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12 rectangular parallelepipeds with edges parallel to axes

Source: ARO 2005 - problem 11.6

April 30, 2005
analytic geometrypigeonhole principlegeometry unsolvedgeometry

Problem Statement

Do there exist 12 rectangular parallelepipeds P1, P2,…,P12P_1,\,P_2,\ldots,P_{12} with edges parallel to coordinate axes OX, OY, OZOX,\,OY,\,OZ such that PiP_i and PjP_j have a common point iff i≠j±1i\ne j\pm 1 modulo 12?
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