Genady

The whole puzzle is in the title. It took a friend of mine, in his own words, "3 pages of factoring (correcting and replacing)". It took me 6 lines. Take a challenge?

It is not necessarily so. It could be different. The puzzle says, that after the swapping and some possible rearrangement of other apples between the piles he gets equal weights again.

It is not necessarily so. According to the puzzle, after removing and inserting he might need to rearrange the other apples between the two sets to get equal weights again.

This is where your problem is. Bye-bye.

This is almost correct. Photon never looses energy in any locality. It's because the space itself expands, it has different energies in different localities, i.e. in different local frames of reference. Consider for example you driving in a car. In your frame of reference the car is not moving and thus has zero kinetic energy. However for a pedestrian on the street, the car is moving with some speed v and has a kinetic energy m(v^2)/2.

Nope, photons don't do that. They don't loose energy. They are created and annihilated in interactions with other particles.

Sorry, didn't notice the change. Corrected. Happy New Year to you as well. BTW, cosmological redshift actually is not a Doppler effect, but for a different reason that has nothing to do with the OP.

The photons are absorbed by gas clouds etc. that are mentioned in the OP.

Because E=hf is energy of one photon. Light consists of many photons, let's say N. The energy of light is E=Nhf. This E decreases because N decreases, not f.

This is incorrect.

[Disclaimer: I don't think time travel is physically possible.] Is splitting of a timeline necessary? I see a scenario without it, I think. A guy travels from year 2022 to year 1922, kills his ancestor, travels from 1922 to 2022, and finds a world, where he never existed. His history line is self-consistent: born, travel, kill, travel. The world's history line is self-consistent: somebody appeared from nowhere in 1922, killed a person, then somebody appeared from nowhere in 2022. No split, just a "loop".

Yes, there is: Astronomy and Cosmology - Science Forums

Here is a related story: "In physical cosmology, the Alpher–Bethe–Gamow paper, or αβγ paper, was created by Ralph Alpher, then a physics PhD student, and his advisor George Gamow. The work, which would become the subject of Alpher's PhD dissertation, argued that the Big Bang would create hydrogen, helium and heavier elements in the correct proportions to explain their abundance in the early universe. ... Gamow humorously decided to add the name of his friend—the eminent physicist Hans Bethe—to this paper in order to create the whimsical author list of Alpher, Bethe, Gamow, a play on the Greek letters α, β, and γ (alpha, beta, gamma). Bethe was listed in the article as "H. Bethe, Cornell University, Ithaca, New York". In his 1952 book The Creation of the Universe, Gamow explained Hans Bethe's association with the theory thus: The results of these calculations were first announced in a letter to The Physical Review, April 1, 1948. This was signed Alpher, Bethe, and Gamow, and is often referred to as the 'alphabetical article'. It seemed unfair to the Greek alphabet to have the article signed by Alpher and Gamow only, and so the name of Dr. Hans A. Bethe (in absentia) was inserted in preparing the manuscript for print. Dr. Bethe, who received a copy of the manuscript, did not object, and, as a matter of fact, was quite helpful in subsequent discussions. There was, however, a rumor that later, when the alpha, beta, gamma theory went temporarily on the rocks, Dr. Bethe seriously considered changing his name to Zacharias. The close fit of the calculated curve and the observed abundances is shown in Fig. 15, which represents the results of later calculations carried out on the electronic computer of the National Bureau of Standards by Ralph Alpher and R. C. Herman (who stubbornly refuses to change his name to Delter)." (Alpher–Bethe–Gamow paper - Wikipedia)

I am not sure what you describe - killing your father before you were born - can't be real. On your worldline, you were born, went back in time, killed your father, then went forward in time (or not.) On your father's contemporary's worldline, somebody appeared from nowhere and killed your father. Is there a contradiction?

In the classical "twin paradox" the twins get between same two points via different spacetime trajectories, and the traveler twin's way is shorter.

I've started watching the videos. Up-to-date and to the point. Bravo, MIT. Lecture Videos | The Human Brain | Brain and Cognitive Sciences | MIT OpenCourseWare

Perhaps it goes too far OT, but I think you are correct -- time travel is not necessarily impossible logically, but it very well can be impossible physically, e.g. as per Kip Thorne's quote above.

Here are the last two paragraphs in the chapter 14, Wormholes and Time Machines: Hawking suspects that the growing beam of vacuum fluctuations is nature’s way of enforcing chronology protection: Whenever one tries to make a time machine, and no matter what kind of device one uses in one’s attempt (a wormhole, a spinning cylinder, a “cosmic string” or whatever), just before one’s device becomes a time machine, a beam of vacuum fluctuations will circulate through the device and destroy it. Hawking seems ready to bet heavily on this outcome. I am not willing to take the other side in such a bet. I do enjoy making bets with Hawking, but only bets that I have a reasonable chance of winning. My strong gut feeling is that I would lose this one. My own calculations with Kim, and unpublished calculations that Eanna Flanagan (a student of mine) has done more recently, suggest to me that Hawking is likely to be right. Every time machine is likely to self destruct (by means of circulating vacuum fluctuations) at the moment one tries to activate it. However, we cannot know for sure until physicists have fathomed in depth the laws of quantum gravity. Thorne, Kip. Black Holes & Time Warps: Einstein's Outrageous Legacy (Commonwealth Fund Book Program) (p. 521). W. W. Norton & Company.

It is here: Black Holes & Time Warps: Einstein's Outrageous Legacy (Commonwealth Fund Book Program) Reprint, Thorne, Kip, Hawking, Stephen W. - Amazon.com

Oh, well... Just to close the case: In terms of linear algebra, the problem can be expressed as the matrix equation A x = 0 where: x is the vector of x1, x2, ... , x17 A is a 17x17 matrix with 0 along the diagonal, and +1 or –1 at the other positions such that each row has eight +1s and eight –1s 0 is the 17-component zero vector From the equation, x is the null space (the kernel) of A. The problem is to show that for all matrices A satisfying the above, the null space x is: x1 = k x2 = k ··· x16 = k x17 = k where k is an arbitrary value. In other words, null space x has rank 1. For this, rank of A has to be 16. That is, if we take a 16x16 minor matrix of A, its determinant is not 0. So, we take a 16x16 matrix with 0s on diagonal and 1s and -1s everywhere else, and consider its determinant. The determinant is sum of products of all possible combinations of matrix elements, one from each row and from each column, with corresponding coefficients 0, 1 or -1. The combinations that include 0s from the diagonal are 0s and don't contribute to the sum. Each row has 15 non-zeroes. There are 16 rows. So there are 15^16 combinations of 1s or -1s. This is an odd number. For every two rows, 14 non-zero pairs are from a same column. These do not contribute to the determinant because the corresponding coefficient in the determinant formula is 0. There are (14 times a number of pairs of rows) of such combinations, which is an even number. After removing the latter even number from the odd number above (i.e. from 15^16) we are left with some odd number of products that do contribute to the determinant. Each product is equal to either 1 or -1 and contributes with a coefficient of either 1 or -1. Thus the determinant is a sum of odd number of 1s and -1s. Here comes the punch line: No sum of odd number of 1s and -1s makes 0 !!! So, determinant of this 16x16 matrix is not 0. QED.

Yes, quantum computation is based on a different logic. The foundation for it is quantum entanglement, which cannot be implemented using usual logic in principle. Thus, it allows to run completely different kind of algorithms. For some problems these algorithms are orders of magnitude shorter than the ones based on usual logic. This is where the quantum computing advantages come from rather than just speed and miniaturization.

No.

Yes, it appears so. He quickly moves on to deeper advantages of flexible learning compared to pre-wiring.

I need a clarification. I can logically show that for any prime number there exists a greater prime number. Is it a negative or a positive?

Like this, I guess. PC, Windows 11, Edge.