Open the doors.
A team of persons went to Hazarduari palace in Murshidabad, West Bengal , India. That's very famous for having doors. They asked one of the travelers who was a mathematician to predict how many door will be open at the end of the game. They played the game as follows. Person opens all the doors. Then person closed doors , ... . And leaves odd numbered doors open. Similarly, person changes the state of the doors , , ..... And leaves other doors untouched. Finally , . changes only the state of the door and leaves all others untouched. Ok mathematician predict how many doors are open at the end?
The answer is 31.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
If d is a factor of n , then person d changes the state of door n . So the number of times door n changes state is equal to the number of factors of n .
The doors that are open at the end are those whose states have changed an odd number of times. The numbers with an odd number of factors are precisely the square numbers (why?), and since 3 1 2 < 1 0 0 0 < 3 2 2 , there will be 3 1 doors left open.