Containment

[limelight]

broken

Quite possible if the students are French, but the doors are sturdy, and the students are all well behaved Aussies. I know of one girl school in Melbourne which carried out the experiment.

[limelight]

A school has exactly 100 students and one hundred lockers. The first student who enters the school opens all the locker doors. The second student enters and shuts every second door. The third student enters and toggles every third door (ie if it is shut he opens it, and if it is open he shuts it). The fourth student enters and toggles every fourth door. And so on until all 100 students have passed through.
What is the state of the doors after the 100th student has passed through?

Ok here's the answer: doors 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 will be open, all others will be shut

But can you explain why?

[limelight]

Before I give the full answer to the 1st puzzle, I want to let you think more about it, I am adding a 2nd puzzle.

Before you give the answer you think is obvious, I suggest you experiment, for instance play the game with your son or daughter, after playing 20 or 30 times one way, then the other, compare the statistics... you may be surprised.

So here we go, this is a game show like we see sometimes on TV. On the show there are 3 doors, the game host tells you that behind one of the doors is a sports car and if you guess the correct door, the car is yours. So you pick one of the doors.

To increase the suspense, the host opens one of the other 2 doors showing there was nothing behind it, he then offers you the possibility to change your mind. What should you do to maximise the probability of winning?

a) You stick with your original choice, believing that the odds are now 50/50 and there is no point in changing, or
b) You change door believing your odds of winning will double.

Which of a) or b) is the correct answer?

I strongly recommend you play the game with a kid, say 15 times without changing the 1st choice and 15 times always changing, and compare which run scored the most wins.

By the way if my puzzles don't interest you, don't hesitate to tell me to shut up

[mAKi]

non non continue!
je suis encore en train de réparer les portes des casiers

Ok here's the answer: doors 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 will be open, all others will be shut
But can you explain why?

y a une histoire d'intervalle
donc entre chaque porte ouverte y a 3, 5, 7, 9, 11, 13, 15, 17 et 19 donc +2 à chaque fois
ce qui s'écrit sous la forme:



et voilà

[mAKi]

c'est quoi la marque de la voiture?

si tu changes de porte y a toujours une chance sur 2 de repartir à

[limelight]

non non continue!
je suis encore en train de réparer les portes des casiers

Ok here's the answer: doors 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 will be open, all others will be shut
But can you explain why?

y a une histoire d'intervalle
donc entre chaque porte ouverte y a 3, 5, 7, 9, 11, 13, 15, 17 et 19 donc +2 à chaque fois
ce qui s'écrit sous la forme:



et voilà

Val! Since you did not define φ this makes no sense whatsoever. But your comment about the interval is interesting.

In fact the doors are visited by students whose number can divide the door number, for instance door 12 is visited by 1, 2, 3, 4, 6, and 12 this can be arranged in pairs, (1 12), (2 6), (3 4) by choosing the pairs whose numbers when multiplied give the door number. This pairing breaks down if the door number is a square, like 36 because while we can pair (1 36), (2 18) , (3 12) , (4 9) , we can't pair 6 with any number but himself (6 6) so the number of students visiting door 36 is odd and the door will be open at the end. and all the doors whose number is a square will suffer the same fate.

[limelight]

c'est quoi la marque de la voiture?

si tu changes de porte y a toujours une chance sur 2 de repartir à

Before the Covid-19 crisis, the car was a Lamborghini Urus, but now times are hard and it has been downgraded to a 2nd hand Citroën 2CV!

And your answer is incorrect, if you change door the odds of leaving on bicycle is only 1/3 not 1/2 as you assume.

[mAKi]

yes I understand your answer, but even with a doubled chance, I'm not sure I guess the right door

what's color of the car?
old 2 CV is very expensive nowadays

[limelight]

what's color of the car?
old 2 CV is very expensive nowadays

Colour? Well it's best described as "very rusty!"

[mAKi]

it was my first car