mathematics

# A Topological Problem

A country is singly landlocked, if there is at least one other country to be crossed in order to come to the sea shore. Austria and Andorra are singly landlocked.

A country is doubly landlocked if there are at least two countries to be crossed in order to come to the sea shore. Liechtenstein and Uzbekistan are doubly landlocked.

Nebraska is a triply landlocked US federal state.

An imaginary island has a 6 times landlocked country and the average number of neighbours of those island countries is at least 6. No country has an exclave like Russia with Kaliningrad or USA with Alaska. All countries are contiguous. A country may have an enclave country as Italy which surrounds Vatican and San Marino.

What is the minimal possible number of countries on this island or continent?

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mathematics

# A Math Question

Not a problem, for it’s too hard to answer and it’s perhaps too vague, but a question:

The concept of a knot arises in 3D. There is nothing like a 2D knot, it is meaningful only in 3D and beyond. You can knot “2D 4D flat” objects in 4D, just as you can knot “1D 3D thin” objects, also known as ropes, inside the 3D world. You need 2 codimensions to do that.

Now, what’s essentially new in 4D space, which is unimaginable in 3D world. What arises there?

Answers like “two chained rings can be unlocked” don’t count. Because we already have interlockings in 2D which  are easily unlockable in 3D.

What is essentially new and fresh inside the 4D world, we can’t imagine here? Or we can’t imagine, but have some clues about that?

Should be at least something. What could that phenomena be?

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futurism

# Nuclear War

I used to believe, that a horrible nuclear winter would come after the nuclear World War IIIin which almost everyone dies. Then there would be so much radioactivity around, that almost every survivor would die. And then, that in a nuclear winter after, almost every double survivor would perish. It was a standard Cold War view and I was also stupid enough to believe it.

I don’t believe that anymore and I have several good reasons. Here’s a few. How come, people in Hiroshima and Nagasaki live normal lives? Why is no atomic desert in Japan where those atomic weapons were actually used? I acknowledge the horrific fate of ten of thousands and even hundred of thousands of people caught there in August 1945. But no nuclear desert there in the aftermath. How so, if the whole Earth would become a nuclear desert after using 10 thousand times more nuclear  material on one million times greater area? How that could be? And then, after so many nuclear tests, when at least a few percent of the nuclear arsenal was already spent in the wild, how is that we can’t see something nasty happening as a result?

Frankly, I believe Carl Sagan was full of shit with this “nuclear winter” propaganda. Of course, we don’t want a nuclear war, but it wouldn’t throw as back to the stone age. Albert Einstein was often full of shit as well.

We had some sobering suggestions even back then. If the nuclear bomb is 1000 stronger, they told us, its killing radius is just 10 times bigger. Good to know, thank you!

And the radioactivity per Mt of TNT ratio falls if you use bigger bombs. Also good to know.

And then we have learned that a sub-kilometer wide asteroid outperforms all the nukes we posses – many, many times over.

I think, that there will be a nuclear war, if everything stays basically the same as it is today. Only a technological breakthrough may prevent it.  This technological breakthrough may result in a swift victory without much fus, but it may also result in an even more horrific war than a nuclear one would be.

But  if everything stays the same, there will be “some nuclear war to end all nuclear wars”. Except there will be sooner or later another nuclear war, a much greater one. This mess may go on for quite a long time, if everything stays the same.

And this stability of circumstances will not happen. The opportunity window for a nuclear war is closing, because of AI at least. Which by itself may trigger one, as Schwarzenegger in Terminator taught us decades ago.

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artificial intelligence

# Friday Thirteenth

Here it comes again. Many years ago I have learned, that if it’s the 13th day of the month, then it is a little bit more likely that it’s Friday, than any other day.

The Gregorian calendar is a bit biased here, there is nothing you can do about it.

Later I wondered if some datamining  algorithm will uncover this oddity. It should have.

It did. But it also showed that there are some even more worrisome biases here. That for example, the 31st day is even more biased toward Wednesdays.

This dataminig story happened in some more innocent times, of course. Perhaps I’ll write more about this at some Wednesday the 31st.

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mathematics

# And Yet Another Geometry Problem

Say, that you have a N-dimensional hypercube with the edge of 1 (unit).

You can then, of course, put 2^N spheres inside, each with a diameter of 1/2.

For example, eight 50 cm diameter spheres, inside a 1 meter cube, fit nicely. As four circles each of 50 cm  diameter fit well inside a unit square.

Now, you squeeze another sphere in the middle, touching every other of those 2^N spheres.

It has been recently noted, that the middle sphere can have a diameter bigger than 1!

Question No. 1: At which dimension X, does this (first) happen?

Question No. 2: At which dimension Y (if at any), the middle hypersphere has bigger hypervolume than that of its “encompassing” hypercube?

Question No. 3: How many 100 dimensional hyperspheres with the diameter of 2, can be squeezed (with no deformation) inside a 100 dimensional hypercube with the edge of 1?

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physics

# Gliese 710

As you probably know, there is a star, 60% the size of the Sun, approaching us at about 15 km per second.

It will be as near as 2 to 3 light months  away in 1.3 million years. Which may be quite annoying due to comets and such.

For all those details I encourage you to Google around, or consult Wikipedia for the latest known facts about this. There is only one thing I have to say here that everyone seems to neglect. It occurred to me during the debate I had with a friend this morning:

Him: You said that it would be dangerous for our planet if some big rock were to hit our Sun. Why do you think so?

Me: If a hydrogen atom from far away free-falls on the Moon, its velocity, relative to the Moon’s surface, will be equal to the Moon’s escape velocity, which is 2.4 km/s. Its temperature would be the same as that of an average hydrogen atom on the Sun’s surface.

Him: That much?

Me: Yes. And if a hydrogen atom from far away falls onto the Earth, accelerated only by Earth’s gravity, its resulting temperature will be about 80 000 Kelvins.

Him: And if a hydrogen atom falls onto our Sun from far away, what would the atom’s temperature be?

Me: Over 100 million Kelvins. As hot as if it had just come from a thermonuclear explosion.

Him: Get out of here!

Me: Indeed!

Him: A lot of hydrogen and other atoms from the Gliese 710 system may be in free fall towards our Sun as we speak, right?

Me: They are still 63 light years away and the gravity between them and our Sun is still incredibly small,  but the final velocity will be over 600 km per second.

Him: It’s even a bit worse than just a free fall, isn’t it?

Me: Yes it’s even a bit worse, they have some initial velocity already. There is a small probability for a central collision of a big rock with our Sun.

Him: But this time it might be different?

Me: Yes. Imagine that a planetoid from G710 manages to fall directly into our Sun. It would be like a small Nova!

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mathematics

# Yet Another Math Problem

Of the Natural numbers from 1 to 10^100, find N such that ABS(SIN(N)) is the smallest one!

We might have a relative winner here. Someone may say SIN(1)=0.841…

Can you do any better?

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