Kogorous Riddle Thread and eworms Deduction Thread

[Rein]

Well, at first he put: 'No Letters used' so I'm assuming he just mistook the S for the E, so I don't really mind who goes next  :D

[miyano_shiho]

Woah, this thread's buried down.
*Just digging it back up*

Riddles, anyone?

[Rein]

Just to say I didn't come up with this XD

A man was found murdered on Sunday morning. His wife immediately called the police. The police questioned the wife and staff and got these alibis: The Wife said she was sleeping. The Cook was cooking breakfast. The Gardener was picking vegetables. The Maid was getting the mail. The Butler was cleaning the closet. The police instantly arrested the murderer. Who did it and how did they know?

[Silver1412]

The Maid. There's no mail on Sunday.

[Rein]

Haha, yeah.
Not a very difficult riddle XD

At any rate, it's your turn Neko-chan :D

[Silver1412]

Hmm... I don't really know any good riddles but... Oh well. I'll give you guys a really simple one.

Sometimes you see me,
sometimes you don't.
I'm in the sky, yet I don't glow.
What am I?

[miyano_shiho]

Spoiler:

Uhm... let's see... a cloud, perhaps?

[Borealis]

moon?

[Silver1412]

Spoiler:

Uhm... let's see... a cloud, perhaps?

Correct. Sorry for not posting earlier. And as I said it was an easy one.

[Felicita]

Shiho, your turn. D:

[miyano_shiho]

Oh sorry, you were waiting. :x I'm still looking for a good one but if you (or anyone else) want to post one, you could do so. I missed this thread too.

[Felicita]

But I don't have any riddle. D:

[miyano_shiho]

Okay, here's one.

There are 100 doors. One day, someone walks through and opens every door. Then he walks through it again and closes the even-numbered doors. On the third trip he stops at every third door and closes the door if it’s open or opens it if it’s closed. And so on: On the nth trip he stops at every nth door, closing an open door or opening a closed one. At the end of the 100th trip, which doors are open?

[Kogorou]

1 4 9 16 25 36 49 64 81 100

Only the doors with square numbers can be open

[miyano_shiho]

1 4 9 16 25 36 49 64 81 100

Only the doors with square numbers can be open

Correct.

Spoiler: Solution

Listing the visits for each door would be helpful.

Door: Visited on trip(s)
1: 1
2: 1, 2
3: 1, 3
4: 1, 2, 4
5: 1, 5
6: 1, 2, 3, 6
7: 1, 7
8: 1, 2, 4, 8
9: 1, 3, 9

It can be inferred that the visits are actually the divisors of the number of each door.
An even total number of visits (divisors) is a closed door, while an odd one is an open door.
The only numbers that have an odd number of divisors are perfect squares. To support this, here's an example:
15: 1, 3, 5, 15
16: 1, 2, 4, 8, 16
Note that a particular pair of divisors multiplied to each other results in the number (1x15, 3x5, 1x16, 2x8).
But perfect squares include one divisor multiplied by itself (4x4), which means that the total number of divisors for perfect squares will always be odd.

Forgive me for the long (and probably confusing) explanation. Every math problem deserves a sound proof.  ;)

Kogorou, your turn.