Brain Teaser No 1

Children!

Mrs Don and myself enjoyed a weekend break recently curtisy Hilton Hotels special two-night DBB from £34.95pppn.

During the weekend we met another couple likewise separated from the kids for the first time ever!. Of course it was less than 5 minutes before Mrs Don was telling Mrs Pete that we had three girls and no boys. Talk about a busman's holiday! Well Mr Pete already knew I'd posted the odd brain teaser on the www.co.uk thingy so quick as a flash beat his wife's reply by saying :-

"Well, we have two children, one at least of which is a girl. What's the chances of the other being a girl as well?"

Now we have established that within a gnat's whisker, the chances of boys and girls are 50/50 and we are ignoring the real life factors of three girls in family so that the chances of a boy/girl are independent events.

So, "What's the chances of the other being a girl as well?"

Cheers

Don

Can we leave the 'well done' or 'try again' acolade for a little while? I've noticed on probabilities, that a few member's intuition sometimes takes over. So lets wait an hour or two, and see if anybody cares to challenge your logic!!

Cheers

Don

Going back to Bam's Books.

I've just realised ( I think!) that the series 1/1 + 1/2 + 1/3 + ....1/n is is not convergent. This would imply, that by stacking enough books, you should be able to cantilever any distance.

Or am I talking nonsense?

Cheers

Don

I have a 12" globe. Sitting (ok, balanced) on this globe I have a large, juicy (but firm) orange, an apple and a lemon. The orange is a perfect sphere of radius 4" (well I did say large!). The apple is 3" radius and the lemon 2" radius. All three just touch each other.

I have a bag of large cherries, also firm, perfect spheres. I want to sit a single cherry on the globe, within the group of orange, apple and lemon, so that it just touches the globe and the other three pieces of fruit.

What size (radius) cherry do I need?


Now I know this teaser has been published before, but the details were spread over two pages and newcomers such as Scott might find it difficult to put the bits together. So I thought i'd make it a bit easier.

Cheers

Don

So I thought i'd make it a bit easier

You could try putting the orange, apple and lemon on a dead flat table instead of on the globe.

This won't help you solve the problem with the globe, but it might be a bit easier to solve, full stop.

Perhaps!

Cheers

Don

...and assuming that having a child of one sex does not have any bearing on the sex of subsequent children (unless you have some proof that it does) then the chances of the second child being another girl are equal to those of it being a boy. That does not mean 50:50 or 1/2 necessarily. I am not a mathematician but in a series of heads/tails do you not multiply each successive probability so that the chances of either heads/tails is 1/2; x 1/2 (1/4); x 1/2 (1/8)etc?

cheers

Nigel

Falling books.

I have found a theoritical solution to the book issue that will allow the books to protrude 11/12 times the length of the books over the edge. This is the point at which the books will be at the point of falling off.

Solution was reached by assuming the top book was at the maximium extension from the second, the second was at the maximium extension from the third (accounting for the position of the top book). This gives the 1/2y rule that Don generated. Therefore there is no limit to the extension from the edge of the desk.

Now on to the fruit cocktail!

Matthew

You said,

I have found a theoritical solution to the book issue that will allow the books to protrude 11/12 times the length of the books over the edge. This is the point at which the books will be at the point of falling off.


Which agrees with what I said on page 33

The first three terms add up to 192.5mm, or 11/12*L, which is pretty close to my first 193mm experimental result!


Great minds..........etc

Cheers

Don

Fred and Wilma have two children, the youngest of whom is a boy. What are the chances that both children are boys? Is the answer the same as the answer to Don's "two girls" puzzle above?

Is the answer the same as to Don's two girls?

Simple answer is "No !"

More involved answer is its 0.5.

Will explain later, unless someone else explains first.

Nice little twist, that was, Scott.

Cheers

Don

Don,

Using a combinaton of very ugly equations and the excel Goal Seek function I have arrived at the result of a cherry with a radius of...

0.5431079 inches

There could of course have been an error in my workings and I will be happy to concede to a more rigorous calculation.

I used only pythagoian principles.

Matthew

Didn't Pythagoras drown one of his followers who pointed out an error in his ways?

Probably something to do with the irrationality of Pi.

Paul

Matthew,

Using a combinaton of very ugly equations and the excel Goal Seek function I have arrived at the result of a cherry with a radius of...

0.5431079 inches


Was this with the fruit on the globe, or the trivial example, he-he-he!!!, with the fruit on the flat table?

Cheers

Don

Nigel C

I am not a mathematician but in a series of heads/tails do you not multiply each successive probability so that the chances of either heads/tails is 1/2; x 1/2 (1/4); x 1/2 (1/8)etc?


This part of your post is absolutely right AFAIK.

So that after two tosses of a fair coin your statement shows that the probability of any particular combination of HH; HT; TH; TT is 1/2*1/2=1/4 and given that HH; HT; TH; TT are the ONLY possible outcomes, the chances of any one, is one in four. QED (well, not really QED, but good enough!)

The next bit is the difficult bit.....I am more persauded by Scott's logic regarding the two girls......but I'm no expert.

Cheers

Don

Paul R,

Didn't Pythagoras drown one of his followers who pointed out an error in his ways?

Probably something to do with the irrationality of Pi.

For some strange reason, your post suddenly made me think of Bam. Can't imagine why !!

Cheers

Don

quote:

---

Well, we have two children, one at least of which is a girl. What's the chances of the other being a girl as well?

---

Surely 0 or 1? There's no chance involved since these particular children already exist and have sexes.

Otherwise the 'chances' seem to be 1/3. Of the set of two children families where one child is a girl 1/3 of the set will have two girls, 1/3 will have old girl young boy and 1/3 old boy young girl.

Paul

Paul R,

Surely 0 or 1? Absolutely right, of course.

But I think the second part of your answer is pretty good, nonetheless !

Cheers

Don

When I wrote 12" globe, I had 12" radius in mind. If anybody has already done the sums with a 12" diameter, just say so when you post your answer, and I will check it out..........

Cheers

Don

i have an expression for d which doesnt involve double square roots. see attached. elegant enough bam?

enjoy

ken

Ken,

I thought it would involve hyberbolic functions (distant memories of A-level further maths) but didn't have all the equations in front of me.

I hear the tinkle of glasses and the popping of corks!

Don,

It was meant to be the globe answer but I think I made an error, am working on the correct solution.

Matthew

OK here is my latest attempt.

Solution was reached by setting up pyramid in cartesian coordinated space with points at
(0,0,0)
(16,0,0)
(5/2,(171^0.5)/2,0)
(3,15/(19^0.5),(288/19)^0.5)

These had to be calculated.

I then solved the four equations for the length between the centre of cherry and points of pyramid using simple pythgorian principles and solving the four equation with four variables (I used excel for this bit).

Solution is now

Cherry radius = 0.836326148038882

I have even developed a spreadsheet that can find the size a sphere that will just fit inside any other four spheres, think I better do some real work now!

cheers

Matthew

[This message was edited by Matthew T on WEDNESDAY 22 May 2002 at 10:39.]

Ken,

I thought it would involve hyberbolic functions (distant memories of A-level further maths) but didn't have all the equations in front of me.


i should actually carry the plus or minus on the solution for d in term of x all the way -- but this doesnt make any difference to the solution

I hear the tinkle of glasses and the popping of corks!

lets wait till we hear from bam...

enjoy

ken

A little teaser to see if all this green house gas stuff is hot air or not!

Electro-magnetic radiation (EMR) is absorded by molecules by exciting the bonds between atoms, in the case of carbon dioxide it is the exciting of a carbon oxygen double bond. This absorbs EMR of a rather specific frequency.

The question

What proportion of the earths atmosphere would need to made up of any one gas to ensure that virtually all the ERM that can be absorbed by that gas is absorbed. Please state assumptions.

Matthew

Ken C,

I've been away for a couple of days, so I've only just seen your solution to Bam's Ladder.

Looks elegant enough for me, although I haven't had time to check it out yet for accurcay !!

You've been very quiet these past few weeks. I hope you were listening to music rather than spending sleepless nights with libraries of maths books!!!!

Well done.

Cheers

Don

Solution is now

Cherry radius = 0.836326148038882


Congratulations. Spot-on (although I only checked the answer to 6 decimal places!) This business of congratulations seems to be getting endemic tonight!!

I have even developed a spreadsheet that can find the size a sphere that will just fit inside any other four spheres, think I better do some real work now!

So, just to test your spreadsheet (which I must say, I am VERY impressed to hear about,) could you do a (all radii) 4" sphere, a 3" sphere, a 2" sphere and a 1" sphere and find the radius of the cherry that sits inside that group of four.

Then see my next post.

Cheers

Don

Matthew T,

Going back to the 4" radius orange; 3" apple and 2" lemon, and this time putting them inside a fruit bowl, any idea of how big the cherry would be that just touched the bowl and the other three bits of fruit?

Usual rule of perfect spheres etc.

Appologies if this is getting too much like an exam or an inquisition!!!!

Cheers

Don