A Fistful of Brain Teasers

With some, if not most, of these it’s easier for me to see the answer than to explain it. After reading the plane seat scenario it just seemed logical to be 50%. I haven’t checked that answer, but it seems like HH did.

 It’s really hard for me to explain things if I can’t see what the problem might be (for others). Not the easiest thing as a part-time teacher.

 I did try it for the coins, though.

Yes Mulberry, it’s takes a lot of effort to see a student’s difficulty and they are all different.

I try to remember my own difficulties as a student pilot and I also try to remember difficulties I detected in my past students.

I do my best to keep explanations clear and concise and to carefully “watch” student responses, and try a different approach when I see that vacant look !

Hi Don,

and quite true. Once I see that vacant look I try to get into a dialogue with the student(s). The class isn’t too big to do this, with about 30 students. This years group is the best I’ve had so far with many of them not afraid to talk/ask.

Train v cyclist !

A road and a railway line run parallel to each other until a bend takes the road over the railway.

A man cycles to work along the road at a constant speed of 12mph. At the bridge he normally passes over a train that has been travelling in the same direction also at a steady speed.

One day, the cyclist was 25 minutes late for work and was overtaken by the train 6 miles before the bridge. The train was on time and travelling at its usual steady speed.

What is the speed of the train ?

Cyclist is travelling at 12/60 = 0.2 miles/minute. It will take 6/0.2 = 30 minutes to reach the bridge. The cyclist running 25 minutes late, the train takes 5 minutes to reach the bridge, traveling 6 miles. Train is travelling at 6/5 = 1.2 miles/minute or 72 miles per hour.

But if it’s British Rail, there was a leaf on the line and the train was late by 4 hours.

Nicely done Mike. And well spotted......no leaves on the line !

Yes, I know ! this one is old as the hills. But at least HH won't have to resort to the internet or tap up his mathematical genius of a son in order to get the answer

Three men order a bottle of wine at £30 to drink with their meal. They each contribute £10 for it.

The waitress asks the manager, who only has a bottle at £25, so he gives the waitress this wine and £5 change. She returns to the table but gives only £3 back, keeping £2 for herself.

Since each man paid £10 and received £1 change he has paid £9. But 3 x 9 = £27, which with the £2 kept by the waitress makes £29.

Where did the pound go? .........(no Brexit jokes please !)

Oh ! go on then. Post as many Brexit jokes as you like !

And for a bit of variety.......and just as easy !

I have two cylinders. The first one made of lead and the other of titanium.

They are identical in physical dimensions and is each painted red, so that you cannot tell which is which.

They both weigh the same, the lead cylinder being hollow and the titanium solid.

The hollow cylinder, being lead, does not sound hollow.

How can you distinguish between the two cylinders without scratching or damaging either and without using any other object ?

Roll them down an inclined plane and the lead one will get to the bottom first.....

best

David

The mystery is because it’s the wrong question (I’m not talking Brexit). The 30 quid is now a red herring, the 3 chaps paid £27, the waitress gave herself a £2 tip, leaving £25 to pay the right price for the wine.

Does a big bucket of water count as an other object ?

TOBYJUG posted:

Does a big bucket of water count as an other object ?

Yes !

But do explain 

I’m not sure a bucket of water helps. The water in deciding how much of itself may be displaced doesn’t know that of the two cylinders, which are the same size and weight, one is hollow....

best

David

Place them both somewhere they will get hot. When you pick them up, the lead will feel hotter, due to its higher coefficient of thermal conductivity.

Alternativley, if they where cooled down, the lead would feel colder.

fatcat posted:

Place them both somewhere they will get hot. When you pick them up, the lead will feel hotter, due to its higher coefficient of thermal conductivity.

Alternativley, if they where cooled down, the lead would feel colder.

 

Or, if you have suitable measuring capability, lead has a higher thermal expansion: a 30cm lead cylinder would increase in length by nearly half a millimetre more than a titanium one if plunged into boiling water to heat from room temperature (this assuming the paint is not damaged by the boiling water). (Yes I had to look that up!)

But much quicker and easier than the above is test with a magnet, as unlike lead, titanium is weakly paramagnetic, and should be attracted to a magnet. It may be easiest to test by bringing  hanging a small strong magnet on a thread and bringing a vertical wall of each cylinder in turn close to the magnet. But that is using another object, so doesn't count. So of course is the boiling water to heat, so Fatcat's answer may be best - they only have to be a few degrees above or below body temp.

David Hendon posted:

Roll them down an inclined plane and the lead one will get to the bottom first.....

best

David

Only if you let go of it first!  Anyway, Don might complain if he is flying the plane...

However I did stop to wonder about this - would the outer ring of denser material in the lead cylinder increase acceleration in a gravity-induced rolling situation? In free-fall it makes no difference. And the rolling resistance is the same with the same surface paint and shape, size and mass of the objects. On that basis I think not

The rolling resistance is the same, but I am suggesting that the acceleration would be different between the two cylinders as they rolled down the plain (not plane!) because the weight is concentrated further from the surface of the plain in one case. There is no rotation in free fall, so that is a different case.

best

David

Plane was fine, (as opposed to ‘plane). 

I’m just not sure about the acceleration. Why, given that the only driving force is gravity: neither weight nor density affect the acceleration due to gravity, with speed in freefall the same if wind resistance the same, so if the outer part of the cylnder is heavier or lighter shouldn’t make a difference, unless there’s some factor of which I’m unaware?

Because the turning moment for the two cylinders is different.....

Or perhaps since they start stationary it’s about inertia, in which case I think, contrary to what I said before, that the solid cylinder will get to the bottom first because gravity acting against its inertia will accelerate the solid cylinder more quickly.

best

David

The speed of rotation is dependent on the distance of the mass from the centre of rotation.

I preface this by clarifying that I am not an expert in mechanics,so could be missing something, but turning moment is force: why would that make it accelerate faster?

That make me think  - it would get over bumps better, so on an inclined plane with transverse ridges  the lead cylinder may roll faster, so maybe that is an answer?

Now, that is running into inertia - but the reverse of your alternative: and I still am inclined to think that on an a smooth surface the two would roll the same.

Mike Sullivan posted:

The speed of rotation is dependent on the distance of the mass from the centre of rotation.

Yes, the dustance varying the  rotational forcebut does that make any difference to acceleration by gravity on an inclined (smooth) plane? 

Yes, if the cylinders are the same diameter, because the circumference is the same and they will travel the same distance at different speeds, arriving at the destination at different times.

 Ow, I’m watching Steven Wilson at the RAH, so can’t be distracted.

Why wouldn’t it make a difference? The cylinders are stationary at the top of the plane and we all agree that left to themselves they will end up at the bottom, except that if they weighed nothing presumably they would stay where they were placed.

This propensity to roll has to be due to gravity acting on them. Gravity is a constant and it acts everywhere equally. So if the mass of the cylinder is distributed differently, as it is in this case, then the acceleration caused by the gravity would be different and one will reach the bottom before the other. I now think that is the solid one.

best

David