Three-box paradox and "Cheshire cat grin": the case of spin-1 atoms

We propose in this work a definite theoretical implementation of the three-box paradox - a scheme in which a single quantum particle appears to be present with certainty in two separate boxes - with spin-1 atoms. We further show how our setup can give rise to a "Cheshire cat grin" type of situation, in which an atom can apparently be found with certainty in one of the boxes while one of its properties (the angular momentum projection along a specifically chosen axis) appears to be in a different box. The significance of our findings are discussed relative to the status of the properties of a system obtained from weak measurements.