Free Will Theorem in a nutshell

Developed by John Horton Conway and Simon Bernhard Kochen

Axioms

The theorem relies on three axioms, namely: FIN, SPIN, TWIN

Later they revised (or strengthened the theorem) by replacing FIN with MIN, we will ignore this and use the axiom FIN instead since it is easier to remember and explain.

FIN

FIN roughly speaking stands for finite. It is based on the assumption that information can not travel faster than the speed of light (299792458 m/s; approximately 300000 km/s).

So the propagation of information is always bound to be at a maximum speed (the speed of light), for example when transferring any kind of information from planet Earth to planet Mars it would take at least around 3 minutes to reach its destiny.

distance from earth to mars

SPIN

In quantum mechanics and particle physics spin is an intrinsic form of angular momentum.

This spin component when squared can be described with a set of three 1s and 0s; whereas it will always have two 1s and one 0, thus it is always a permutation of (1,1,0) (i.e. (1,0,1) or (0,1,1)).

this part needs some more explanation - i don’t understand where this 101 rule comes from and is it really required for my explanation here? needs to be explained why it is no matter which axis gets measured later

The spin of the particle is not determined until we actually measure it; this can be explained with the Stern-Gerlach experiment (here a Medium post).
Briefly explained:

Measuring Z in sequence:

Diagram Z axis

Measuring Z and X alternating:

Diagram Z X Z

Conclusion: observing a quantum mechanical system changes the system - so it is never predictable in which state the system is until we measure it; it is not determinable.
When this happens, we say that the component of the spin in the z direction and the component of the spin in the x direction are incompatible observables: it is not possible to know both of their values at the same time.

TWIN

It is possible to entangle two particles, so even when they are separated by a far distance they give the same result when measured in parallel directions. They are just like twin particles.

So speaking the ‘information’ of these two particles is consistent to another - measuring direction A will give 0 or 1 (as spin) and the entangled particle will always have the same spin (0 or 1) in that direction.

Theorem

Now let us assume we have an entangled particle which gets observed by two independent scientists. One of them is doing the experiment on Earth and the other is doing the same experiment on Mars. They do not have to choose the same axis to measure, that is up to the independent scientists to choose by their own freedom.
Let us remember, information would require at least 3 minutes to go from Earth to Mars (or vice versa) - FIN - and both particles are in an entangled state - TWIN - also the result of the observation is not predetermined, it is only ‘decided’ by the particle as soon as we start measuring - SPIN.

It is not possible for any information to be transferred fast enough between the two particles (due to the vast distance) it is impossible for them to synchronize by exchange of information. Yet they still give the same result and this result is not determined until we measure.

Hence the particle does not rely on any events that happened before - no determination - and it is independent of the location, the choice is just made up on the spot at the moment of measurement; until then the particle did not decide what state it is in. This is the free will exercised by the particle