4th door,
and you continue this process, at the toggling the state of every nth door at the n'th step.

At the end of this process (when n=1000), how many doors are open?

Here is a diagram:
http://img504.imageshack.us/img504/1967/door2rc2.gif
 
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I think I found the result numerically...

969 are open

is that right? Now let's see if I can derive it without computer :smile:

[edit]Cool, I didn't know spoiler tags worked here![/edit]
 
Yep numerically that's right. Now what if there are 10^100000 doors? ;)
 
maze said:
Yep numerically that's right. Now what if there are 10^100000 doors? ;)

For N doors, the number of open doors is: N-FLOOR(SQRT(N))

DaveE
 
correct indeed! nice one