# Brain Teasers are Back!

Once i’ve figured out (it’s a brain teaser in itself) how to import pictures, notes and format mathematical symbols I have a few new ones.

Meanwhile, there’s nothing to stop YOU from posting YOUR brain teasers. The more the better !

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The Explorer

An explorer set off on a journey.

He walked a mile south, a mile east and a mile north. At this point he was back at his start. Where on earth was his starting point?

OK, other than the North Pole, which is pretty obvious, where else could he have started this journey?

Hi Don

Glad you’ve got Brain Teasers back again. Makes me wonder whether other long-running topics will re-emerge.

On the old “explorer” chestnut, (s)he could start from anywhere on a line (circle?) of latitude just over a mile from the South Pole, walk a mile south to point X, then east all the way round a line of latitude back to X and finally a mile north back to the start. Or, even closer to the pole, go a miles south to Y then twice (sorry struggling to get italics) round a line of latitude back to Y and thence to the start. Etc, etc.

Roger

Nicely done Roger.

Down here in rural Wiltshire, Paul, the retired MD of a local HiFi manufacturing company which shall remain Naimless, has bought a circular field and a goat.

Paul wants to tie the goat to the fence surrounding the field, and leave just enough slack in the rope so that the goat can graze half the area of the field.

Should the rope tether be equal to the radius of the field; shorter than the radius; or longer than the radius ? (you can ignore the distance between the end of the tether and jaws of the goat)

And yes ! I shall leave the inevitable follow-up teaser to a bit later in the thread

In this image, where did the white square in the top image disappear to in the bottom image?

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Can the tether move around the fence or not? If not, it needs to be longer than the radius. If so, it needs to be shorter than the radius. He would have been better tethering it to a stake in the middle.

The four colored shapes are indeed identical in both figures. The difference is that the red and blue triangles have different slopes ( 3/8 and 2/5 respectively) so neither overall figure is a triangle.

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Well done sir!

Ok here is a more physics-y one.

In a high rise building there is an elevator which has the same speed on the way up as in the way down. Suddenly a brick falls from top of the elevator shaft down onto the elevator.

Is the risk of damage to the elevator from the impact of the falling brick largest if the elevator is travelling upwards or downwards?

At a given elevation when impact occurs, I would say when the lift is going up as the relative velocity will be greater.

If you mean a given height of cabin when the brick starts to fall, it is more complicated.

A puzzle which has occupied me for many years -

You have to go from A to B a distance x. If it is raining, will you get less wet if you run?

I don’t have an answer, but it is an interesting question.

I’ve wondered about the rain problem. Assume the same amount of rainfall throughout the journey. Approximate your head as a circle (or any other shape) perpendicular to the rain, which is approximated to fall vertically. Approximate the front of your body as a planar surface parallel to the rainfall. Various elementary school classes have demonstrated that your head gets less wet if you run, and I think that’s intuitively obvious. How many more raindrops the front of you encounters is going to be a function of the velocity of the rainfall and how fast you run. The relative size of the two planar surfaces comes into it as well.That’s as far as I’ve been able to get.

This is a classic and much argued over problem. The types of analysis most of us would head for are formalised here:

http://www.people.hbs.edu/dbell/walk%20or%20run%20in%20the%20rain.pdf

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You are right that relative velocity is the correct measure of damage (basically the kinetic energy at impact).

You still have some head scratching to do to get to the right answer. All the information you need is in the question and the answer will likely surprise you ( as it did me ).

I don’t think the velocity of the rain matters. I think it is the related to the volume of space you travel through in unit time and the rain flux. I see IanG has provided one analysis.

Assuming the tether is fixed and can’t slip around the fence the tether should be a bit longer than the radius - if it is exactly the radius it will inscribe an area less than the semicircle. I’ve tried to work out how much longer but don’t see a way to calculate it without some ugly and therefore tedious maths which is no fun. Hoping for some inspiration on the drive home.

Ah! You anticipated my next teaser…

This is a (badly) drawn fulcrum of a square base right pyramid.

The base is a 4 x 4 square and the top is a 2 x 2 square.

The perpendicular height is 6.

What is the volume of the fulcrum ?

(Sorry Ian, I will post the (really hard) Goat in a Field teaser shortly. But you have a head start, since you have visualised the question)