SOLUTION: There are 210 control panels lined up in a row in a long room. Each control panels has its own switch and is currently switched off. •The room has an entry door and an exit door

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Question 975412: There are 210 control panels lined up in a row in a long room. Each control panels has its own switch and is currently switched off.
•The room has an entry door and an exit door. There are 210 engineers lined up outside the entry door. Each control panel is numbered consecutively from 1 to 210. So is each Engineer.
•Engineer No. 1 enters the room, switches on every control panel, and exits. Engineer No. 2 enters and flips the switch on every second control panel (turning off control panels 2, 4, 6, …). Engineer No. 3 enters and flips the switch on every third control panel (changing the state on control panels 3, 6, 9, …). This continues until all 210 Engineers have passed through the room.
What is the final state of control panel No. 144?
•How many of the control panels are left working after the 196th has passed through the room?
•How many of the control panels are left working after the 210th has passed through the room?
•Make a list of the panels which are turned off after each engineer passes through. (Excel solution only
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


Each engineer will touch those and ONLY those switches where the number of the switch is divisible by the engineer's ordinal number. Every integer has an even number of even divisors, except perfect squares which have an odd number of even divisors.

If is a divisor of , then . But that implies that as well. Hence, divisors come in pairs, except in the case of a perfect square where there is one value of that divides so that there is a such that AND creating the odd divisor.

Therefore all of the switches with non-perfect square numbers will be touched an even number of times, and since they started in the off position, they will end up in the off position.

But the perfect square numbers...

John

My calculator said it, I believe it, that settles it