Another puzzle

Ah, the intellectual thread.

Here are a couple that I find quite intriguing:

a) Imagine a rope stretching around the equator of a symmetrical spherical earth and touching the surface at all points.  How much longer would the rope have to be such that it is now raised exactly one metre above the earth's surface at all points?

a) Imagine a rope stretching around the equator of a symmetrical spherical earth and touching the surface at all points.  How much longer would the rope have to be such that it is now raised exactly one metre above the earth's surface at all points?

Pi x 41 : - )

Units, please, Irving?  Units?

See me in my study after prep (whatever that is).

There are 4 marbles in a bag. One is red, one is yellow and the other two are blue. If I pick out two marbles at random, look in my hand and show you a blue marble, what are the odds that the other marble is also blue?

Just over 28% chance that the other marble is blue.

Having thought about this again, it doesn't matter if he shows you one of the marbles or not.

The chances of scooping 2 blues in a single dip into the bag is 1 in 6, and this is not altered by the fact that you are shown the colour of one afterwards.

I would say the chances of scooping two blues in a single swoop is 1 in 4. (R & B, Y & B, R & Y, B & . If after you have scooped two balls and show 1 and it is blue the chance of the second one being blue is 1 in 3.

There are 4 marbles in a bag. One is red, one is yellow and the other two are blue. If I pick out two marbles at random, look in my hand and show you a blue marble, what are the odds that the other marble is also blue?

Just over 28% chance that the other marble is blue.

Having thought about this again, it doesn't matter if he shows you one of the marbles or not.

The chances of scooping 2 blues in a single dip into the bag is 1 in 6, and this is not altered by the fact that you are shown the colour of one afterwards.

I would say the chances of scooping two blues in a single swoop is 1 in 4. (R & B, Y & B, R & Y, B & . If after you have scooped two balls and show 1 and it is blue the chance of the second one being blue is 1 in 3.

I prefer 1 in 5.

When you remove two balls at random, there are six possible combinations (R+Y, R+B1, R+B2, Y+B1, Y+B2, B1+B2).  On revealing the fact that one of the balls removed is blue, you eliminate the R+Y possibility, hence any of the remaining five combinations is possible, giving a 20% chance of each.

I disagree.  Imagine if one of the blue balls had a tiny white dot on it.  Do you still make R+B1 and R+B2 the same?

OK, now try with 100 blue balls.  Do you still make R+B1, R+B2 ... R+B100 equal to the same single combination?

What about the rope around the world question?

OK, so let me check, FOD ...

As we know, the original length of the rope is somewhere in the region of 40 million metres.  So, you are telling me that, by extending it by just over six metres (2pi) to around 40,000,006 metres, the entire rope will now be raised to a height of one metre above the surface of the earth?

*checks around the class for smirking from school chums*

Half of the way - Final Answer!

What about the rope around the world question?

OK, so let me check, FOD ...

As we know, the original length of the rope is somewhere in the region of 40 million metres.  So, you are telling me that, by extending it by just over six metres (2pi) to around 40,000,006 metres, the entire rope will now be raised to a height of one metre above the surface of the earth?

*checks around the class for smirking from school chums*

What about the rope around the world question?

OK, so let me check, FOD ...

As we know, the original length of the rope is somewhere in the region of 40 million metres.  So, you are telling me that, by extending it by just over six metres (2pi) to around 40,000,006 metres, the entire rope will now be raised to a height of one metre above the surface of the earth?

*checks around the class for smirking from school chums*

I am. I think it would be the same length if you were to wrap a string around a ping pong ball.

Excellent, FOD!  I like your style.  You are indeed correct ... although the result is somewhat surprising, don't you think?

Irving ... you would do well to take a leaf out of FOD's book.  You don't see him with his nose stuck in some popular culture magazine.

I was surprised when I worked it out, but I was also confident and strong!

Admirable qualities, FOD.

There may be a place available at the prefect table in the near future.  I'll keep you in mind.

Taking the diameter of the earth at 12,756km (Google) add 000 to make metres then adding two (one metre either side of the earth on the diameter) we get figures of 40074155.8 before raising the rope and 40074162.1 after raising it. So it would be 7 metres longer.
Your faithfully,
Mr A Nal.

The diameter is only lengthened by two metres (one at each end) the circumference of that diameter increases by 6.3metres (rounded up to 7) in my calculations.

Nur nur de nur nur :-)

Impressive that you two wrestle with this long after your chums have arrived at the correct answer, long after the exam has finished, long after the exam room has been darkened and closed for the night, long after the school has been pulled down and converted into a retail park.

Now, who's first up to tell me what's wrong with the faster runner/slower runner scenario?

What's wrong is it assumes the faster runner stops each time he reaches the next point.