Brain Teaser No 3

quote:

Sorry I thought I did - I'll try again.

You did, but I thought Steve might like to have another go first........hope you didn't mind too much?

I think a lot more people will be able to follow your later explanation, which is very clear. Nicely done.

Cheers

Don

quote:

Originally posted by Don Atkinson:
.....hope you didn't mind too much?


Don

Not at all - at least we're out of that fuel-less desert where I was failing miserably

Ian

OK a couple of quick and easy ones (well, with you two around they all seem quick and easy........)

Missile interception!!

Two missiles are heading directly towards each other (head-on), one at 8,500 mph the other 21,500 mph. They start 1,760 miles appart.

How far appart are they one minute before they collide?

Cheers

Don

How Old is the Red Rose City ? (and why is this an unlikely tale?)

Two professors were having drinks in the faculty bar, one a professor of English, the other of mathematics.

The English professor spoke "Its curious how some poets can write one immortal line, and nothing else of lasting value. Take John William Burgon? for example. A total failure, who nobody reads, yet he penned one of the most marvellous lines in English poetry 'A red-rose city, half as old as Time'.

The maths prof, who liked improvising brain teasers came back almost instantly with...

"A red-rose city, half as old as Time
One billion years ago the city's age
was just two fifths of what time's age will be
a billion years from now, Can you compute
how old the crimson city is today?"

Can you help the poor English professor out?

Cheers

Don

If my arithmetic is right the missiles are 500 miles apart one minute before they collide.

And if I've understood correctly the Crimson city is 7 billion years old.


Ian

quote:

If my arithmetic is right the missiles are 500 miles apart one minute before they collide.

And if I've understood correctly the Crimson city is 7 billion years old.

Both your arithmetic is right and you correctly understood the Crimson city.

8 minutes is an embarrasingly short time between posts on this tread, especially when that's between the question and the the answers..........

Cheers

Don

Here is one about probabilities. A facetious Naim dealer has hidden a CD5 blanking plug under one of three cups that are placed upside down. If you guess correctly under which cup the plug is, you get it for free. I know I want such a plug.

The dealer asks you to choose a cup by placing your hand on it without moving the cup.
Once you have placed your hand on the cup of your choice, the dealer picks up another cup to show that there is no plug under it.

Then the dealer offers you the choice to change your mind and opt for the remaining cup instead of the one you've got your hand on.

Do you improve your chances by switching your choice to the other cup, or not?

I don't think anyone gave a general solution for the tibetan walk. I would start by drawing a rectangle, or more general, a trapezoid.

quote:

Do you improve your chances by switching your choice to the other cup, or not?

As my wife does - always change your mind.

I'll explain later, but your chances will ******.

Cheers

Don

I haven't quoted the better odds, to give others a chance to think about this one.

You should change your mind to improve your chances of getting the blanking plug...

When you had your first choice you had a 1 in 3 chance of picking the correct cup. Should you change your mind once one cup has been removed you have a 1 in 2 chance of being lucky. If you stay with your original choice you're stuck with your original probability of a 1 in 3 chance. That's probabilities for you.

I guess this means that you should always swap your box on "Deal or no Deal" should you ever get offered the chance!

Dear boys

Let's cut straight to the heart of the matter.

If you really wanted to get married and move in with your girlfriend, you wouldn't be having all these doubts. Surely your feelings are sending you a warning? It's quite possible to love someone but not to want a permanent, committed, live-in relationship. If that was what you both wanted, there'd be no problem, but it doesn't sound like you're convinced you'd be happy together.

An age-gap isn't wrong or right. It just is. The age-difference doesn't have much of an impact now, while you're both relatively young. If you were to get married, think of things in twenty years time. She'd be in her sixties, already a pensioner, and you'd still be at work. You might be lucky to have an elderly wife in good health but you might have to take care of her if she had a stroke, say, or advanced arthritis. How would you manage this while working full-time? There are relationships with a bigger age-gap than this which have worked out well. As for your parents, they realise that you are now an adult and they accept you as you are. The fact that you were hoping they'd help you make up your mind is another sign that you're not sure. If you were sure you wanted to be with this woman, the age-gap would be a small price to pay.

But you're not sure, and that's OK. Your feelings are sending you a message. Are you willing to go along with them? Don't your feelings matter as much as your girlfriend's? You could decide to call it a day now, or give yourself a cut-off point of three to six months in which to make up your mind. The choice is not between this woman and no-one. There are plenty of other women out there with whom you could be happy and not have any doubts about it. In the same way there are plenty of men with whom your girlfriend could go on and form a good relationship. If you are going to split up, wouldn't it make more sense to do it sooner rather than later? That way you both have the maximum chance of finding someone you are sure about.

I wish you all a good relationship whether that's with each other or someone else. Good luck!

quote:

Originally posted by Stubby:
You should change your mind to improve your chances of getting the blanking plug...

When you had your first choice you had a 1 in 3 chance of picking the correct cup. Should you change your mind once one cup has been removed you have a 1 in 2 chance of being lucky. If you stay with your original choice you're stuck with your original probability of a 1 in 3 chance. That's probabilities for you.

What if, having had one cup uncovered, you were given the chance of selecting either of the remaining 2 cups and chose the one you origianly picked? Are the odds now 1 in 2 rather than 1 in 3?

We applaud Auntie Rasher for going straight to the heart of the matter, Don gets the prize for his phenomenal feat of self restraint, Stubby gets the medal for effort and commitment, and Nigel will be recommended for his ruthless peer reviewing skills. I hope I didn't leave anyone out. Meanwhile there is an additional challenge. Explain in such a way that noone is left in doubt in the end.

quote:

Explain in such a way that noone is left in doubt in the end.

Before attempting that feat (please don't wait for me, just have a go - anybody), let me add, that in my calculation, if you change your mind, your chances of winning are actually two-thirds.

I didn't quote that before, (substitute "double" for the six* in my previous post) - in case it gave too much away.

Cheers

Don

Three cups (x, y, z) and one puck.

The probability of finding the puck depends on how many cups you are allowed to choose:-

a. If you can choose all three cups, the probability of finding the puck is 1.0 (ie a dead cert)
b. If you can choose two cups, the probability is 0.67
c. If you can choose one cup, the probability is 0.33
d. If you can't choose any cups, the probability is 0

By choosing one cup, say x, and sticking with it come what may, you are putting yourself into option c with a probability of 0.33

If you are able to choose two cups, say y and z, you are putting yourself into option b with a probability of 0.67

You can deliberately force the cup-shuffler to reveal both y and z, by initially nominating cup x. The shuffler will reveal either y or z., assume its y, (you have now got your first cup revealed). When given the choice of x or z, choose z. (you have now got both y and z revealed, which was your strategic aim)

What was in your mind, or what was in the mind of the shuffler, isn't relevent.

Debate.........

Cheers

Don.

If you had the choice of three, then each is worth a third of a certainty. Left with two the two remaining each one has a half chance of certainty. If one knows all but one of the uncertainties, then one has certainty.

Fredrik

quote:

Left with two the two remaining each one has a half chance of certainty.

Fredrik, this is the bit that most people have difficulty with.

Try the following excercise first.

Assume there were ten cups and one puck, well shuffled.
Assume that the shuffler knows where the puck is.
Assume that you know the shuffler will reveal eight empty cups before offerring you the opportunity to change your mind.

What would you do?

The correct answer is to change your mind.
The reason is that your chance of winning is 0.9

Once this concept has been graspped, we can move towards the puzzle as published by Alexander.

Cheers

Don

Perhaps a helpful way to think of this perplexing puzzle is to focus on the probability of the puck being the other two cups (2/3) , as opposed to the one you have chosen (1/3).

Now when one of the cups is revealed as empty there is a 2/3 probability that the puck is under the other unrevealed cup. And still the same probability of 1/3 that it is under your chosen cup. So it is better to switch.

Ian

Ian,

I think that's a nice description

Cheers

Don

I can hardly add anything here that hasn't been said yet.
You need familiarity with probability theory in order to be 'misled' into using a wrong model.
Impeccable logic is not very effective at changing your mind once you're in the wrong model.

Actually going through the physical motions of the 10 cup example might
appeal to the intuition well enough to conclude that the chances for the two remaining cups are wildly different.
The 'words only' alternative can be an exaggerated 1000 cup example.
When you pick one cup, the chance of picking the right one is negligable so you can assume you've got the wrong cup.
The dealer then removes 998 other cups. Intuitively it's virtually certain that the plug is under the remaining cup.
The actual choice the dealer offered you then was "do you think the plug is under your cup or under one of the 999 others?".

Segue to an explanation to bring intuition and logic together for the 3cup model - as Don proposed.

It would be nice to know if everyone is convinced now of the 1/3 vs 2/3 solution.

Dear Alexander and Don,

It might be 9,998 cups removed with two remaining. The chances between the two remaining cups left is still 50:50!

ATB from Fredrik

PS: The chance between any number of cups is statistically equal. Reduce the number of cups and the chance grows [four leads to a quarter, three leads to a third, two leads to a half and one is certain], and each is still statisticaly equal. Any other logic, in a game of chance, is the road that leads to bankruptcy...

Sorry Fredrik but what seems to you simple logic and irrefutable is in fact not right.

The trouble is that probability is not an immutable thing but changes with increased knowledge. Remember the man removing the cups is not doing so at random - he is removing cups he knows to be empty.


In the beginning with 3 cups we have no knowledge about where the puck is and so the probability is 1/3 that it is in any given cup. and 2/3 that it is NOT in that cup.

Now we pick a cup and nothing changes one time in three we'll have picked the right one and twice in 3 we'll have chosen the wrong one.

Then the other chap removes a cup WHICH HE KNOWS TO BE EMPTY. This provides you with extra information about the situation. The probability that you were right is still exactly as before 1 in 3 and the probability that you were wrong is still 2 in 3. But now crucially there is only one way in which you can be wrong, namely that the puck is under the remaining other cup. So the probability that the puck is under the other cup is 2 in 3 - so you're better off if you switch.

Compare this with the situation where the chap removed a cup from the 3 but you were not told whether the one he removed contained the puck or not. Then it makes no difference whether you change your choice as the chances are then still 1 in 3 for both remaining cups.

Hope this helps.

Ian

quote:

Left with two the two remaining each one has a half chance of certainty.

Fredrik, this is the bit that most people have difficulty with.

Try the following excercise first.

Assume there were ten cups and one puck, well shuffled.
Assume that the shuffler knows where the puck is.
Assume that you know the shuffler will reveal eight empty cups before offerring you the opportunity to change your mind.

What would you do?

The correct answer is to change your mind.
The reason is that your chance of winning is 0.9

Once this concept has been graspped, we can move towards the puzzle as published by Alexander.

Dear Fredrik,

I set out the above on the previous page. It is specifically designed to take us through this brain teaser, one step at a time. Have you had a chance yet to work through it?

In addition, you might like to try a practical experiment with three cups, a friend and a puck (you had better read that again carefully!!!) Make sure the friend knows where the puck is and make sure he ALWAYS shows you a cup WITHOUT the puck.

Run the test say thirty times where you always stick with your first choice (you will find the puck about 10 times

Then run the test another thirty times but always change your mind (you will find the puck about 20 times)

Later, I'll describe how to run this test when your friend doesn't know where the puck is.

Cheers

Don

Diagonal AC

The rectangle is inscribed within a quadrant.

Given the unit dimensions shown, accurately calculate the length of the diagonal AC.

Time timit is one minute

No need to post the length of the diagonal (just yet), simply post how long it actually took you - be honest!

Cheers

Don

Geometry..........

I can't concentrate when I'm shuddering.