The present invention discloses a polyhedral block with a plurality of intermittently magnetic sides. The magnetic intermittency is caused by a freely movable magnet that resides in and traverses in an inner cavity, exerting its pull-force through the side of the block to which it is nearest. The cavity is shaped to facilitate magnet movement in response to encountering an external magnetically attractable surface. As a user rotates the block on a magnetically attractable surface, the contained magnet is caused to move within its cavity, ultimately resting in a user-selected position on the magnetically attractable surface. The block contains one or more non-magnetic connection elements on its sides. These non-magnetic connection elements serve to connect the block with compatible, non-magnetic blocks and other objects.