What is the proper way to explain the twin paradox? The paradox in the twin paradox is that the situation appears symmetrical so each twin should think the other has aged less, which is of course impossible.
There are a thousand explanations out there for why this doesn't happen, but they all end up saying something vague like it's because one twin is accelerating or you need general relativity to understand it.
Please will someone give a simple and definitive explanation for why both twins agree on which twin is younger when they meet for the second time?
 A: Introduction
This is the third (and last) in a series of posts explaining time dilation, and it is going to assume you’ve read the preceding posts What is time dilation really? and What is time, does it flow, and if so what defines its direction?. Much of what follows won’t make sense unless you’re familiar with the topics discussed in the previous two questions. This is also going to be the hardest of the three posts by quite some way, but it just isn’t possible to gain a real understanding of the twin paradox without exploring some hard ideas. You have been warned!
In what follows I’m going to assume I am the stationary twin i.e. I remain on Earth while you go zooming off on your return trip in your spaceship. Remember that when you see me or my it refers to the stationary twin and you and your refer to the accelerating twin.
So as not to keep you in suspense, I'm going to explain that the asymmetry arises because the geometry of spacetime looks different for the two twins. To calculate the elapsed time we need a function called the metric, and in the coordinate system of an accelerating observer the metric looks different from normal flat spacetime. When we take this into account both twins agree about their respective ages.
My version of events
In the question on time dilation I explained what we mean by time dilation and how we calculate it. In particular I showed this spacetime diagram:

Figure 1
This shows our two trajectories through spacetime using my rest coordinates i.e. the coordinates in which I remain stationary at the origin. In these coordinates I remain at $x=0$ and simply travel up the time axis from the starting point $A$ to the finishing point $B$ as shown by the black arrow. You go hurtling away along the $x$ axis from point $A$, then stop, reverse and scream back to meet me again at point $B$, as shown by the red arrows. So the red line shows your trajectory through spacetime as measured using my coordinates.
From the time dilation question we know that the elapsed time shown by a clock carried by an observer, $\Delta\tau$, is related to the length of the observer’s trajectory, $\Delta s$,  by:
$$ \Delta s^2 = -c^2 \Delta\tau^2 $$
And we know that the length $\Delta s$ is calculated using a function called the metric. In flat spacetime this function is the Minkowski metric, and it tells that if you move a distance $\mathrm dx$ along the $x$ axis, $\mathrm  dy$ along the $y$ axis and $\mathrm dz$ along the $z$ axis in a time $\mathrm  dt$ then the total distance you have moved in spacetime is given by the Minkowski metric:
$$\mathrm  ds^2 = -c^2\mathrm  dt^2 + \mathrm dx^2 +\mathrm  dy^2 +\mathrm  dz^2 $$
Since it’s hard to draw 4D graphs it’s usual to assume all motion is on the $x$ axis, so $\mathrm  dy =\mathrm  dz = 0$, in which case the metric simplifies to:
$$\mathrm  ds^2 = -c^2\mathrm  d\tau^2 = -c^2\mathrm  dt^2 + \mathrm  dx^2 \tag{1} $$
To calculate the length of the red curve we use the cunning trick of noting that velocity is defined by $v = \mathrm dx/\mathrm  dt$ so $\mathrm dx = v\,\mathrm  dt$, and if we take equation (1) and substitute for $\mathrm  dx$ we end up with:
$$ \mathrm  d\tau = \sqrt{1 - \frac{v^2(t)}{c^2}}\,\mathrm dt $$
So the elapsed time $\tau_{AB}$ is given by the integral:
$$ \tau_{AB} = \int_{t_A}^{t_B} \, \sqrt{1 - \frac{v^2(t)}{c^2}} \,\mathrm  dt \tag{2} $$
where $v(t)$ is your velocity as a function of time. The exact form of $v(t)$ will depend on how you choose to accelerate, but since $v^2$ is always positive that means the term inside the square root is always less than or equal to one:
$$ 1 - \frac{v^2(t)}{c^2} \le 1 $$
And therefore the integral from $t_A$ to $t_B$ must be less than or equal to $t_B-t_A$. This means your elapsed time $\tau_{AB}$ must be less than my elapsed time $t_{AB}$ i.e. when we meet again you have aged less than I have.
So far so good, but the paradox is that we could draw the spacetime diagram in figure 1 using your coordinates, i.e. the coordinates in which you are at rest, to give something like:

Figure 2
In these coordinates you remain stationary so your trajectory shown by the red line goes straight up your time axis, while my trajectory shown by the black line heads off in the $-x$ direction before returning. If we use the same argument as above we would conclude that I should have aged less than you, but we can’t both have aged less than each other.
And that’s the paradox.
Your version of events
The resolution to the paradox turns out to be very simple. When I calculated the length of your trajectory in the previous section I used the Minkowski metric, equation (1),  and after some algebra ended up with the equation for your path length in equation (2):
$$ \Delta t_\text{you} = \int_{t_A}^{t_B} \, \sqrt{1 - \frac{v^2(t)}{c^2}}\,\mathrm  dt $$
The resolution to the paradox is simply that in your rest frame the metric is not the Minkowski metric, and therefore the equation you have to use to calculate my path length is not the same as equation (2):
$$ \Delta t_\text{me} \ne \int_{t’_A}^{t’_B} \, \sqrt{1 - \frac{v’^2(t)}{c^2}}\,\mathrm dt’ $$
and that’s why when you calculate my path length we both agree that my path length is longer than yours i.e. we both agree that I age more than you do.
So what is your metric?
The form of your metric will depend on exactly how you accelerate, and in general will not be a simple function. However there is a special case that is reasonably simple, and that is what I’m going to assume for the rest of this answer. I’ll assume that your acceleration (or rather deceleration) is constant, so your motion consists of the following:


*

*at time zero you pass me with some positive velocity $v$ and constant deceleration $a$ - constant deceleration means you are accelerating towards me and in the opposite direction to your velocity

*the constant deceleration eventually slows you to a stop at some distance $x$ away from me

*you maintain the constant deceleration and now you start moving back towards me i.e. your velocity becomes negative

*eventually you pass me again now with a velocity of $-v$
For motion at constant acceleration your metric is a function called the Rindler metric:
$$\mathrm  ds^2 = -\left(1 + \frac{a\,x}{c^2} \right)^2 c^2\mathrm  dt^2 +\mathrm  dx^2 \tag{3} $$
For now I won’t attempt to justify this (I may do so in an appendix) I’ll just make a few comments on it before showing how to use it to calculate the trajectory length.
The Rindler metric doesn’t look completely different to the Minkowski metric that I used before. Indeed at the point $A$, where we part company, the value of $x$ is zero for both of us, and if we set $x=0$ the Rindler metric reduces to:
$$\mathrm  ds^2 = -c^2\mathrm  dt^2 +\mathrm  dx^2 $$
which is just the Minkowski metric. Likewise if we take the acceleration $a$ to zero, the equation (3) just reduces to the Minkowski metric. However when $a \ne 0$ and $x \ne 0$ the two metrics are different, and the further $a$ and $x$ are from zero the more different the metrics become.
OK let’s attempt the calculation
Now we can calculate my elapsed time in your rest frame using the correct metric i.e. the Rindler metric. Let’s remind ourselves of the spacetime diagram:

In your frame I pass you at time zero with a negative velocity, and I head off to negative $x$ before turning round to come back. What is perhaps not obvious is that the acceleration $a$ is negative. This is because $a$ is your acceleration. In the diagram above my acceleration relative to you is obviously positive so your acceleration relative to me must be negative.
We start as before by writing down the metric:
$$\mathrm  ds^2 = -c^2\mathrm  d\tau^2 = -\left(1 + \frac{a}{c^2}x \right)^2 c^2 \mathrm dt^2 +\mathrm  dx^2 $$
And we use the same trick of substituting $\mathrm dx = v(x)\mathrm dt$. After rearranging we end up with:
$$ \Delta t_\text{me} = \int_{t_A}^{t_B} \, \sqrt{\left(1 + \frac{a\,x(t)}{c^2}\right)^2 - \frac{v^2(t)}{c^2}}\,\mathrm dt \tag{4} $$
This is actually pretty similar to the equation (2) that I used to calculate your elapsed time, apart from that extra term $a\,x(t)/c^2$. But it’s that extra term that makes the difference. To see why consider the leftmost point on my trajectory in figure 2. At this point my velocity is zero so the term in the square root becomes:
$$ 1 + \frac{a\,x(t)}{c^2} $$
But the product $a\,x(t)$ is positive, which means $1+ax(t)/c^2 \gt 1$ and therefore at this point $\mathrm d\tau \gt\mathrm  dt$. Doing the integration in this region gives my elapsed time as greater than your elapsed time.
And this is the key to understanding the twin paradox. When you use equation (4) to calculate the length of my trajectory you’re going to find that my elapsed time is greater than your elapsed time, which is exactly what I found when I did the calculation in my frame. The resolution to the paradox is that the metric you use to do the calculation is not the same as the metric I use to do the calculation.
A: It seems to me that the existing answers contain far more information than the average novice is looking for when s/he asks about the twin paradox.  
So with that novice in mind:
Alice stays home on earth.  Bob flies from earth to Betelgeuse, and then from Betelgeuse to earth.  The rookie mistake is to think that there are two relevant frames here --- Alice's and Bob's.  Instead there are three --- Alice's, Outbound Bob's and Inbound Bob's.
In Alice's frame, Bob leaves earth at noon, arrives at Betelgeuse at (say) 3:00, having aged two hours, and arrives back home at 6:00, having aged another two hours.  Bob's clock, obviously, runs slow by a factor of 2/3.
In Outbound Bob's frame, Bob leaves earth at noon, and arrives at Betelgeuse at 2:00, just as Alice's clock is striking 1:20.  Alice's clock, obviously, runs slow by a factor of 2/3.
In Inbound Bob's frame, Bob leaves Betelgeuse at 2:00, just as Alice's clock is striking 4:40, and arrives home at 4:00, just as Alice's clock strikes 6:00.  Alice's clock, obviously, runs slow by a factor of 2/3.   
The thing many beginners overlook is that Outbound Bob and Inbound Bob disagree about what Alice's clock says at 2:00.    
A: I would like to add something on the simpler version known as the Symmetric Twin Paradox without acceleration, which has been mentioned in a comment and always pops up sooner or later anyways. 
Although it is constantly pointed out that the "inertial" setup is not physically consistent, for reasons of acceleration/deceleration, non-Minkovsky metric etc, beginners are often left with a nagging sense that it should be working as a toy model, hence something is missing etc, and the discussion continues ad infinitum. I think there is a useful lesson to be gained in making explicit the source of inconsistency, and some additional insight on the main question, so I'd rather add this here. 
For convenience of notation, I restate the problem as follows:
Let twins A and B move at the same speed $v/c = \beta$ in opposite directions relative to inertial observer O: A in the negative direction of O's $x$-axis, B in the positive direction. Both synchronize their clocks to O when they pass by O's origin at $x = 0$, such that for this event $ct = ct_A = ct_B = 0$. Subsequently A and B continue on their respective ways towards planets $P_A$ and $P_B$, both at rest wrt to O, at locations $x(P_A) = - x_0$, $x(P_B) = x_0$. As soon as they reach their planets, both twins jump ship into new inertial frames A' and B' moving at the same speed $\beta$ wrt O, but in opposite directions. That is, A' moves now in the positive direction of O's $x$-axis, while B' moves in the negative direction. At the moment of the transfer, A' synchronizes its clock to A, such that $ct_{A'} = ct_A$, while B' synchronizes with B, such that $ct_{B'} = ct_B$. When A' and B' pass again by the origin of O, everybody compares clocks. The question is, as before, if A' and B' report the same time when they meet again at O, how is this compatible with the assertion that A/A' and B/B' must see each other undergoing time dilation and conversely?
The answer lies in a serious discontinuity brought in by the transition from frames A and B to frames A' and B'.
One way to see this is to note that although coordinate transforms between O, A, and B are immediate, given their standard Einstein synchronization, other frame pairs no longer synchronize in this usual way and the corresponding transforms are slightly different. For instance, we know that A' synchronizes with A when both pass by $P_A$, but neither clocks time at $t=0$ for this event. So how do we write the coordinate transformation between A and A'? Simple: we just account for the coordinate shifts between the respective origins by means of Poincaré transformations:
Take frame O of coordinates $(x, ct)$ and frame O' of coordinates $(x', ct')$ moving at relative velocity $\beta$. If an event observed by O' at coordinates $(x'_0, ct'_0)$ is observed by O at coordinates $(x_0, ct_0)$, then the (Poincaré) transformations between O and O' read simply
$$
x' - x'_0 = \gamma\left[\left(x-x_0\right) - \beta\left(ct - ct_0\right) \right]\\ 
ct' - ct'_0 = \gamma\left[\left(ct-ct_0\right) - \beta\left(x - x_0\right) \right]
$$
and
$$
x - x_0 = \gamma\left[\left(x'-x'_0\right) + \beta\left(ct' - ct'_0\right) \right]\\ 
ct - ct_0 = \gamma\left[\left(ct'-ct'_0\right) + \beta\left(x' - x'_0\right) \right]
$$
for $\gamma = 1/\sqrt{1-\beta^2}$ as usual. All we have to do now is identify the correct relative reference coordinates and velocities of the various frames. 
It is easy to show that at the final rendezvous O-s clock shows $ct = 2ct_0 = 2x_0/\beta$, while the clocks of A' and B' both show $ct_{A'} = ct_{B'} = 2ct_0/\gamma$, as expected from symmetry and time dilation in O's view. But let us consider for example the way A and A' observe B and B':


*

*Between the "departure" at O and the "turning" at $P_A$, A observes B moving in the positive x direction at velocity (see relativistic addition of velocities) $\bar\beta =  \frac{2\beta}{1+\beta^2}$, and the corresponding Lorentz transforms are 
$$
 x_A = {\bar\gamma}(x_B + {\bar\beta} ct_B) \;\;\; ct_A = {\bar\gamma}(ct_B + {\bar\beta} x_B)\\
 x_B = {\bar\gamma}(x_A - {\bar \beta} ct_A) \;\;\; ct_B = {\bar\gamma}(ct_A - {\bar \beta} x_A)
 $$
with $\bar\gamma = \frac{1}{\sqrt{1-\bar\beta^2}} = \frac{1+\beta^2}{1-\beta^2}$. When A reaches $P_A$ at $ct_A = ct_0/\gamma$, it observes B at location $x_A(B) = \bar\beta\; ct_A = \bar\beta\; ct_0/\gamma$. However, according to the 2nd of the above transforms, at this point B-s clock only shows time $ct_B = \frac{1}{\bar\gamma}\frac{ct_0}{\gamma} < \frac{ct_0}{\gamma}$. This is the expected time dilation as observed by A, which is good. But this also shows that when arriving at $P_A$, A observes that B has not yet reached $P_B$.

*Between the "turning" at $P_A$ and the final rendezvous at O, A' observes B' moving in the negative x direction at relative velocity $\bar\beta=-\frac{2\beta}{1+\beta^2}$. Since we know that the 2nd rendezvous takes place at coordinates $(x_{A'} = 0, ct_{A'} = 2\frac{ct_0}{\gamma})$ for A' and, likewise, at $(x_{B'} = 0, ct_{B'} = 2\frac{ct_0}{\gamma})$ for B', the coordinate transformations between A' and B' read
$$
x_{B'} ={\bar\gamma}\left[x_{A'} + {\bar \beta} \left(ct_{A'} - 2 \frac{ct_0}{\gamma}\right)\right], \;\;\;ct_{B'}-2 \frac{ct_0}{\gamma} = {\bar\gamma}\left[\left(ct_{A'}-2 \frac{ct_0}{\gamma}\right) + {\bar \beta} x_{A'}\right]\\
x_{A'} = {\bar\gamma}\left[x_{B'} - {\bar \beta} \left(ct_{B'}-2 \frac{ct_0}{\gamma}\right)\right],\;\;\; ct_{A'} - 2 \frac{ct_0}{\gamma} = {\bar\gamma}\left[\left(ct_{B'}- 2\frac{ct_0}{\gamma}\right) - {\bar \beta} x_{B'}\right]
$$
But now we can check that when A' "takes over" at $P_A$ for $ct_{A'} = \frac{ct_0}{\gamma}$, it observes B', $x_{B'} = 0$, at location $x_{A'}(B') = {\bar\beta}\;\frac{ct_0}{\gamma}$, and sees B'-s clock showing time $ct_{B'} = \left(2 - \frac{1}{\bar\gamma}\right)\frac{ct_0}{\gamma}$. Since for B' the time difference to the final rendezvous is $2\frac{ct_0}{\gamma} - \left(2 - \frac{1}{\bar\gamma}\right)\frac{ct_0}{\gamma} = \frac{1}{\bar\gamma}\frac{ct_0}{\gamma} $, this is consistent with time dilation as observed by A'. But at the same time we find that $ct_{B'} = \left(2 - \frac{1}{\bar\gamma}\right)\frac{ct_0}{\gamma}  \ge \frac{ct_0}{\gamma}$, which means that according to A' as it passes $P_A$, B' has already passed by $P_B$!! Conversely, due to the time jump introduced this way, the time A' observes on B'-s clock at the final rendezvous is $\left(2 - \frac{1}{\bar\gamma}\right)\frac{ct_0}{\gamma} + \frac{1}{\bar\gamma}\frac{ct_0}{\gamma} = 2\frac{ct_0}{\gamma}$, consistent with both its own clock and the point of view of O. Needless to say, the same result obtains for A and A' as seen from B an B', and even for O as seen from either A/A' or B/B'. So we are left with the following conclusion:

A/A' does observe B/B' undergoing time dilation, and conversely, as expected. But, the fortuitous switch of inertial frames to simulate a $180^o$ reversal of velocity introduces an artificial discontinuity in worldviews and time measures for the frames involved. For example, at the turning point $P_A$, A claims that B has not yet turned at $P_B$, while A' claims that B' passed past $P_B$ long ago. Neither A nor A' ever witness the actual turning of B/B' at $P_B$ (it can also be verified that A observes B' before reaching $P_B$, while A' observes B past $P_B$). Technically, it is this discontinuity of worldview that allows twins A/A', B/B' to display the same time at the final rendezvous at O, despite mutual time dilation. 

As an exercise, check that a similar discontinuity is responsible for the discrepancy between the clocks of O and A'/B' at the final rendezvous, although O does appear to undergo time dilation as observed from both A/A' and B/B'. 
A: The short answer is that the situation is not symmetrical- one twin remains in their (inertial) reference frame through-out, while the other starts in one frame and ends in another. That asymmetry resolves the paradox.
By contrast, if each twin were to accelerate at the same rate to some immense distance at some immense speed, and then return to meet again where they started, provided their respective journeys were truly symmetrical they would have aged by the same amount.
The confusion arises because of inconsistencies and misunderstandings in the way different authors explain why the asymmetry causes the age difference. 
The definitive answer is that spacetime is not Euclidean but Minkowskian, and the longest proper time interval between two events is that along a path that is stationary in space. However, that hardly counts as a simple explanation that is likely to be readily grasped.
The simplest non-mathematical explanation for those who are familiar with the key features of SR, especially the concept of simultaneity, is that the change is accounted for by the changing reference frame of one twin when they reverse their direction of motion to return to the other. The consequence of the change is that their plane of simultaneity rotates, and intersects the time axis of the other twin further in the positive time direction of that other twin. The shift in the point of intersection accounts for the time difference.
Arguments can arise from this, as some commentators assume that acceleration is needed to bring about the change of reference frame, while others cite the Lord Halsbury illustration with three travellers. The arguments are at cross-purpose, since what counts is the change of reference frame; how it is brought about is immaterial.
At this point I should say that those commentators who claim that GR is needed to explain the paradox are emphatically wrong. Accelerations can be straightforwardly taken into account in SR. What SR cannot do is to allow us to calculate the effects of different paths in areas in which gravity cannot be neglected.
Readers should also bear in mind that the effect is not dependent upon travellers, rockets etc, but is a property of spacetime. The elapsed time measured between two events A and B at a single spatial location in one reference frame is always greater than the sum of the elapsed times measured along other paths from A to B.
A: There is a mental picture I've always used for this.
There are two people, the traveler and the couch-potato.
As the traveler is moving away at constant velocity, he sees the couch-potato's clock advancing more slowly.
Then suppose there is an event where the traveler's velocity reverses, like bouncing off a big trampoline in space.
In that instant, the couch-potato's clock is seen by the traveler to suddenly jump forward.
That's what acceleration does - it makes the clocks of unaccelerated objects appear to run faster, because the lines of simultaneity are changing direction.
Then, during the return trip, the couch-potato's clock appears to run more slowly than the traveler's clock, but not enough to make up for the turnaround in the center.
So at the end of the trip, the couch-potato's clock is ahead (the couch-potato is older).
A: Twin Paradox Spacetime Diagram (calculus not required)
Below I will present some arguments below that
the spacetime diagrams produced by the non-inertial observer are distinctly different from those by an inertial observer.
While there are a lot of words to setup the diagrams,
you can jump to the end to see main argument.

But first:
In the Twin Paradox, there are two issues:

*

*The Clock Effect. The elapsed proper time (wristwatch time) between the separation-event (call it O) and reunion-event (call it Z) depends on the spacetime-path (worldline) taken from event O to event Z. (The spatial analogy in Euclidean space is: the odometer-distance from one point O to another point Z depends on the path taken.)(There is no Clock Effect in Galilean relativity, which could be regarded as an extrapolation of what we observe with our human-scale clocks when we travel short distances over short times at slow velocities.)
In the usual setup, "the stay-at-home twin" remains inertial from O to Z, whereas the traveling-twin visits another event Q (not on the inertial worldline OZ) on route to Z via the inertial worldline OQ, followed by another inertial worldline QZ--- however, the worldline OQZ is non-inertial. The wristwatch along inertial worldline OZ elapses more time that the wristwatch along the non-inertial worldline OQZ.


*The "Twin Paradox". Assuming the Clock Effect is established, the so-called paradox is the attempt to study the problem from view of the non-inertial observer, somehow claiming equivalence with the inertial observer by invoking the principle of relativity. If successful, then this would invalidate the clock effect--leading to no route-dependence of elapsed proper time from O to Z.

MY PLAN:
With spacetime diagrams,

*

*Establish the Clock Effect in the inertial stay-at-home frame.

*Attempt to study the problem from the non-inertial frame. I will argue that the non-inertial observer is not equivalent to the inertial observer, and so the principle of relativity can't be invoked for the non-inertial observer.
I will show that the spacetime diagrams produced by the non-inertial observer are distinctly different from those by an inertial observer.

I will draw spacetime diagrams on rotated graph paper so that we can more easily visualize the ticks along worldlines and lines-of-simultaneity. (This is based on my article "Relativity on Rotated Graph Paper", Am.J.Phy. 84, 344 (2016); https://doi.org/10.1119/1.4943251 .)
In the diagram below, we draw the spacetime diagram of inertial observer Alice (at rest in this frame) and her light-clock. (Time runs upwards.) The light-rays trace out on a spacetime diagram "Light-Clock-Diamonds", and this establishes a coordinate system for Alice. Her worldline is along the timelike-diagonal and her lines of simultaneity are parallel to the spacelike-diagonal.
Using Alice's grid, we draw Bob's worldline OQ (with $\beta_{Bob}=(PQ/OP)=8/10$ (for arithmetic convenience)).


TIME DILATION
What do Bob's light-clock-diamonds look like?
(See the diagram below)

*

*Their edges must be parallel to the grid (since the speed of light is independent of the velocity of the source).

*It turns out that the area of Bob's light-clock-diamond must be equal to that of Alice's (since the Lorentz Transformation has determinant 1, and thus preserves area, as well as the lightlike directions). [This implies that Alice measures the width of Bob's light-clock to be shorter than hers.]

So, we get $6$ ticks along Bob's segment OQ.
Indeed, $$10^2-8^2=6^2.$$



(from the diagram above) One tick after separation, each sends a light-signal to the other. Each receives the signal three ticks after separation---that these reception times are equal is in agreement with the Principle of Relativity. By the way, for $\beta=8/10$, the Doppler factor $k=\sqrt{\frac{1+\beta}{1-\beta}}=3$.

(see the diagram below) Alternatively, one can do a Bondi k-calculus radar measurement: send a light-signal at time $T$ after separation and wait for its echo. Since the echo is received at $k^2T$, one can determine $k^2$ (here $k^2=9$.. so $k=3$). There must be $k$ ticks on the segment on Bob's worldline since separation. Divide that segment into $k$ parts and draw in the light-clock diamonds, and construct a coordinate system for Bob.



(see the diagram below) Yet another way, draw the causal diamond of $OQ$ and count the number of Alice clock-diamonds in it. In this example, there are $18\times 2=36$ of Alice clock-diamonds. This number is the square-interval of OQ. So, OQ has $\sqrt{36}=6$ ticks. (By the way, $k^2=(width)/(height)=18/2=9$. So, Alice would determine $\beta_{Bob}=(k^2-1)/(k^2+1)=8/10$. Further, Bob would count $6\times 6=36$ of his diamonds... and determine $\beta_{Bob}=0$.)



CLOCK EFFECT
Let's assume that Bob returns at the same speed.
So, Alice can splice two time-dilation problems.
We count: inertial Alice from O to Z ages 20 ticks,
while piecewise-inertial (but-still-non-inertial) Bob along OQZ ages 12.

Just a reminder, we are just drawing a spacetime diagram of their ticking light-clocks.


TWIN PARADOX
Suppose non-inertial Bob tries to draw a spacetime diagram (as Alice did earlier).
For concreteness, Alice's diagram could be thought of as a splicing of two time-dilation problems: O-to-P and P-to-Z, where OPZ is along Alice's inertial worldline.
It seems natural that Bob along OQZ will try the same idea, using his lines of simultaneity, which are parallel to the spacelike diagonals of his clock-diamonds.


So, let's try to draw the diagrams from Bob's frame.
I'll draw the frame of the last leg QZ, then the frame of the first leg OQ.



Then, I'll cut and splice them together.
And now we see some odd features in this Frankensteined spacetime-diagram:

*

*Portions of Alice's worldline are missing, for example, event P.
Bob's spacetime-diagram is not a full map of spacetime, whereas Alice's spacetime-diagram is.

*Some events in spacetime appear twice on Bob's diagram, like event X.

*Maybe... maybe, something can be done to give Bob a better, more reasonable looking spacetime diagram.... but Alice had no such issues.
One can't claim that Bob is equivalent to Alice.

*There is no Lorentz transformation that will straighten the kink in Bob's non-inertial worldline OQZ.


Thinking physically, 
if there was a ball on a frictionless table in Alice's frame and in Bob's frame.

The ball in Alice's frame would not have moved from O to Z.

However, by the Law of Inertia, the ball in Bob's frame would have moved when Bob turned around at event Q.... since Bob traveled non-inertially along OQZ.
Although we can view the world from Bob's "rest" frame, 
that does not mean that we can treat Bob's frame as an "inertial" one.


Now to make Bob's diagram look even more odd...

Let's consider an asymmetric trip:
the outgoing leg has velocity $\beta_{Bob,out}=8/10$,
but the incoming leg has velocity $\beta_{Bob,in}=-8/17$.

Here is Alice's spacetime diagram, followed by Bob's spliced spacetime diagram.

In Bob's diagram, Alice's worldline is clearly discontinuous.


Bob is certainly not equivalent to Alice.

(This method of graphical calculation is developed in my paper
"Relativity on Rotated Graph Paper", Am.J.Phy. 84, 344 (2016); https://doi.org/10.1119/1.4943251 .)
A: $\let\D=\Delta$
I only recently became aware of this post. An important contribution
to an ever-ready question. Incidentally, it always makes me wonder to
see how hard is this matter, how many questions and answers it still
raises after over a century. At least in part, this shows IMHO that
there should be much to be thought about the teaching of relativity,
apparently worldwide. But let me leave general reflections and come to
the real motivation of my post.
There are some points, not of minor importance, which could be
improved in John' post. I'll try to give some contribution.

1) I read a phrase

"the geometry of spacetime looks different for the two twins.

I would like to mark a distinction between geometry and metric, or
better between metric and the form it takes in different coordinates. I'm afraid it may be taken as a puristic criticism, but I believe that the lack of a clear distinction may harm understanding and also be a cause (one of several) to make the subject difficult.
Geometry and metric - an element of the whole structure we call geometry - are intrinsic properties of spacetime (let alone more general mathematical structures). They can and should be defined independently of reference frames and coordinates. As an example: euclidean geometry implies a special metric, and both can be defined without introducing cartesian coordinates.
For our subject this becomes relevant as an easy misunderstanding
should be avoided: that geometry (and metric) of spacetime be different
if it is viewed from an inertial frame or from an accelerated one. To
be more precise: spacetime of SR is flat (Minkowskian) and such
remains even though we reason and measure in an accelerated frame. To
be sure, John never says anything different, but not even says that
explicitly. There is a concrete risk that a not well-educated reader
may draw a wrong conclusion.
To be honest, I'm not sure I myself never incurred in the same flaw.
It's rather common among physicists, who are generally accustomed not
to pay much attention to issues of language accuracy. It's usually
understandable as a shortening of discourse. But in critical points it
can have unwanted consequences.

2) Another general point. Once you have shown that what we are doing is
just measuring lengths of two distinct paths, there is nothing more to
be proven. Spacetime length is invariant (John is very clear on
that) and we are free to choose the most easy way to measure it. In
the present case an inertial frame, with its "natural" coordinates
$(x,t)$, is the right one. 
Nothing forbids, of course, to choose another way, i.e. a different
(accelerated) frame. As an exercise it's very useful. Unfortunately
John can only mention how it may happen that the relevant integral, in
spite of its intimidating look, will give the same result. He can't go
further, as he didn't give - with good reason! - $x(t)$ and $v(t)$ of
the standing observer as seen from the accelerated one...
Unfortunately this weakens the argument. The reader has to take John's
word. Sure, he's not writing a book. But this leaves a question open
about relativity teaching, not to be dealt with here.

3) Now for a technical (but highly physical) issue. John in his Appendix
explains Rindler's "metric". All OK but... 

We'll do the race again but this time you start at rest and
  accelerate with a constant acceleration a.

Unless I have missed the point John never says what a constant
acceleration is (we know that it's the proper acceleration). To be
sure, he writes

It isn't hard to prove that the accelerating twin's metric has the
  Rindler form, but unless you are a fan of algebra the proof isn't very
  exciting

This is not a matter of more or less algebra. There is an important
piece of physics involved and I would have appreciated a deeper
approach. To understand proper acceleration is far from trivial and
requires thinking about basic aspects of SR.
Another point. John shows figures where Rindler's coordinates are
named $(x,t)$. He doesn't write the transformation equations from or
to $(x,t)$ of an inertial frame and this is understandable, as those
equations involve exponential (hyperbolic) functions and logarithms.
But to use equal names is objectionable. Not only because they are in
any case different beasts, but above all because their physical
meaning is different. More precisely: whereas Rindler's $x$ and
inertial $x$ both measure proper lengths of standing objects (standing in respective frames) this  true for the $t$'s. 
Not a word about that. Rindler's $t$ isn't the time a clock in the
accelerated frame marks. Equal $\D t$ at different $x$ don't
correspond to equal proper times of standing clocks (this is written
in the metric). It has a far-reaching consequence: a sort of
"gravitational redshift" in the accelerated frame. Clocks in different
positions can't be synchronized with each other.

4) About time dilation. I have an objection to John's post [1]. In a first part he defines time dilation as the difference in length between two paths sharing their extremes (figs 1,2). But later (fig. 4) he changes his mind and time dilation becomes the difference between AB and AC. IMHO the latter is the right meaning of the term "time dilation". It refers to the difference of time interval between two given events when one measures it in the frame where both events happen in the same place (proper time $\D t_0$) and when the measurement is done in any other frame (generic time interval $\D t$). It's widely known that in these conditions we find 
$$\D t = \gamma\,\D t_0.$$
The former effect - different lengths (proper times) of paths with
common extremes - is related to the twin paradox and also more
general thereof. Actually time dilation may also be redrawn as a case of path difference (please excuse me if I don't show how). But the link is rather artificial and I find it better to keep the distinction.
[1] https://physics.stackexchange.com/posts/241773
A: Special relativity has no problem describing accelerated observers. However, it's only in the case of a non-accelerating (inertial) observer that we can do all the things ordinarily assumed in an introductory treatment of SR:


*

*Construct an orthogonal coordinate system (called Minkowski coordinates) covering all of spacetime and adapted to the observer.

*Write down Lorentz transformations with the usual form.

*Calculate proper time intervals using the usual equation $\Delta \tau^2=\Delta t^2-\Delta x^2$ in terms of that observer's Minkowski coordinates $(t,x)$.
Now it's not actually necessary to do any of these things in order to do SR. You can do SR without any coordinates at all, if you like. For a treatment in this style, see Bertel Laurent, Introduction to spacetime: a first course on relativity. But regardless of whether you use coordinates, the distinction between inertial and noninertial motion is baked into the structure of SR. For example, in Laurent's presentation, one of the axioms is that proper time is maximized for inertial motion. That is, the twin "paradox" is an axiom: that among all world-lines connecting events A and B, there exists a world-line whose length (proper time) is maximal.
The form of this axiom is very similar to the form of the axiom in Euclidean geometry saying that two points determine a line. The only difference is that a line is a curve of minimal distance.
It makes sense to take this maximal-time property as an axiom because we verify it in experiments. For example, see Chou et al., Science 329 (2010) 1630, in which an atomic clock was accelerated in a tabletop experiment and shown to observe the time dilation predicted by special relativity.

There are a thousand explanations out there for why this doesn't happen, but they all end up saying something vague like it's because one twin is accelerating [...]

This is the correct explanation. The only reason it could be criticized as vague is because, as written, it doesn't spell out why it matters that one twin is accelerating. It matters for the reasons given above.

or you need general relativity to understand it.

This is simply wrong.
A: The paradox is about twins, one of which goes on a journey to outer space where he is observed by the Earth twin to be moving with great speed. Finally, the traveler twin  returns to meet his twin on Earth. According to special relativity, when the twins meet, the traveler twin has aged much less than the Earth twin.
The paradox is supposedly in that both of the two twins see the other one undergo the same kind of fast travel. If both see the other one move similarly, how is it possible that the twin that got off the Earth and got back aged much more?
Answer: equations of special relativity refer to quantities and coordinates measured in inertial frames only. Consequently the calculation of proper time elapsed on the moving watch corresponding to measured coordinate time can be done only if this coordinate time was measured in an inertial frame. 
Only the Earth twin can be in an inertial frame all the time, while the traveler twin can't, as he needs to accelerate and decelerate to get back to the Earth.
So there is actually no paradox - although the two observations of speed are the same, other things are not and this destroys the symmetry of the twins. The traveler twin has no way to use the time dilation formula because he has no coordinate time measurements that would qualify. 
Given the trip took $T$ seconds of the Earth time, time dilation formula implies that total proper time that the traveler experienced on his trip is ($v$ is speed of the traveler):
$$
\int_0^T \sqrt{1-\frac{v^2}{c^2}}\,\mathrm dt
$$
which is lower or equal to $T$.
Morale of the story: if you don't want to age quickly, get yourself moving.
A: Appendix - why the Rindler metric?
After reading my answer you could be forgiven for feel a bit cheated because it all depends on my claim that the accelerating twin observes a spacetime described by the Rindler metric not the Minkowski metric and I did kind of pull this out of the air.
It isn’t hard to prove that the accelerating twin’s metric has the Rindler form, but unless you are a fan of algebra the proof isn’t very exciting (if you are a fan of algebra see this article or Gravitation chapter 6). What I’m going to do instead is demonstrate a reason why the accelerating twin’s metric cannot be Minkowski, and in the process hopefully illustrate just how fascinating special relativity can be.
Suppose you try to outrun a light beam by travelling at a constant velocity $v$, where obviously your velocity has to be less than the speed of light. We’ll give you a head start by allowing you to start at $x = d$ while the light has to start at the origin. The spacetime diagram for the race looks like this:

Figure 1
Note that in this diagram the $y$ axis shows $ct$ not just time. I’ve done this because for a beam of light $x = ct$ and therefore on my diagram the trajectory of a beam of light is a line at 45°. Anything travelling slower than light follows a line at an angle $\theta$ greater than 45°.
Hopefully it’s obvious that you cannot outrun the light and it will always catch you eventually. The faster you go the nearer the angle $\theta$ of your line gets to 45° degrees but since you can never reach $c$ the angle $\theta$ must always be more than 45° and therefore the world line of the light ray must eventually cross yours. And this makes sense. In your inertial frame light travels at $c$ so it doesn’t matter what distance $d$ the light ray starts, it will always reach you at a time $t = d/c$ so the light will always catch you.
But now the fun starts. We’ll do the race again but this time you start at rest and accelerate with a constant acceleration $a$. As before we’ll give you a head start and this time we’ll start you at $x = c^2/a$. You’ll see why I chose this starting point in a moment.
Both of the articles I linked above give the equation for your trajectory in my rest frame. If we start you at $x(0) = c^2/a$ the equation for your world line is:
$$ x = \frac{c^2}{a}\sqrt{1 + \left(\frac{at}{c}\right)^2} $$
This time the spacetime diagram looks like this (this is a real calculation for a constant acceleration of 9.81 m/s$^2$):

Figure 2
As before the light travels along a straight line at 45° but this time your line is a curve because of course you are accelerating not travelling at constant velocity. What’s more, your world line is a curve that tends asymptotically to the line $x = ct$ so your world line and the world line of the light ray never meet.
Hang on, let’s take a step back for a moment, your world line and the world line of the light ray never meet so you can outrun a ray of light.

An observer accelerating with constant acceleration $a$ can outrun any ray of light starting any distance greater than $c^2/a$ behind them

And that means in your coordinates there is an event horizon at a distance $x = c^2/a$ behind you. Your spacetime geometry contains an event horizon just like a black hole does, and this fact alone shows that your spacetime cannot be described by the Minkowski metric.
If we go back to the Rindler metric we can show how the event horizon arises. The Rindler metric is:
$$ ds^2 = -\left( 1 + \frac{ax}{c^2}\right)^2c^2dt^2 + dx^2 $$
A light ray follows a null geodesic that has $ds^2=0$, and if we set $ds^2=0$ and rearrange the equation above we can get an expression for the velocity of light:
$$ \frac{dx}{dt} = c\left(1 + \frac{ax}{c^2}\right) $$
So in your rest frame not only is the velocity of light not constant but it goes to zero at $x = -c^2/a$. That’s why there is an event horizon there.
Proof that you can outrun light
just for completeness let's prove you can outrun the ray of light. The equation for your trajectory is:
$$ x = \frac{c^2}{a}\sqrt{1 + \left(\frac{at}{c}\right)^2} $$
We take a factor of $at/c$ out of the square root to get:
$$\begin{align}
 x &= \frac{c^2}{a}\frac{at}{c}\sqrt{1 + \left(\frac{c}{at}\right)^2} \\
   &= ct\sqrt{1 + \left(\frac{c}{at}\right)^2}
\end{align}$$
For large times $t \gg c/a$ we have $c/(at) \ll 1$ so we can approximate this using the binomial theorem:
$$ x \approx ct\left(1 + \tfrac{1}{2}\left(\frac{c}{at}\right)^2\right) $$
The trajectory of the light is given by $x_\text{light}=ct$ so the distance between you and the light beam is:
$$\begin{align}
 x - x_\text{light} &\approx ct\left(1 + \tfrac{1}{2}\left(\frac{c}{at}\right)^2\right) - ct \\
                    &\approx \frac{c^3}{2a^2t}
\end{align}$$
And this shows $x - x_\text{light} \gt 0$ is always greater than zero; i.e., the light can never catch you.
A: 
What is the proper way to explain the twin paradox?

By explaining that time is merely a measure of local motion, that time dilation is a reduced rate of local motion, and that being separated by relative motion can be likened to being separated by distance.   

The paradox in the twin paradox is that the situation appears symmetrical so each twin should think the other has aged less, which is of course impossible.

When we're separated by distance, we are subject to perspective, such that you look smaller than me and I look smaller than you. But we don't claim that this is impossible We do not cry paradox. In similar vein when we pass each other at some relativistic speed, I say that the light in your light clock appears to be moving zigzag fashion like this /\/\/\/\/\ at a slower rate than my own, which appears to be moving straight up and down like this ||. However you say that the light in my light clock appears to be moving like zigzag fashion like this /\/\/\/\/\ at a slower rate than yours, which appears to be moving straight up and down like this ||. There is no paradox. It's just a form of perspective, caused by the fact that we are separated by relative motion rather than distance. See the simple inference of time dilation on Wikipedia where you can find these images by user  Mdd4696:
 

There are a thousand explanations out there for why this doesn't happen, but they all end up saying something vague like it's because one twin is accelerating or you need general relativity to understand it.

No, there are correct explanations for this, which succinctly and clearly describe why there is no paradox at all. The Wikipedia explanation isn't wrong, but it's so long-winded it rather bores the reader to death.    

Please will someone give a simple and definitive explanation for why both twins agree on which twin is younger when they meet for the second time?

No problem. But that isn't the paradox. The paradox is that two twins in relative motion each claim that the other's clock is running slower than his own. As I said, this is merely the "perspective" associated with relative motion. 
Anyway, when one of the twins slows down and turn round and comes back to Earth, both twins concur that he was the one who was moving, that he was the one who accelerated and decelerated, and that his light "really" moved like this /\/\/\/\/\ in his light clock, which ran at a lower rate than that of the stay-at-home twin. Due to the wave nature of matter the same effect occurred in the body and brain of the travelling twin, who didn't notice this time dilation locally, and who has fewer grey hairs as a result. He suffered time dilation because the rate of local motion is of necessity reduced by the macroscopic motion through space, because the maximum rate of motion is c. Not because "the geometry of spacetime looks different for the two twins". To calculate the elapsed time we do not need a function called the metric, all we need is Pythagoras's theorem, wherein the hypotenuse is the light path, and the base is the speed as a fraction of c, and the height is the Lorentz factor:
$$\Delta t' = \frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}$$
It really is rather simple. Just focus on the motion and the light-paths, and you can't go wrong.    
A: The solution is relatively simple. One twin has to initially accelerate to move "away from" the other twin. Although it may "appear" to the moving twin, that the stationary twin is moving away, that is an illusion. We may always "appear" stationary in our own reference frame, but that doesn't mean that we "are stationary".
The moving twin experiences time dilation. (he ages slower) 
A: Which twin aged more than the other IS the paradox.
Both twins aged the same amount of time. But the problem is, that space and time are both relative - therefore, we can't draw any absolute conclusion based on relative intervals.
Thus, we need to have a way to put all events of the Universe in causal order - and that can only happen by inventing some kind of "time 2.0" called " supertime" that tracks a universal "now" that's diffrent from the one that relative time measures.
That means, that no object (and in this case, no twin) can just leap out of spacetime and move "towards the future faster (or slower) than someone else". That breaks causality.
Therefore the solution to the paradox, is that they both aged the same amount of supertime. Sure one's skin has more wrinkles than the skin of the other - but that is just because the rate in which the laws of physics affected twin A didn't affect twin B equally as fast (or as slow - depending on who you take as a reference frame).
Supertime is going to be intrically connected with the speed of light - since its the only absolute unit of measurement that we axiomatically agree upon. And supertime is going to be the rate in which we move inside spacetime - which is - the speed of light - 1ls/s.
Supertime should also have its own universal reference frame, and all objects in spacetime should move in it, relative to supertime. Thus relative to supertime, every object's speed is 1)constant, 2) invariant and 3) equal to 1ls/s. As long as, the more you accelerate in space the more you decelerate in time and visa versa so that your speed in supertime is always equal to 1ls/s.
So in supertime there is no acceleration. As we said, if you accelerate in space you decelerate in time - so your "net" acceleration relative to supertime is zero. So we arrive to the conclusion of the paradox withought invoking acceleration or curved spacetime.
So how many ls/s passed for each twin? The same amount.
But twin A moved in space faster than twin B (relative to supertime), thus the laws of physics applied to him faster than the other twin did. If we look at the same example the other way around, twin A did in fact saw twin B (or the Earth) accelerate in space so shouldn't twin B be older?
No, because relative to supertime, it was A who did the movement and not B, B has a fictitious point of view - his speed was NEVER allocated in space, therefore the laws of physics didn't apply on him any faster than before.
Thus the symmetry brakes and we can have a Universe were events don't contradict eachother. Any other way around this, leads to conspiracy theories than real physics.
A: Most of the answers argue that the travelling twins are not in an inertial frame, hence SRT (special relativity theory) only applies to the stationary twin in an inertial frame.
But we can introduce a third traveller in the standard twin paradox to completely eliminate acceleration:
https://bit.ly/3t3CmIh
In the above spacetime diagram by Brent Meeker, all the triplets are in inertial frames! Hence SRT can be applied to each triplet, as there is no acceleration involved.
The POV (point of view) of the outbound triplet is given by the period 2007-2012 in the spacetime diagram below on the vertical time axis:
https://bit.ly/3zoCkMd
and the POV of the inbound triplet is given by the period 2012-2017 in the spacetime diagram above on the vertical time axis. It can also be seen that Earth initially moves away from the outbound triplet, then approaches the inbound triplet on the return journey.
The spacetime diagram from the POV of the moving triplets indicates that the clock on Earth must be running slower, hence the triplet on Earth must be younger when they reunite!
This is in direct contrast to the POV of the triplet on Earth, who sees the clock of the travelling triplet running slower!
In [1] they mention 3 scenarios that are perfectly symmetrical.
[1] Has the Twin Paradox Really Been Resolved?
https://www.scirp.org/pdf/jamp_2021090215032696.pdf
Two triplets travel in opposite directions from a triplet on Earth, then return to the third triplet at the same speed. Can the above paradox be resolved using special relativity alone?
Assume everything is completely symmetrical, so the velocity (and acceleration) of the two spaceships are identical during the entire journey.
The triplet A will observe the triplet B moving firstly away (with a velocity given by the law of addition of velocities). Then triplet A will see triplet B moving towards him/her (with a velocity given by the law of addition of velocities).
According to SRT (special relativity theory), from the viewpoint of triplet A the clock of triplet B must really be running slower! This is a prediction made by SRT.
When the triplet A and triplet B are united with triplet C, however, triplet A discovers that the clock of triplet B is NOT running slower! This contradicts the prediction made by triplet A, which is embarrassing.
A similar conclusion will be reached by triplet B!
The only triplet to make a correct prediction will be the triplet C in the stationary frame on Earth! As SRT claims any triplet in an inertial frame can be considered at rest, and triplets A and B both make wrong predictions, it begs the question if SRT is only valid in a preferred inertial reference frame?
Some might argue that triplets A and B are in non-inertial frames during brief moments of time, hence SRT does not apply.
This objection can be countered by considering 5 travellers. Traveller A approaches traveller C at rest on Earth with speed v. When flying past Earth traveller A and C resets their clocks to zero.
Traveller B comes from the opposite direction and also resets his/her clock when passing by traveller C on Earth.
So during launch, travellers A and B are all in inertial frames and SRT can be applied.
During the turnaround point, traveller A continues to travel at speed v but is met by traveller D, who also travels at speed v, but in an opposite direction to traveller A. As traveller A and D pass each other, traveller D records the clock reading of traveller A and also resets his clock to zero. When traveller D meets the triplet (traveller) C on Earth without stopping or landing, traveller D passes on his clock reading and also that of traveller A.
A similar scenario applies to traveller B and E.
During the complete journey, travellers A to E were all in inertial frames! Hence SRT must apply. Yet, the predictions of travellers A,B,D and E are incorrect!
A spacetime diagram can also be constructed to show the symmetry between triplet A and B.
The conclusion seems to be that SRT only gives the correct answer when viewed from a preferential reference frame corresponding to the stationary triplet C! This violates SRT, which claims that any reference frame can be considered to be at rest.
A: In search for a clear and simple explanation, it is best to turn to the Lorentz ether theory. It is often said, that this theory is empirically equivalent to the SR. Since the same mathematical formalism occurs in both, it is not possible to distinguish between Lorentz Ether Theory and Special Relativity by experiment. The introduction of length contraction and time dilation for all phenomena in a "preferred" frame of reference, which plays the role of Lorentz's immobile ether, leads to the complete Lorentz transformation.
This paper simulates all kinematic effects of Special Relativity – time dilation, length contraction, relativistic velocity addition, relativistic Doppler Effect, reciprocal Lorentz transformations, Twin paradox, Bell‘s spaceship paradox e.t.c. on the on the simplest examples of the movement of barges, shuttles, and boats in an aquatic environment.
It should be noted, that there is no "paradox" related to the Lorentz theory. In the Lorentz theory, the paradox that has arisen in the depths of special relativity is resolved by means of elementary algebraic methods, staying within the same frame of reference and not taking into account the acceleration or deceleration.
So, let's consider resolution of the "paradox" in the framework of the Lorentz theory.
1)  Let’s consider what would happen if one of two twins who are at rest in the ether at one point, flies at speed $v$ to a distant point and then after a while returns to twin $A$ remaining at rest.
If for the twin flying in the ether his “local time” characterizing the rate of physical processes in his body and the pace of the movement of his clock on both segments of his flight (there and back) slows down due to interaction with the ether, then the lapse of his “local time” will be $1/\sqrt {1-v^2/c^2}$ times less than for the twin at rest in the ether, and the “travelling” twin will get less “old”. The turn of the travelling twin, provided it is virtually instantaneous, has no practical effect on the ratio of times of both twins.
2)  Now let's  calculate, what will happen if the two twins are flying side by side in the ether at speed $v$ – with their “local time” passing slower – then one of them stops, staying at rest in the ether for some time, then catching up with the travelling twin.
The twin who continued his flight in the ether with no information about the fact of his motion in the ether perceives this maneuver of his brother as a round trip to a distant point.
An obvious answer is that, since according to the ether theory after the twin’s stop in the ether his time will pass faster than the “local time” of his twin brother who continues his flight, and then when the twin stopping in the ether after some time catches up with the missing brother, he will age more than the latter. The “local” time of the twin catching up with his flying brother will actually flow slower than for the flying brother. This is due to the faster speed of the twin catching up with his brother. As a result, the brother making a stop in the ether will age not more, but less than his twin brother who has not interrupted his flight.
Let us demonstrate that if the proper times of the motion there and back of the non-inertial twin relative to the inertial twin are equal, then for the non-inertial twin it will take $1/\sqrt {1-v^2/c^2}$   times less time than for the moving inertial twin, and the non-inertial twin will age less.
Let at the time of stop of one of the twins in the ether the clocks of the parting twins show zeros. Suppose that after making a stop for some time, the twin who has lagged behind, at the moment $t_1$ of the ether time when his clock (because of the stop) was showing this time, left at speed $u$, such that $v<u<c$, following his brother flying away from him. The distance between the twins at the start of the twin who has left behind is equal to $vt_1$. Setting out, the twin left behind will catch up with the twin flying at a constant speed $v$ at the point in time $t_2$, having spent the time equal to $vt_1/(u-v)$. During this period, by the clock of the twin following the flying away brother at speed u, there will be a lapse of proper time, which is  $1/\sqrt {1-v^2/c^2}$ times less than the ether time and equals $vt_1\sqrt{1-(u/c)^2}/(u-v)$. Let us assume the velocity $u$ such that the proper time $t’_2-t’_1$ of the catching up twin is numerically equal to the time $t_1$ of his stay at rest relative to the ether, i.e.  $t’_2-t’_1=t_1$ or
$$t_1=vt_1\sqrt {1-(u/c)^2/(u-v)} (1)$$ 
This equation meets the condition under which the twin spends the same proper time on a trip to a distant point and back. By elementary transformations of the equation (1) we can obtain the value of velocity $u$, which is equal to $\frac {2v}{1+(v/c)^2}$ . Substituting this value in the expression for the time $vt_1/(u-v)$ required for the return of the twin, and summing the time $vt_1/(u-v)$  and the time $t_1$, we obtain the ether time spent by the lagging behind twin on the stop and return to the flying twin. This time is equal to $2t_1/(1-v^2/c^2)$. Since the clock of the inertial twin flying at a speed $v$  go $1/\sqrt {1-v^2/c^2}$   times slower than the clock at rest in the ether, the flying twin will determine the time spent by the lagging behind twin on the stop and return to the flying twin as a quantity meeting the equality:
$$t’_2=2t_1/\sqrt {1-(v/c)^2}$$ 
Since the time elapsed for the non-inertial twin by the moment of his return is numerically equal to $2t_1$, and the time of the inertial twin is numerically equal to $2t_1/\sqrt{1-(v/c)^2}$, then the lapse of time for the non-inertial twin is $1/\sqrt {1-v^2/c^2}$   times shorter, and he has aged less than the inertial twin has.
This way we can see, that non–inertial twin (or clock) will show $1/\sqrt {1-v^2/c^2}$ less time than inertial one, despite of direction of its motion in ether. 
We get absolutely the same result as in the Special Relativity, but resolution of the paradox is very simple.
A: According to me twin paradox can be thought of as illusion in explaining time dilation.Imagine two brothers A and B.
One brother is on earth and the other travelling away from him to a distant planet.
Now after one year both the brothers flash a light.
Now the age of brother A will reach instantaneously to some one on earth assuming the person is very near.
Now consider person B because he is travelling the distance between B and the observer is increasing,now because the speed of light which is the medium of age information is finite,the more the distance betweenB and observer,the more the time taken by the light.Say if B has travelled a distance of 0.1 light year,it would take 1.1 light years for the observer to know that the age is one year,So he is less than his age by 0.1year for every 1.1 year.
This is just miracle of nature.
In reality the person B also ages at the same rate as A,it is only when they share their information to second person,the medium makes it feel as if one is aging fast or slow.
