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100 Bulbs

Puzzle:

There are 100 light bulbs lined up in a row in a long room. Each bulb has its own switch and is currently switched off. The room has an entry door and an exit door. There are 100 people lined up outside the entry door. Each bulb is numbered consecutively from 1 to 100. So is each person. Person No. 1 enters the room, switches on every bulb, and exits. Person No. 2 enters and flips the switch on every second bulb (turning off bulbs 2, 4, 6, …). Person No. 3 enters and flips the switch on every third bulb (changing the state on bulbs 3, 6, 9, …). This continues until all 100 people have passed through the room.

What is the final state of bulb No. 64? And how many of the light bulbs are illuminated after the 100th person has passed through the room?

For Solution SCROLL DOWN...

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Solution:

Light Bulb 64 is on. The total number of bulbs which are on including #64 is 10.

First think who will operate each bulb, obviously person #2 will do all the even numbers, and say person #10 will operate all the bulbs that end in a zero. So who would operate for example bulb 48: Persons numbered: 1 & 48, 2 & 24, 3 & 16, 4 & 12, 6 & 8 ........ That is all the factors (numbers by which 48 is divisible) will be in pairs. This means that for every person who switches a bulb on there will be someone to switch it off. This will result in the bulb being back at it's original state.

So why aren't all the bulbs off? Think of bulb 36:- The factors are: 1 & 36, 2 & 13, 6 & 6 Well in this case whilst all the factors are in pairs the number 6 is paired with it's self. Clearly the sixth person will only flick the bulb once and so the pairs don't cancel. This is true of all the square numbers.

There are 10 square numbers between 1 and 100 (1, 4, 9, 16, 25, 36, 49, 64, 81 & 100) hence 10 bulbs remain on.

35 comments:

Jayant said...
This comment has been removed by the author.
Jayant said...

All perfect squared bulbs will be ON
Reason : Because they have odd number of factors,so they go from ON->OFF->ON...->ON

So, 64th bulb : ON
No. of bulbs ON = 10 (1,4,9,16,25,36,49,64,81,100)

saahith said...

yes.......64th bulb ON.....and all perfect square bulbs ON

Anonymous said...

64 - On
Toatl 10 bulbs on

pravesh said...

yes ,above answers are right.

Anonymous said...

factors of 64=1,2,4,8,16,32,64
ie on-off-on-off-on-off-on
so bulb no.64 will be on

ankit_blacki said...

only perfect squares have odd number of factors...
thus.. all perfect squares till 100 will change their state.
hence, 10 on and 90 off.
since 64 is a perfect square, it will be on

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DavidJ said...

I really enjoyed this nice mix of logic and mathematics.

I got bulb 64 is ON and at the end there are 10 bulbs on (all perfect squares).

To figure out whether bulb 64 was on or off I looked at which people would hit bulb 64's switch:

Person 1 (on)
Person 2 (off)
Person 4 (on)
Person 8 (off)
Person 16 (on)
Person 32 (off)
Person 64 (on)

And after person 64 no one would touch bulb 64's switch. Therefore it's on.

Now, the REAL question of this puzzle is how many lights are on! I deduced that bulbs that were hit an odd number of times would be off (like bulb 64). So the question rephrased is how many bulbs were hit an odd number of times?

I was still kind of stumped at this point on how to quickly figure it out (I could do it individually for 100 cases, but that wouldn't be solving the puzzle correctly), so I looked at what bulbs would be on if there were just 10 bulbs.

I got that bulbs 1, 4, and 9 would be on.

1, 4, and 9?!? Those are perfect squares! I looked at a few cases. 16 would be on, but 15 and 14 would be off. 64 would be on, but 66 would be off, etc...

So I reasoned that all bulbs that were numbered with a perfect square would be on. That's 10 bulbs.

Bulbs 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.

Great puzzle. It made me feel dumb until I figured it out!

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