With these two definitions laid out, Conway explained what he meant by free will.  Free will, for the purpose of this theorem, is the ability of an experimenter to decide in which direction he or she will measure the particle spin.  If an experimenter has no free will, then when an experimenter attempts to measure a specific angle of spin, he will be guided by magic to the angle he is determined to measure. The proof of the theorem depends on the fact that the experimenter can choose what he measures and when.

ments of the triple numbers from the SPIN axiom above.  This means both particles, no matter the order the angles are measured in, will give the same number, 1 or 0, if measured from the same angles.  This means, if we measure angles a, b, and c for particle one and angles, b, c, and a in particle 2, we will get corresponding digits for a, b, and c.
With these theories and a resounding, "Ok, big boy, if I may call you that," Conway launched into the proof.  Take two twin particles and separate them.  Choose a time difference in measuring the triple of the two particles faster than the maximum communication speed allowed by the FIN axiom. Therefore, we can say that the only possible influences these two particles have on each other is their past history.  Note that both particles have an individual history we'll call I_L and I_R respectively. Note these also have a shared history called I_C.  We can therefore form a function describing the particle movement based on its triple number t, it's individual history and it's joint history with the other particle.  We can write these as f_L (t, I_C, I_L ) andf_R (t', I_C, I_R ) respectively.  Note by the twin

      Some points of
orthoganality of a cube

Using the SPIN data, Conway asked us to pretend that any individual particle has already determined its spin in every direction.  He proceeded to describe to us a cube with 33 points of orthogonality on it wherein we could insert the determined particle with the (1,0,1) spin requirement given above and measure where the particle must give a 0 and where it must give a 1.  After a little combinatorical mathematics it was shown the particle could not sit inside this cube as we hit a (0,0,1) orthogonal triple, but then Conway told us this result, while interesting, relied on the uncheckable axiom that orthogonal directions commute.  Conway promised to do better and give a solution that relied on no uncheckables.
The last axiom, TWIN, the final piece of the puzzle, was given by the idea that if two particles are brought together, you can measure their total angular momentum.  The range of this data is from 2 to -2, but, if you're lucky, you can get a total momentum of zero, which implies you have a pair of particles with opposite spins.  So we then know that, if you carefully separate these particles and square their spin, we will get identical measure

          Conway as a particle?

                                                              8