There are 104 students at The School of Hat Design. There are 104 gerbils in a row out on the playfield. Shockingly, none of them is wearing a hat. So student 1 goes by and puts a hat on every gerbil. But student 2 goes by and slyly removes the hat from every other gerbil, starting with gerbil 2. Then student 3 goes by and changes the hat state of every third gerbil, starting with gerbil 3. (That is, if the gerbil is wearing a hat, he removes it. If the gerbil has no hat, he supplies a hat.) Student 4 goes by and changes the hat state of every fourth gerbil, starting with gerbil 4.
Imagine that this continues until all 104 students have followed the pattern of student N changing the hat state of every Nth gerbil, starting with gerbil N.
At the end, how many gerbils will be wearing a hat?
This is also known as the "locker problem"
The number of gerbils who will be wearing hats at the end will be number of perfect squares between 1 -104 inclusive = 10
To see this...let's suppose there are only 4 gerbils and 4 students
H = hat on O = hat off
Sudent 1 H H H H
Student 2 H O H O
Student 3 H O O O
Student 4 H O O H
Note that after student 4 passes......the number of gerbils wearing hats are just the perfect squares from 1-4 inclusive = 2
This result can be generalized to any N