All e⁻ paired: diamagnetic, sample weakly repelled by magnetic field. Sample in a Gouy or Faraday balance [google] and it will apparently lose mass when magnetic field applied. To actually lose mass I think would violate a Law of Thermodynamics.
One or more unpaired e⁻: paramagnetic, sample is attracted into a magnetic field and will apparently gain mass. An e⁻ has “spin” and angular momentum that causes it to have a magnetic moment. For metal cmplxs the angular contribution is largely quenched and magnetic moments close to the spin-only values are found. From quantum mechanics μ = √n(n+2) BM (Bohr magnetons) where n is the number of unp e⁻.
[Fe(CN)6]^4- This is Fe(II) 3d^6 (CN^-). As previously stated, CN^- is a strong field ligand and the d orbital splitting diagram is: t2g (↑↓)(↑↓)(↑↓) ( )( ) eg The ion has no unp e⁻s and is .: diamagnetic: repelled by a magnetic field.
[Fe(CN)6]^3- This is Fe(III) 3d^5.
The d orbital splitting diagram is: t2g (↑↓)(↑↓)(↑) ( )( ) eg The ion has one unp e⁻ s and is .: paramagnetic, attracted into a magnetic field. Magnetic moments are easily determined by the Gouy or Faraday (Evans) technique and it is found that the moment is ~2.3 BM (some orbital contribution) rather than 1.73 BM (√3)
F. A. Cotton, G. Wilkinson, C. A. Murillo, M. Bochmann
Advanced Inorganic Chemistry 6th ed (1999). p784, 790