Genady

How will the answer change if the slips are randomly reshuffled each time before the next prisoner enters the room?

Yes, this is the strategy. +1

Perfect. +1

There is a group of four prisoners, Al, Bill, Chuck, and Dick. After one year in prison, they have a chance to be released. It works as follows. There is a room with a row of four boxes. Slips with the prisoners' names are randomly placed in the boxes, one per box. Each prisoner enters the room, one at a time, checks one or two boxes of their choice, leaves the room without changing anything, and goes to his cell without any communication with other prisoners. Next prisoner enters the room. Etc. If each prisoner finds his names in the boxes he checks, all four are released. If even one of them does not find his name, all four stay in prison for another year. A year later, they get this chance again. Of course, the slips are placed randomly again then. How long are they expected to stay in prison? The calculation is straightforward. Each prisoner has 1/2 chance to find his name by checking two out of the four boxes. A chance that all four will find their names is 1/2 x 1/2 x 1/2 x 1/2 = 1/16. Thus, they are statistically expected to stay in prison for 16 years. It turned out that there is a strategy which shortens this time. Significantly. Down to 2-3 years! What is the strategy? No tricks. Pure strategy. They can strategize only before entering the room.

Bravo! +1

It doesn't matter. They can define this rule as they wish. (Of course, the tokens should not fall off the table ) All good questions.

Right, it is a consequence of the coordinate system choice, called synchronous coordinates. Perhaps, in other choice of coordinates, the time would vary, e.g., conformal time. But physics does not depend on coordinate system. A different coordinate system will not be physically distinguishable from the current one, but it might allow to get some results easier than getting them in the current one. Convenience / symmetry / simplicity / usefulness are the only criteria for the choice of coordinates.

Fixed.

Depending on conditions, different dimensions behave differently. For example, in the spherically symmetric case with central mass, the temporal and the radial dimensions vary while the two spherical dimensions do not. Similarly, in the homogeneous and isotropic case, the spatial dimensions vary while the temporal does not.

One can understand without math various observational consequences of GR. One cannot understand without math the principles, reasons, and derivations of these consequences.

This is an illustration / clarification of a possible state during the game:

Not allowed

There is a round table and an unlimited supply of round tokens. Two players place in turn one token at a time anywhere on the table without overlapping other tokens. The last player to place a token, wins. What is a winning strategy?

Yep, I needed algebra as well: +1

You are right that if we want to know what they see with their eyes, we need to use Doppler formula. And of course the result is consistent with Lorentz transform because it is the same SR. However, to explain what happens, as opposed to what is seen, is straightforward: In the inertial frame A, B's time is dilated 1.25 times. Thus, while B's clock has advanced by 2 years, A's clock has advanced by 2.5 years.

That's how I thought about it, too.

Yes. +1

There is no problem, because the cda does not say that a constructed sequence does not appear in any list, but in the given list. Cantor's diagonal argument - Wikipedia

The list is very long. The very first use of calculus was to calculate motion of bodies and geometric properties of curves and shapes, such as their lengths and areas.