Brain Teasers ? or 50 Years On........... ?

A rhetorical question is not a declarative statement with a question mark at its end. Is it?

Whatever!?!

I win.

Hi Mark,

Lionel's link and others, indicate that air bubbles in water, do not accelerate as they rise. I haven't experimented this myself, so I am relying on the integrity of web links.

Hi Winky,

Drag also depends on speed, usually speed squared. So any tendency for an air bubble to accelerate would be tempered to some extent by the speed it attained. In other words, it might very quickly reach a steady speed after detachment from its point of nucleation.

A bubble forming in a solution of carbonated water, might well be attracting more CO2 out of solution and into the bubble, thus rapidly adding to its volume and bouyancy, and thereby overcomming the speed-related drag.

Generally

I think we need a link to a few more sites with reliable authors describing audited experiments

If I got Lionels link right, bubbles in water do not accelerate, because they do not grow (grow=add carbon dioxide). This should only be true for non-carbonated water...

Yep, that's what I thought, but since I only looked at Lionel's link, I thought it might be as well if somebody came up with other links to substanciate the evidence in Lionel's link. I an inclined to accept Lionel's link, it is quite intuitive, but Mark seems to consider that all bubbles accelerate.

I popped over to France today, flying more or less East-West or vice versa and it reminded me of the following mathematical calculation............

A pilot flies from West Town  to East Town  in a straight line, making good a ground track from West to East 090°. There is a steady, uniform wind bowing from 210° at a speed of Vw. At the top speed of the aeroplane the pilot makes a speed over the ground of 3Vw/2 and the journey tales precisely one hour.

If the wind speed is unchanged, what is the minimum time it will take the pilot to make the return journey ?.

If he relies on dead-reckoning what compass bearing (heading) should he fly.

(Assume that Grid North, True North, Magnetic North and Compass North are all the same.)

Apologies for accidentally misdirecting the thread. I'd enjoyed the previous teasers and simply wanted to suggest a couple more that i've come across and enjoyed over the years. The bubble question was one which I liked and the claims of which matched my own observations (in champagne, as it happens, not of air in water).

It was intended simply as a question requiring consideration of the rate of growth of a cubic factor outrunning a square factor, as at least one of you quite rightly spotted. What I wasn't trying to do was spark a discussion of whether bubbles actually do accelerate or not, in which liquids, and what original research papers may or may not have been published on the matter!

I'll get my coat.

Mark

Mark

by no means leave. in fact sort the above problem which is beyond my mathematical capability.

i would argue that there is no such thing as a straight line on a sphere....

Mark,

One of the main problems I find when drafting puzzles, is getting the words and meanings succinct and unambiguous. It's almost impossible. So I don't mind when others seek clarification and I also feel disappointed with myself when I realise i've made a right horlicks of the question.

You have nothing to apologise for. You asked a perfectly good question and related it to fizzy bubbles. Some of us simply expanded the question out of pure curiosity to expand our own knowledge. None of us would have given the matter any thought at all if you hadn't raised the initial problem.

Take a look at my puzzle regarding the fence around the lorry park. Steved expanded the possibility to other shapes - which hadn't really crossed my mind. I then expanded it to the possibility of an ellipse v semi-circle, only to recall that the equation for the perimeter of an ellipse isn't very simple, I was hoping others might take it up for me !!

No need to get yer coat - just try the aeroplane one, the time answer is very neat. (The heading answer is also neat in terms of trigonometry but isn't anything special when turned into degrees from North)

Remember, Sin 30 = 1/2; Cos 30 = (Sqrt3)/2

I don't suppose that many would disagree.

I presume a straight line connects opposite sides of a sphere and passes through the center of the sphere.

Makes you think more carefully about how we define things in this universe.

Any one have a teaser that would challenge emotional intelligence ?

I Q Tests are rather two dimensional as a sum up of oneselfship.

You mean something like..........

"How can you get Putin to smile ?" or

"How do you get Cameron to make friends in Europe?"

I enjoy watching Brian Cox and Dara O'Briain when they are together. They both try to be "pub-discussion" friendly, but only one of them really succeeds.

Strictly speaking, a sphere is hollow. It is just the surface defined by points equidistant from a single point. The surface is of zero thickness and has no straight lines longer than 0 length. A "ball" is the solid version, and contains infinitely many straight lines.

I added a couple of words winky. I hope you don't mind ?

I hi-jacked them from wikipedia, which also suggests that "sphere" is derived from a Greek word meaning "globe" or "ball".

So to the intelligent, emotional layman, I suppose that a sphere could have a thick surface or even be solid, rather than be a two-dimensional soulless construct. ?

I don't mind at all.

Before I forget...........the circumference of an ellipse "c"................to save you  looking it up in wikipedia

C = 2Пa [1 – (1/2)^2(e^2) – (1.3/2.4)^2((e^4)/3) – (1.3.5/2.4.6)^2((e^6)/5) …..]

a = semi-mjor axis

e = eccentricity

..........so can we get a bigger lorry park ?

I thought I'd make this one a bit more prominent. It seemed to merge in with some other general discussion and might have been over-looked. It really isn't that difficult !

Hoop La (Not our usual Hopeless)

A thin hoop, diameter "d" is randomly thrown onto an infinitely large chess board, the squares of which each have side "L"

What is the probability that the hoop will lay across more than one square ?

I don't know how to work out the probability except that if it does cross boundaries then if it crosses only one then there are 4 ways for that to happen; if it crosses across a corner then there are also 4 ways? But if it lands squarely within one square?

So, 1 in 17?

Is it - infinite?

If not - then counting down from said number with increasing desperation.

There is an (?) interesting and possibly similar problem arising from the old observation of users of Road Atlases. Why does the place I am interested always appear in the gutter or edge of the page? I think someone from OS explained.

Because in the places you want to go, no one wants you to get there?

I think you need to look at the probability that the hoop is fully inside a square. And then the probability that it crosses a boundary is 1 minus the above probability.

If I have my sums right I think the probability that the hoop crosses a boundary is d/L * (2 - d/L)

What is that in numbers? i.e. 1 chance in x?