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Let's say we have 100 light bulbs.

Well, let me actually just draw them.

So I have one light bulb there.

I have another light bulb.

And I have 100 of them.

100 light bulbs.

And what I'm going to do is — Well actually, before I even

start turning these light bulbs on and off, let me let

you know that they are all off.

So they start off.

Now, the next thing I'm going to do is I'm going to number

the hundred light bulbs.

I'm going to number them one through 100.

So the first light bulb is light bulb one.

The second light bulb is light bulb two.

All the way to light bulb 100.

And what I'm going to do is, first I'm going to go and I'm

going to switch essentially every light bulb.

So if they all start off, I'm going to turn them all on.

So let me do it.

So on my first pass — let's call this pass one — so in

pass one, I'm going to turn all of these on.

On, on, on.

They're all going to be turned on.

On.

And then in pass two, what I'm going to do is I'm going to

switch only every other light bulb.

So, for example I'll say, OK, I won't switch light bulb one.

I'll only switch light bulb two.

So light bulb one will stay on.

Light bulb two will be off.

Light bulb three will be on.

Light bulb four will be off.

And so essentially, every light bulb — if you look at

their numbers — that is a multiple of

two, will be switched.

So 100 will be switched, so that'll also be off.

Then I'm going to come — and ignore this right here — and

I'm going to switch every third light bulb.

So what's going to happen?

This one's going to — let me switch colors arbitrarily —

this one's going to stay on.

This one's going to stay off.

And I'm going to switch the third light bulb.

So this one was on.

Now this one will be off.

The fourth light bulb will stay off, because I'm not

touching it.

The fifth light bulb would have been on

and it'll stay on.

Now the sixth light bulb, in this case we switched it off,

and now it'll be on again.

But I think you get the point.

Every third light bulb, or if we look at the numbers of the

light bulb, every numbered light bulb would that is a

multiple of three is going to be switched.

And if it was a multiple of three and two, it would have

been switched on the first time and then

off the second time.

But I think you're getting the point.

But what I'm going to do is, I'm going to do 100 passes.

So the first pass, I switch every light bulb.

They all started off, so they're all going

to be turned on.

The second pass, I switch every other light bulb or

every second light bulb.

The third pass, I do every third one, or that's a

multiple of three.

And my question to you is, after 100 passes, how many

light bulbs are still on?

Or how many are on, period?

And that is the brain teaser?

How do you figure out, of the hundred, which ones

are going to be on?

You should be able to do this in your head.

You don't have to make an Excel spreadsheet and actually

do all the on and off switches.

So the first question is, how many of these are going to be

on after I do 100 passes?

And just to make it clear, what's the 100th

path going to be?

Well, I'm only going to switch every 100th light bulb.

So whatever this light bulb was already doing, I'm just

going to switch it.

If it was off, it'll come on.

If it was on, it'll become off.

So the first question is, how many of these are going to be

on after 100 passes?

And then the bonus question is, which of these

are going to be on?

Well, let me actually just draw them.

So I have one light bulb there.

I have another light bulb.

And I have 100 of them.

100 light bulbs.

And what I'm going to do is — Well actually, before I even

start turning these light bulbs on and off, let me let

you know that they are all off.

So they start off.

Now, the next thing I'm going to do is I'm going to number

the hundred light bulbs.

I'm going to number them one through 100.

So the first light bulb is light bulb one.

The second light bulb is light bulb two.

All the way to light bulb 100.

And what I'm going to do is, first I'm going to go and I'm

going to switch essentially every light bulb.

So if they all start off, I'm going to turn them all on.

So let me do it.

So on my first pass — let's call this pass one — so in

pass one, I'm going to turn all of these on.

On, on, on.

They're all going to be turned on.

On.

And then in pass two, what I'm going to do is I'm going to

switch only every other light bulb.

So, for example I'll say, OK, I won't switch light bulb one.

I'll only switch light bulb two.

So light bulb one will stay on.

Light bulb two will be off.

Light bulb three will be on.

Light bulb four will be off.

And so essentially, every light bulb — if you look at

their numbers — that is a multiple of

two, will be switched.

So 100 will be switched, so that'll also be off.

Then I'm going to come — and ignore this right here — and

I'm going to switch every third light bulb.

So what's going to happen?

This one's going to — let me switch colors arbitrarily —

this one's going to stay on.

This one's going to stay off.

And I'm going to switch the third light bulb.

So this one was on.

Now this one will be off.

The fourth light bulb will stay off, because I'm not

touching it.

The fifth light bulb would have been on

and it'll stay on.

Now the sixth light bulb, in this case we switched it off,

and now it'll be on again.

But I think you get the point.

Every third light bulb, or if we look at the numbers of the

light bulb, every numbered light bulb would that is a

multiple of three is going to be switched.

And if it was a multiple of three and two, it would have

been switched on the first time and then

off the second time.

But I think you're getting the point.

But what I'm going to do is, I'm going to do 100 passes.

So the first pass, I switch every light bulb.

They all started off, so they're all going

to be turned on.

The second pass, I switch every other light bulb or

every second light bulb.

The third pass, I do every third one, or that's a

multiple of three.

And my question to you is, after 100 passes, how many

light bulbs are still on?

Or how many are on, period?

And that is the brain teaser?

How do you figure out, of the hundred, which ones

are going to be on?

You should be able to do this in your head.

You don't have to make an Excel spreadsheet and actually

do all the on and off switches.

So the first question is, how many of these are going to be

on after I do 100 passes?

And just to make it clear, what's the 100th

path going to be?

Well, I'm only going to switch every 100th light bulb.

So whatever this light bulb was already doing, I'm just

going to switch it.

If it was off, it'll come on.

If it was on, it'll become off.

So the first question is, how many of these are going to be

on after 100 passes?

And then the bonus question is, which of these

are going to be on?

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