Lockers and Bulbs

Task 1 (Source: Illustrative Math)

The 20 students in Mr. Wolf's 4th grade class are playing a game in a hallway that is lined with 20 lockers in a row.

• The first student starts with the first locker and goes down the hallway and opens all the lockers.
• The second student starts with the second locker and goes down the hallway and shuts every other locker.
• The third student stops at every third locker and opens the locker if it is closed or closes the locker if it is open.
• The fourth student stops at every fourth locker and opens the locker if it is closed or closes the locker if it is open.

This process continues until all 20 students in the class have passed through the hallway.

1. Which lockers are still open at the end of the game? Explain your reasoning.
2. Which lockers were touched by only two students? Explain your reasoning.
3. Which lockers were touched by only three students? Explain your reasoning.
4. Which lockers were touched the most? 
   

 Task 2: (Source: Engineering materials)

You have 100 light bulbs lined up in a row. You walk past them and turn them all on in the first pass. Then, starting at the same end, you go along and switch every other light bulb off... so all even-numbered light bulbs are now turned off.

Now, you go back again and switch every third light bulb. But, any light bulb that is off is turned back on, and any light bulb that is on, is turned off.

You go back again and switch every fourth bulb, if it is on you turn it off and vice-versa.

Then you repeat this process for every fifth bulb, and every sixth bulb, and every seventh bulb... up to 100 passes.

So, clever clogs, how many light bulbs are on after 100 passes?