Problem 611

Problem 611

Hallway of square steps

Peter moves in a hallway with N+1 doors consecutively numbered from 0 through N. All doors are initially closed. Peter starts in front of door 0, and repeatedly performs the following steps:

  • First, he walks a positive square number of doors away from his position.
  • Then he walks another, larger square number of doors away from his new position.
  • He toggles the door he faces (opens it if closed, closes it if open).
  • And finally returns to door 0.

We call an action any sequence of those steps. Peter never performs the exact same action twice, and makes sure to perform all possible actions that don’t bring him past the last door.

Let F(N) be the number of doors that are open after Peter has performed all possible actions. You are given that F(5)=1, F(100)=27, F(1000)=233 and F(106)=112168.

Find F(1012).



  • 首先,他移动到距离初始位置为一个正平方数的门。
  • 然后,他移动到距离新位置为一个更大的平方数的门。
  • 他改变现在所面对的门的状态(如果是关着的就打开,如果是开着的就关上)。
  • 最后他回到标有0的门。