First a definition of terms and numbers we'll be using:
Examples of "half speed" or "half" the relativistic velocity combination are:
.6c relative velocity is the relativistic velocity combination of two frames at .33c.
.8c relative velocity is the relativistic velocity combination of two frames at .5c.
.8824c relative velocity is the relativistic velocity combination of two frames at .6c.
.9756c relative velocity is the relativistic velocity combination of two frames at .8c.
.3846c. relative velocity is the relativistic velocity combination of two frames at .2c.
.2c relative velocity is the relativistic velocity combination of two frames at .1c.
The slopes of the lines of simultaneity x/t for each of these half speeds is the inverse of the slope of their velocity lines x/t so:
.33c half speed slope of simultaneity line is 3.
.5c half speed slope of simultaneity line is 2.
.6c half speed slope of simultaneity line is 5/3.
.8c half speed slope of simultaneity line is 5/4.
.2c half speed slope of simultaneity line is 5.
.1c half speed slope of simultaneity line is 10.
What's interesting to note is that the slope of the half speed perspective's lines of simultaneity intersect the same proper times on the full relative velocity lines as shown in this spacetime diagram (STD):
The thin red line is .6c relative velocity depicted in a Minkowski STD. Alice red is depicted as moving and Bob blue is depicted as stationary when actually both are moving at .6c relative to each other. The horizontal blue lines are Bob's lines of simultaneity from his perspective and the thick red lines are Alice's lines of simultaneity from her perspective.
The thin green line is the half speed (.33c) of .6c. The slanted green lines are the half speed lines of simultaneity which happen to intersect Bob and Alice's proper on-board times. This gives us an indication that, from the half speed perspective, Alice and Bob are ageing at the same rate simultaneously. In fact relativity dictates that any constant relative velocity between two frames yields no age difference between the two participants Bob and Alice. Age difference being a permanent phenomenon is not the same thing as reciprocal time dilation which is due to perspective.
Now let's say Alice stops at the 3 ly mark. She sends a light signal to Bob informing him she has stopped. During the 3 yrs it takes for the light signal to reach Bob, his relative velocity to Alice remains at .6c and her relative velocity to Bob has changed to 0c. This imbalance in relative velocity is verified by the Doppler shift ratio between them. Alice sees a Doppler shift ratio of R=1 and Bob sees R=1/2 which correspond to v=0 and v=.6c respectively.
Upon receiving the news, Bob will be able to calculate Alice has permanently aged 1 yr less than Bob even though they never re-unite. They are separated by distance and separation has a time value associated with it just by the very fact it takes time to traverse a distance. (There is a mathematical formula for converting separation into a time value but I won't discuss it here.)
So let's calculate the age differences for various velocities for when Bob and Alice re-unite or co-locate in an instantaneous present (effects of the delay of the speed of light will be negligible between them). I know, I know, when do I get to the puzzle? I'm still laying the groundwork and setting up the parameters.
If you draw the STD, which I will show later, you get the following results for age difference due to co-location:
At .6c return, Alice's v line intersects Bob's at t=10 and t'=8 so the age diff is 2 yrs less for Alice.
At .8c return, Alice's v line intersects Bob's at t=8.75 and t'=6.25 so the age diff is 2.5 yrs less for Alice.
At .8824c return, Alice's v line intersects Bob's at t=8.4 and t'=5.6 so the age diff is 2.8 yrs less for Alice.
At .9756c return, Alice's v line intersects Bob's at t=8.075 and t'=4.675 so the age diff is 3.4 yrs less for Alice.
This method makes it very hard graphically to establish where the .3846c and .2c return velocities would intersect with Bob to establish their age difference. Well let's just use the way Alice establishes her age difference when she stops and can't co-locate with Bob, with the info contained in her light signal. It turns out this method employs where the half speed lines of simultaneity (green) intersect Alice's return velocity lines (red).
The green lines of simultaneity are labelled with their corresponding half speed velocities which of course have corresponding relative velocities between Bob and Alice.
.9756c (half speed .8c) has a simultaneity line slope of 5/4 and intersects the red v line at t'=4.6. 8-4.6=3.4
.8824c (half speed .6c) has a simultaneity line slope of 5/3 and intersects the red v line at t'=5.2. 8-5.2=2.8
.8c (half speed .5c) has a simultaneity line slope of 2/1 and intersects the red v line at t'=5.5. 8-5.5=2.5
.6c (half speed .33c) has a simultaneity line slope of 3/1 and intersects the red v line at t'=6. 8-6=2
.3846c (half speed .2c) has a simultaneity line slope of 5/1 and intersects the red v line at t'=6.4. 8-6.4=1.6
.2c (half speed .1c) has a simultaneity line slope of 10 and intersects the red v line at t'=6.7. 8-6.7=1.3
There's 1 more line of simultaneity to consider, but it's not for a returning velocity or even a stopped one, it's a separating velocity where Alice does not make a velocity change.
.6c (half speed .33c) has a simultaneity line slope of 3/1 and intersects the red v line at t'=8. 8-8=0 age difference.
So we see we got all the right answers plus 3 more right answers. But we can get so many more using this method. There are areas of this graph that seem incomplete. What happens if Alice just slows down a little when she makes her velocity change. What happens to the age difference if she speeds up? These questions are mathematically solvable, but does relativity allow them to be solvable? That's the puzzle.
Here's the STD with velocities where Alice slows down and doesn't stop or return.
We can see that separating velocities also yield results for age difference:
.2c (half speed .1c) has a simultaneity line slope of 10/1 and intersects the red v line at t'=7.3. 8-7.3=.7
.3846c (half speed .2c) has a simultaneity line slope of 5/1 and intersects the red v line at t'=7.6. 8-7.6=.4
Would anyone believe this nice clean hyperbolic line joining the points of intersection is some meaningless coincidence? If this method yields final age difference results, can it predict age difference results as they unfurl for each year Bob ages? (The answer is yes mathematically but what does relativity have to say about it? What happens if you extend the hyperbola to separating velocities of greater than .6c? Is that allowed under relativity? Here's the STD:
So if Alice accelerates to .8c away from Bob she will turn out 8-8.6=.6 yrs older than Bob.
And if Alice accelerates to .8824c away from Bob she will turn out 8-9=1 yr older than Bob at the time he gets her message that she is accelerating away.
Does anyone have a problem with this because you should.