Can you solve the locker riddle?

Suo as always with a pen and paper in her hand is waiting eagerly for the weekly riddles show on  the radio. Mr Bean is again back with a a wonderful puzzle for the kids.  Mr Bean starts with this week’s riddle.

Mr Bean begins:

Kids , there was a very rich man Mr. Benz who love her niece a lot and wanted to have her as the heir of his wealth. But he knew he couldn’t simply name her otherwise all other relatives would pester that girl.

Mr Benz was confident of his niece’s puzzle solving ability and banking on her skills he came up with a riddle.

All the 99 relatives and the little girl were  informed by the lawyer that if they solve this riddle together then they can divide the wealth evenly or else who soever solves the puzzle first will get the inheritance.

The Riddle:

The 100 relatives were taken to a secret room with 100 lockers with one word each. Heir one will open every locker, heir two will then close every second locker , heir three will change the status of every third locker( open it if it’s closed and close it of its open). This pattern will be repeated until all 100 of them would have gone.

The words that you get to know from the open lockers will provide you the cue to open the safe.

Think of the answer: Suo finds this riddle very interesting and eagerly waits for  Mr Bean to share the answer.

The Answer:

Before anybody could even start the legwork. The little niece comes up to lawyer and says she knows which of these lockers will remain open.

She explains further:  Actually the number of times a locker is touched is tantamount to number of factors of the locker number.

For example:  For the locker number 6 , person one will open  it, person  two will close it, person three will open it and person six will close it. The numbers -1, 2,3,6 are factors of six . So the lockers with even number of factors will remain closed and ones with odd number of factors will remain open.

And we know that the only lockers with odd number of factors will be perfect squares since these numbers will have one factor which when multiplied by itself gives back the number.

Therefore the lockers that will remain  open have perfect square as numbers. Now when the girl read out the words  from the lockers she came to know that the code is the first five lockers touched only twice.

The girl knew that the code then has to be 2, 3, 5 ,7 and 11. Since the lockers to be touched twice will be prime numbers with only two factors -1 and the number itself. So finally uncle’s niece gets her inheritance.

The End:

Suo enjoyed listening to today’s show and decides to ask the amazing riddle to her friends.

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