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

George Fredrik Fiske posted:

Don! You really are a naughty boy! But your posts lighten my day.I thought the Tin Mines were in Cornwall!

Today I serviced the Bottom bracket on a fifty year old Claude Butler cycle of exquisite condition, though the set-up needs finessing ...

I added to the mix a beautiful gipiemme crankset of the era just for the pleasure of matching one classic with another!

I am sure people are bored by my appreciation of British made cycles from the pre-EEC era! 

I can be naughty also!

ATB from George

Ah ! The tin mines were in Cornwall, but tin was exported to South Wales to places like Aberdulais on the River Neath. Aberdulais has a significant waterfall, so plenty of energy to establish a tin-plating factory. And "George" needed tin-plate for his custard pie factory.

Nice pictures of your Carlton and your friend's Butler BTW, and although the details of each cycle are lost on me, i'm sure that others reading this thread will also appreciate and enjoy them.

I did post a note on your "Wall"

Wall replied!

Best from G

steved posted:

Baking tin:

Rough calculations suggest 440 sqcm of tin. Height "t" 6.83cm. Radius also 6.83 cm, so diameter "d" 13.66cm

Steved, your calculations might be rough, but your maths is spot-on !

My answer for d is.... 20 ÷ ³√π      (twenty divided by the cube root of Pi)

t is d/2

And for the Area ..... (πd²) + (4000/d)

These tie-in with your answers to 3 sig.figs

I was day-dreaming this morning about gold.

 Its relative rarity endowing it with a value that means it’s primarily used as a “currency” rather than for its industrial properties. It is very ductile, highly resistant to corrosion, highly reflective, very dense, and a very good conductor of electricity. Its main use in industry seems to be in electrical contacts, especially in computers.

 A little “Googling” revealed that about 186,000 tonnes of the stuff has been mined to date and its density is about 19.3 tonnes per cubic metre.

I figured that a typical small office is about 5m x 4m x 3m (16’ x 13’ x 10’) and that allowing space for access to store and retrieve the commodity, three quarters of such an office could be used as a Gold Store. A typical hifi listening room might be of similar size.

 Assuming that each Country in the World had an equal share of the gold mined to date, how many typical small offices would each country need, to store its gold ?

I'm not sure that gold is primarily used as a currency, though whether all as it seems is another matter: one conspiracy theoty is that most of the gold in Fort Knox isnt. With an almost identical dentity, gold plated tungsten would be indistinguishable in normal routine checks, so is entirely plasuible...

before calculating the number of rooms needed to hold the entire world's mined gold, consider haw impossible it would be for people to carry bage  sometimes shown in films, and the effecton a Mini car if the boot was filled...

Innocent Bystander posted:

I'm not sure that gold is primarily used as a currency, though whether all as it seems is another matter: one conspiracy theoty is that most of the gold in Fort Knox isnt. With an almost identical dentity, gold plated tungsten would be indistinguishable in normal routine checks, so is entirely plasuible...

before calculating the number of rooms needed to hold the entire world's mined gold, consider haw impossible it would be for people to carry bage  sometimes shown in films, and the effecton a Mini car if the boot was filled...

.............or the effect on a bus, overhangging a cliff in Italy........

At 19.3 t/m³ you're right, it's pretty heavy stuff.

With the brainpower on this thread, I'm confident that one of you could come up with an equation that would determine the optimal size for a loudspeaker driver as a function of room dimensions. There is a relationship, but I've never seen it expressed as an equation...

Innocent Bystander posted:

...consider how impossible it would be for people to carry bags  sometimes shown in films...

The Hollywood gold-bag carrying idiocy was perhaps at its most egregious in Three Kings. Clooney, Marky-Mark and the rest of the gang slinging large duffles full of gold bars as if they were full of styrofoam.

winkyincanada posted:

Innocent Bystander posted:

...consider how impossible it would be for people to carry bags  sometimes shown in films...

The Hollywood gold-bag carrying idiocy was perhaps at its most egregious in Three Kings. Clooney, Marky-Mark and the rest of the gang slinging large duffles full of gold bars as if they were full of styrofoam.

At first I had thought they were special bags (more like suitcases than duffles?) padded out with hydrogen, even though we didn't see the gas filling, but after a quick calculation (density of H2 0.07) I realised that even if only half full each of gold and gas the weight would have been too much to carry, so my conclusion was that poor George and chums had been fooled, and had indeed taken bars gold-foil-wrapped styrofoam.  The question is, had the Saddam Hussein's mob tricked them, or had it been the original Kuwaiti(?) owners who had been clever with fake gold there to be stolen?

Just to tidy up the last question.................

• 186,000 tonnes of gold would occupy a volume of 9,637m³.
• Each room is 5x4x3 = 60m³ of which 45m³ is usable (¾)
• Number of rooms required is 9,637 ÷ 45 = 214
• Number of countries in the world is about 200 (UN membership is about 195)
• Hence each Country would need (technically) 2 rooms
• But in broad terms it's a little bit more than 1 room per Country.

Well, I thought it was an interesting visualisation.....

Cheers, Don

What would be the "value" of each room full of gold. The price today was about $US 1,222 per oz troy.

And if it was replaced by Chinese steel at US$ 50 per metric ton (down from $500 per tonne in 2012)

And what would the gold be worth if it was stored in a vault in Panama ?

And the steel if it was in Port Talbot ?

The circumference of the Earth can easily be recalled if ………

• You know that “one minute of arc” along the Earth’s surface equals a Nautical Mile.
• There are 360 degrees in a circle and 60 minutes of arc in one degree
• Hence circumference = 360 x 60 x 1 = 21,600 nm

Or

• The French, not being happy with good old Nautical Miles, Statute Miles and the Greenwich Meridian, decided – quite arbitrarily - that the distance from the North Pole to the Equator along the Meridian that ran through Paris (which co-incidentally also runs through Greenwich) should be called 10,000 kilometres. Thus making the N/S circumference 4 x 10,000 = 40,000 km

Simplifying everything by assuming that the Earth is a perfect sphere of 40,000 km circumference, let’s assume that the French have planned to lay out a telephone cable around the equator on the surface of the earth. Yes, I know it’s difficult to imagine the French being this clever, but for sake of EU harmony……………(and let’s assume there is “land” all the way around the equator……….)

They would have ordered 40 million metres of cable.

Now imagine they decided it would be better if the telephone cable was suspended on poles 33’ high – ok, ok lets keep it simple and call them 10m high all around the earth….. and ASSUME there is no sagging of the cable between these poles, which are fairly closely spaced.

How much more cable would they need to order ?

62.8m.

Hungryhalibut posted:

62.8m.

Spot-on HH.

2πR_e   where R_e is the additional radius ie 10m

So for a circular cricket pitch, if the local team's playing area was increased by an additional 10m strip around the perimeter, the new Boundary rope would also need an additional 62.8m.

It doesn't seem quite intuitive, somehow !

I'll fess up - it was my son Henry who told me the answer - took him about ten seconds. He's doing maths and further maths at A level and is planning to read maths at Uni. 

Hungryhalibut posted:

I'll fess up - it was my son Henry who told me the answer - took him about ten seconds. He's doing maths and further maths at A level and is planning to read maths at Uni. 

 HH

Just a heads up regarding further maths and university.

When my daughter was studying for her A levels, one of her teachers told her to take further maths, as it would give her a better chance of getting into a good university. Adding she could always drop it once she’d been accepted, if her work load got to much.

However, her teacher was wrong and right. She was accepted by a good university,  but her acceptance letter stated if she didn’t get grade A in further maths the offer would be withdrawn.

My daughter was not best pleased with her teacher. Although it turned out Ok as she got an A.

Thanks for the tip, Fatcat. What we found is that if an applicant takes further maths, they are offered slightly lower grades overall. Henry has been offered AAB. His firm choice has said they may flex if he gets A* in maths - so A*BB, or A*AC, and the insurance choice will definitely flex. He got 100% in maths AS, so hopefully can get the A*, meaning that there would be some flexibility if things go awry elsewhere. This sounds like one of Don's brain teasers...

Hungryhalibut posted:

Thanks for the tip, Fatcat. What we found is that if an applicant takes further maths, they are offered slightly lower grades overall. Henry has been offered AAB. His firm choice has said they may flex if he gets A* in maths - so A*BB, or A*AC, and the insurance choice will definitely flex. He got 100% in maths AS, so hopefully can get the A*, meaning that there would be some flexibility if things go awry elsewhere. This sounds like one of Don's brain teasers...

This sounds far more complicated than any of Don's brain teasers..........

For those of you who don't have a son called Harry..........

• The circumference of a circle c = 2πr where r is the radius.
• The circumference of a bigger circle C = 2πR where R is the bigger radius
• The difference in circumference δ = C - c = 2πR - 2πr
• This can be re-written as δ = 2π(R - r)
• From the initial question we know that R - r = 10m
• So δ = 2*π*10 = 20π
• π = 3.14 (more or less)
• So δ = 20 x 3.14 = 62.8 Voila !!

Harry would have done all this in his head in about 10 secs

Henry, please! All he did is take the extra diameter (2 X 10m = 20m) and multiply by pi. 20 times 3.14 comes to 62.8.

Another one, which if you have a son called Henry, might be a doddle......especially if they are aware of the elegant  solution

Its (pretty) obvious that the maximum number of different sized circles (excluding infinitely large circles that have become straight lines!) that can be made to 'just touch' one another is four. (each and every circle just touches the other three).

In the attached diagram, the three circles A, B and C have radii of 1, 2 and 3 respectively. What are the radii of the two 'fourth' circles, that can be made to just touch each of A, B and C. (note the two 'fourth' circles will not touch each other, ‘cos that would break the
maximum rule!!)
Oh! BTW, there is a standard way and an elegant way to solve this little problem.

The above text is for those whose eyesight has difficulty reading the Flickr version above

Hungryhalibut posted:

Henry, please! All he did is take the extra diameter (2 X 10m = 20m) and multiply by pi. 20 times 3.14 comes to 62.8.

That's right.

Mrs D still has difficulty reconciling the idea that increasing the radius of a cricket pitch by 10m (say from 40m to 50m) requires the same amount of extra boundary rope (62.8m.......up from 251.2m to 314m) as the French telephone cable when raised from the ground to 10m high telegraph poles (up from 40 million metres to 40,000,062.8m).

Mind you, she's not interested in cricket, so we'll let that one pass

It is a very weird concept I agree. The way I reconcile it is that in the scale of the world, it's a microscopic increase, whereas with the cricket pitch it's a lot more relatively. 

Henry can't do the circles - well, he's done the small one but not the big one. He's getting on with revision, which is far more useful at the present time. 

Hungryhalibut posted:

It is a very weird concept I agree. The way I reconcile it is that in the scale of the world, it's a microscopic increase, whereas with the cricket pitch it's a lot more relatively. 

Henry can't do the circles - well, he's done the small one but not the big one. He's getting on with revision, which is far more useful at the present time. 

Absolutely. This term, with its crucial exams, is vital. It could affect the rest of his life.

He will need a bit of relaxation time, but there are far better ways to relax than tackling the Naim Forum brain teasers. Plenty of time for that when he retires or winds down to 3 or 4 days a week as retirement approaches in 50 years time.............