To show the twins must have different ages, all you need is Doppler effect, and Einstein's 2nd postulate - which Johan has been silent about.
Apply "c is constant" postulate to light pulses sent between the two twins. It is about the level at which 16 year olds are taught physics. No Lorentz Transform is needed.
I gave this argument above, it was ignored, so here is a better version with pictures.
This is very similar to what Teahive posted, but maybe explained in more detail.
[pic will expand if clicked, how can i make it bigger always?]
This picture plots the position (horizontal axis) versus time (vertical axis) of the two twins.
I am indebted to Johan for the idea of measuring twin age by requiring each twin to emit a pulse of light, say every second, which is received by the other twin. We can compare total numbers of pulses received when the twins get back together to get the ages.
There will be nothing weird in this argument except that one "c must be constant" rule.
Looking at the pictures, both are of the same situation, with a lefthand stationary twin and a righthand twin moving first away and then back with constant velocity v.
The only difference between the lefthand and righthand picture is that light rays between the two twins are shown in opposite directions. It could all be on one diagram but the light rays would then be difficult to see.
We note the following:
(1) Every light pulse emitted will be received by the other twin, so we can work out twin age by counting number of
received pulses.
(2) Doppler effect alters the received frequency of the pulses. The frequency depends on whether the relative velocity of the twins is inward or outward.
(3) We need to be careful about
what time to measure relative velocity. For a given light ray it is obviously the transmit-time velocity and the receive-time velocity which must be compared.
(4) The Left-hand stationary twin has no velocity. To work out Doppler we use the Receive velocity for rays going to the travelling twin, and the earlier time (because the rays are transmitted before received) Transmit velocity for rays coming from the travelling twin. This is a crucial difference.
(5) Looking at the pictures, you can see that the rays received by either twin are initially low frequency, and swap over to high frequency for the inbound journey.
(6) Looking at the pictures, it is clear that received rays in the two directions are different. Rays
from the traveller will be lower frequency (outbound journey Doppler) for a longer time T1 than rays
to the travelling twin, which are low frequency for only T1'. Conversely the high frequency time T2 is shorter than T2'.
(7) Here is where we get relativistic. Unlike sound, the speed of light is constant so the doppler effect due to light speed is
the same for light pulses in both directions:
1 + v/c for the inbound journey (higher frequency).
1-v/c for the outbound journey (lower frequency).
This
symmetry combined with the asymmetry from (4) will give us the surprising result. For sound rays, and a non-relativistic picture, the difference in wave velocity between the travelling and stationary twin would cancel with the asymmetry caused by the differing low and high frequency times.
(8) Doing the calculation for the number of pulses from each twin:
Stationary twin: f1T1 + f2T2
Moving twin: f1'T1'+f2'T2'
Let us also label total time:
T1 + T2 = T1' + T2' = T
(9)
f2 = f2' = (1+v/c)f
f1 = f1' = (1-v/c)f
[f is the frequency at which pulses are emitted. It will turn out we need to scale this by a time dilation factor. But that will be the result of our calculation, we do NOT assume this]
Difference in age is: f(1-v/c)(T1-T1') + f(1+v/c)(T2-T2') =
f[ (T - T) - v/c( T2' - T2) - v/c(T1- T1') ]
The first term cancels and (by inspection) T2' > T2, T1 > T1' so we must have the stationary twin older than the moving twin.
If we wanted we could go further and
derive the gamma factor which is the (real) time dilation of the moving twin. That would need more care because once we have time dilation, we must check which reference frame each quantity is measured in. But without doing this we can see that the ages must be different.
All we have used to get this result is speed of light being constant, and physics remaining the same in any inertial frame.
The nice thing about this proof is that it can be done, and shows the relativistic effect, entirely without frames of reference, Lorentz transform, or any other complex math.