Windows problem

frenchX · December 12th, 2010, 9:09 am

Here a small brainteaser I found on the Net. There are 100 closed windows and 100 people. The first person opens every window. The second person closes every 2nd window. The third person visits every 3rd window; if the window is closed, he opens it; if it is open, he closes it. So on and so forth. After the 100th person, how many windows are open?Additionnal question find a formula for N windows with N people

ruairih · December 12th, 2010, 10:39 pm

The pattern is of perfect squares (even number of factors). eg 4, 9, 16 etc

DevonFangs · December 14th, 2010, 11:08 am

It sounds right. For every number you have to take the divisors, but only with square numbers you are sure that 2 people visit every window.So the answer is: the number of perfect squares lower than N (so the floor of \sqrt{N})

anilmag · April 12th, 2011, 1:33 pm

You missed window #1 in your answer. Plus those windows which are open has odd number of factors, e.g., 1 (1), 4 (1 2 4), 9 (1 3 9), 16 (1 2 4 8 16), ... so that they would be open at the end (they were closed to start with)

animeshsaxena · April 13th, 2011, 2:27 pm

10 windows closed...as per divisors...logic 90 windows open

anilmag · April 13th, 2011, 2:51 pm

QuoteOriginally posted by: animeshsaxena10 windows closed...as per divisors...logic 90 windows openNope, 10 windows open, 90 windows closed. Think of Window #1, only the 1st person 'touches' this window and opens it, so it would be open at the end (it was closed to start with).

Cuchulainn · March 26th, 2025, 12:34 pm

Apology over inside-out windows on tower blocks

https://www.bbc.com/news/articles/cgm107x2wexo

They are obviously not members of this forum.